TWIST1 has different interactions between PDGFRA and CDKN2A, which have high frequency for proneual subtypes of Glioblastoma tumors.

our covariance-specified CRF network may indicate that these two aberrations are the most salient in interacting with pairs of genes that include the gene TWIST1.


Using this view, we were able to show that several of the above methods are subsumed by the Normalized Cut (NCut) segmentation algorithm of [9] in a sense that will be described.

in the following, we will focus on the NCut algorithm and will adopt the terminology of image segmentation (i.e.


Assuming n, m Ñ 8, each of (i)-(iii) implies that the B ˚which maximizes ( 5) is nearly rank one, or equivalently B ˚« r b r (see Supplementary Information §S4 for intuition).

the tightening procedure initialized at r will produce a global maximum of the PDA objective.


• If j belongs to Λ 14 \ B 14 (1) -i.e., ⊕ region of Figure 1(b)-We have ∇f j (x 15 ) = ∇f j (x 14 ) because x 15 − x 14 ≤ ξ and j ∈ B 14 (1).

we can safely retrieve g j = ∇f j (x to represent ∇f j (x 15 ).


To solve this optimization, one needs to compute the likelihood function for n k 2 possible bicluster indices.

the computational complexity of the MLE is exponential in n. Theorem 5.


Once a part has been drawn from the distribution, there is no way of affecting the "base level" for the value of the part.

we assume for the remainder of this paper that c i ≥ 0 for 1 ≤ i ≤ s + 1 and that c i = 0 for some value of i (between 1 and s + 1).


Generating unnatural images usually results in a high diversity score.

to investigate whether the variation of generated images is meaningful, we need to evaluate the visual realism of the generated samples as well.


S is decomposed into p subsamples S 1 ,. .

., S p . Each subsample S r , r ∈ [1, p], contains m columns and is used to define a rank-k Nyström approximation K r . Dropping the rank subscript k in favor of the sample index r, K r can be written as K r = C r W + r C ⊤ r , where C r and W r denote the matrices formed from the columns of S r and W + r is the pseudo-inverse of the rank-k approximation of W r . The learner further receives a sample V of s columns used to determine the weight µ r ∈ R attributed to each expert K r . the general form of the approximation of K generated by the ensemble Nyström algorithm is The mixture weights µ r can be defined in many ways.


For example, even if the required separation is σ max K 1 /2 then if we look at the Voronoi tessellation around the centers of the Gaussians, each cell will contain at least 1 − (2π) −1 K 3 /4 exp (− K /2) of the mass of the Gaussian.

when K is large, the supports of the Gaussians are almost disjoint.


We then performed 5-fold crossvalidation, the crossvalidation error is given by the negative logarithm of the data (spike or gap) in the test sets: CV error = − log(P (spike|t)) .

(20) we measure how well the PSTHs predict the test data.


There are at most k − 1 eigenvectors corresponding to nonzero eigenvalues, since the rank of the matrix S b is bounded from above by k − 1.

the reduced dimension by classical LDA is at most k − 1.


These points are identified using the condition 0 < η j,i < 1 nνj .

ρ j can be recovered from a point sv satisfying this condition by Figure 1 shows the resulting estimates of our q-OCSVM method for 4 hierarchical MV-sets with α = 0.2, 0.4, 0.6, 0.8 3 .


Let be the total number of packets in all the input queues, let be the number of new arrivals, and let ¡ F ¡ be the number of packets sent.

the total wait of all packets is increased by the number of packets that remain: .


Theorem 1 If V (Q V ), defined as above with respect to some POMDP model of a dynamical system, is a square matrix, i.e., the number of e-tests in Q V is the number of states |S| (in that POMDP model), then Q V constitutes a linear EPSR for that dynamical system.

is square then it is invertible because by construction it has full rank, and hence for any history h, b w t for some weight vector w t . Q V constitutes a linear EPSR as per the definition in Equation 2.


For any arms a = e|S ∈ A, we define the unit gap of a (i.e., the gap for one phase) as For any arms a = e|S ∈ A, we define the sunk-cost gap (irreversible once selected) as where for two feasible sets A and B, A ≺ B means that A is a prefix of B in some decision sequence, that is, there exists a decision sequence σ = S 0 = ∅, S 1 , . . , S k such that S k = B and for some j < k, S j = A.

∆ * (e|S) means the largest gap we may have after we have fixed our prefix selection to be S ∪ {e}, and is upper bounded by ∆ max .


The intuition here is that when the sample points are too "sparse," some cells will have large size when the tree leaves are non-empty (satisfied when trees are fully-grown).

when a query point falls into a leaf cell, with high probability, it will be far away from the training data in the same cell, violating locality (see the right panel of Fig.


The bounds are not tight (they involve absolute constants), but they demonstrate some key features: i) the solution to the constrained LASSO x is a good estimate of x 0 up to the same constant of proportionality µ that appears in Brillinger's result.

ii) x − µx 0 2 2 is a natural measure of performance.


Interestingly, we will show that their result is incorrect and this approach can never lead to a consistent joint distribution in the vein of [17,18,19].

there currently does not exist a graphical model distribution for high-dimensional multivariate count data that does not suffer from severe deficiencies.


The local feature extraction through Multi-CNN [6] encodes contextual information due to large field of view (FOV); the FOV for 1, 1/2 and 1/4 scaled input images is 47 × 47, 94 × 94 and 188 × 188 pixels, respectively.

we designed the experiments under single and multi scale settings to assess rCPN's contribution.


However, these algorithms are exceedingly slow when d s is large and they have not been extended to handle flexible orientations.

here we derive an alternative optimization procedures that expands upon ideas from [8,12], handles arbitrary/unknown dipole orientations, and converges quickly.


However, the same arguments discussed above apply to regression.

we present an algorithm that solves the ranking problem directly, without first computing the rating.


where N i is the set of vertices which are connected with vertex i in G. Intuitively, larger W D value indicates larger weight, less conflict with other vertices or both.

the obtained x t is likely to contain vertices with large weights and have large potential room for improvement.


Training the CNN improves the validation error for a while but the model eventually starts to overfit.

we found it critical to keep track of the validation error and freeze the model before it overfits.


Moreover, the performance of the 'Noisy case' is also close to that of the 'Perfect case ' for most data sets listed in Table 3.

we can conclude that our 'EM+Prior Model' is able to take advantage of the pnor distribution on class labels even when some of the' guidance' is not correct.


The problem arises from the fact that pseudo-dataset methods are functionally local in the sense that each pseudo-datapoint sculpts out the approximate posterior in a small region of the input space around it [10].

when the range of the inputs is large compared to the range of the dependencies in the posterior, many pseudo-datapoints are required to maintain the accuracy of the approximation.


In case of at least one positive eigenvalue (which exists if β is chosen large enough), there is a unique stable fixed point w * : The vector b appearing in (12) is the eigenvector of C corresponding to the largest eigenvalue µ.

a stationary unimodal 4 distribution p(w) of the weights w is predicted, which is centered around the value w * .


Each noisy input S i 0,⌫ from the ⌫th cluster, equals Si 0,⌫ ( ¯ S i 0,⌫ ) with probability (1 + m 0 )/2, ((1 m 0 )/2) respectively.

the average overlap of the noisy inputs with the central pattern, say µ = 1 is ranging from m 0 = 1 denoting the noiseless limit, to m 0 = 0 where the inputs are uncorrelated with the centers.


[9] showed that it is possible to achieve state-of-the-art denoising performance with a plain multi-layer perceptron (MLP) that maps noisy patches onto noise-free ones, once the capacity of the MLP, the patch size, and the training set are large enough.

neural networks indeed have a great potential for image denoising.


One option to achieve this task is to use the unlabeled examples to estimate the true mixture density 1 2 p 1 + 1 2 p 2 , however number of unlabeled examples required to estimate mixture density using non parametric kernel density estimation is exponential to the number of dimensions [10].

for high dimensional data this is not an attractive option and also such an estimate does not provide any clue as to where the decision boundary is.


In the event that Φr * (x) − T Φr * (x) is large for only a small number of states (that is, the Bellman error of the approximation produced by r * is large for only a small number of states), we thus expect to have a choice of θ for which l(r * , θ) + 2θ l(r * , 0).

theorem 1 reinforces the intuition (shown via Figure 1) that the SALP will permit closer approximations to J * than the ALP.


The first, denoted C 3 Q3 , is the empirical classification error that counts the number of times the predicted (discretized) angle was different than the true angle, that is, The second evaluation measure, denoted ¢ 1 , is the average absolute difference between the predicted angle and the true angle, ¢ 1 © 1 @ Q @ ' !B 1 W d¥ ' © ¥ ' W . It should be kept in mind that the test data was obtained from the same direction set as the training data.

¢ 1 is an appropriate evaluation measure of the errors in our experimental setting.


Some reconstruction algorithms require a stronger guarantee of bounded coherence µ(A) ≤ O(1/k), where µ(A) defined as It is easy to check that the Gaussian, Bernoulli, and Hadamard-based random matrices given above have coherence bounded by O( (log d)/m) with high probability.

one can take m = O(k 2 log d) to guarantee 1/k coherence.


The power of the REINFORCE learning rule is that the weight change is equal to the gradient of the expected return J(w w w) = E[R {θ} ] over all possible threshold sequences {θ}.

a single-trial learning rule performs like stochastic gradient ascent averaged over many trials.


Equivalently, the query probabilities P n should have inverses 1/P n bounded above by some deterministic quantity r max (which, again, may depend on The a priori upper bound r max on 1/P n can be pessimistic, as the dependence on r max in the final deviation bound will be very mild-it enters in as log log r max .

our goal is to prove a bound on | that holds with high probability over the joint distribution of Z 1:n . To start, we establish bounds on the range and variance of each term the (conditional) range and variance are non-zero only if E i [W i ] = 0.


From a different perspective, each submodule U-Net can be viewed as a "super layer".

the kU-Net is a "deep" deep learning model.


In this range, large similarity values are very sensitive to space variations, and their red level curves are almost vertical, meaning that the similarity information is largely carried by space coordinates.

the visualization of neighborhoods is relatively accurate: visually nearby points are indeed similar; proximity in a neighborhood is informative regarding p(Y ).


One important feature of the deep forest model proposed in [Zhou and Feng, 2018] is that the model complexity can be adaptively determined according to the input data.

it is interesting to integrating several mGBDT layers as feature extractor into the deep forest structure to make the system not only capable of learning representations but also can automatically determine its model complexity.


In a dual-microphone array it is usually assumed that the difference in the two channels is limited to a small time delay (or linear phase in frequency domain) and therefore the cross-correlation is peaked at the the time corresponding to the delay.

methods that search for extrema in cross-correlation waveforms are commonly used [2].


In the realm of GAN applications, it makes sense to consider that P (the true distribution) can be extremely complex.

even when deformed exponential families are significantly more expressive than regular exponential families [25], extra care should be put before arguing that complex applications comply with such a geometric convergence in the parameter space.


The main tradeoff is that Monte Carlo error bounds are probabilistic, though the bound probability is a parameter to the algorithm.

we believe the Monte Carlo approach is superior for all situations that can tolerate minor stochasticity in the approximated output.


Second, we investigate whether attention might equally modulate the gain of all visual neurons responding to the stimulus, but in a task-dependent manner.

the goal of the present study is to determine, using new computational simulations, whether the modulatory effect of attention on early visual processing might be explained by gain-only modulations, if such modulations are allowed to be sufficiently complex (affecting only select visual neurons, or task-dependent).


Decoupling trains two networks In each mini-batch data, each network samples its small-loss instances as the useful knowledge, and teaches such useful instances to its peer network for the further training.

the error flow in Co-teaching displays the zigzag shape.


On the other hand, we have the following proposition about the gradient of f (L).

taking a first order Taylor expansion for f (L t ) and g t (L t ), we can show that the gradient of f (L t ) equals to that of g t (L t ) when t tends to infinity.


Since log-sigsoftmax is composed of a nonlinear function, its output vectors can be greater than d + 1 linearly independent vectors.

we have the following theorem: Theorem 3.


Their capacity can be controlled by varying their depth and breadth, and they also make strong and mostly correct assumptions about the nature of images (namely, stationarity of statistics and locality of pixel dependencies).

compared to standard feedforward neural networks with similarly-sized layers, CNNs have much fewer connections and parameters and so they are easier to train, while their theoretically-best performance is likely to be only slightly worse.


For example, it can be shown that Naive Bayes text classifiers return probability estimates that are monotonic in the frequency of a word-for the class in which the word is most common.

if we are trying to discriminate between documents about New York and Boston, then we expect to find that the Naive Bayes feature measuring the frequency of "Giants" in the corpus is an MF for the class New York, and likewise for "Patriots" and Boston.


The only power iteration step of Algorithm 3 can be thought of as K = kL(R + 1) queries directed to a private data sanitizer which produces f 1 (T; u) = T(I, u, u) or f 2 (T; u) = T(u, u, u) each time.

the 2 -sensitivity of both queries can be separately bounded as where T = T + symmetrize(e i ⊗ e j ⊗ e k ) is some neighboring tensor of T. applying the Gaussian mechanism [9] we can (ε, δ)-privately release one output of either f 1 (u) or f 2 (u) by where = 1, 2 and w ∼ N (0, I) are i.i.d.


Another experiment showed that after many exposures to the same moving stimulus, the sequence of spikes triggered in different neurons along the motion trajectory was also triggered in the complete absence of motion, again indicating that motion signals in cortex may be generated internally from lateral connections [27].

we see a number of reasons to propose that direction selectivity in the cortex may indeed develop and be computed through a mechanism analagous to the one we have developed here.


For training, we used the filtered CB6133 dataset [13,26], which has filtered out proteins in the original CB6133 dataset having 25% or higher similarity with some proteins in CB513.

the filtered CB6133 dataset includes 5534 proteins.


By using experimental data from three different classes of neurons, we demonstrate that biological neurons utilize noise-shaping.

the spike-generation mechanism can be viewed as an oversampling and noise-shaping AD converter.


According to [22], optimal seed s and optimal diffusion matrix L −1 are mutually determined.

we choose to let model learn seed and similarities on its own instead of providing supervision on affinity as [15].


We note that if the parameters of Exp and Poly were set to be a function of n, instead of N , then, then their performance would also not depend on the replication factor (the peformance would be the same as the N/n = 1 case).

the degradation in performance of Exp and Poly is solely due to the suboptimality in setting their parameters.


Throughout this paper, we consider the most commonly used 2 regularization.

the solution of the binary classification problem is given by minimizing the following objective , over a hypothesis class F. When F is the RKHS of some kernel function, the algorithm described above is called kernel support vector machine.


Then the resulting classification rule is  that is, the pi are in the same order as the p * i .

p are sufficient if one only requires the classification rule.


Note that since the number of samples is very small at the beginning, m 0 = 124, and the regularizer is very large, the condition number of problem is very small.

gradient descent is able to converge to a good neighborhood of the optimal solution in a reasonable time.


For this purpose we define the sets A = {X 1 ≤ r 1 , .

. . , X d ≤ r d } and The probability of a realization r is given by we can compute the probability mass of a realization r using only the cdf of X.


. . , x s , By Hoeffding's inequality, which is less than δ 2 .

by Theorem 5, P sup P sup  The core of the uniform bounds in Theorems 1 and 2 is the following uniform bound on the minimum of k linear functions on a ball in R m . Theorem 5.


The effects of NE on efferent areas are modulatory in nature, and are well captured as a change in the gain of efferent inputs so that neuronal responses are potentiated in the presence of NE [9].

in phasic mode, the LC provides transient facilitation in processing, time-locked to the presence of behaviorally salient information in motor or decision areas.


Note that the accumulated error of ultimate network output can be linearly bounded by layer-wise error as long as the model is feed-forward.

l-OBS is a general pruning method and friendly with most of feed-forward neural networks whose layer-wise Hessian can be computed expediently with slight modifications.


As discussed in the introduction, there is evidence that most real life networks present scale-free behavior.

when considering simulated results for validation, we use the model by Barabási & Albert (1999) to generate graphs with this property.


The owl measures the interaural time difference for each frequency of the input sound as an interaural phase difference.

multiple directions in space that differ in their associated interaural time difference by the period of a tone at that frequency are consistent with the same interaural phase difference and can not be distinguished.


The result holds with high probability for all x ∈ X , and for all r µ > r > r n , where r n n→∞ − −−− → 0.

as n grows, the procedure is eventually adaptive to the Hölder parameters in any neighborhood of x.


As another useful property of LARS/Homotopy algorithms, it is worthwhile to mention that changes of the active set only happen at the breakpoints [17,7,15].

given the parameters corresponding to a pair of adjacent breakpoints, e.g., λ (i) and λ (i+1) , the active set for λ ∈ (λ (i+1) , λ (i) ) is the same as λ = λ (i) .


This procedure is then repeated for the remaining M − 1 sums, yielding a total computational effort of O(M T 2 ).

initialize the array subE 0 [k] := getIEC(0, k, 0), and iterate for all m = 1, . . . , M : A close look at eqn.


First, through layer-wise coordination, the information from the source and target sentence will meet earlier, starting from the low-level representations.

the decoder can leverage more fine-grained source information when generating target tokens, instead of only using high-level representations outputted by the encoder in the previous model structure.


Specifically, it is not possible to compute the required integrals analytically.

one has to resort to approximate inference.


Indeed, since Λ is admissible, by step 3, λ(φ ǫ ( β)) ∈ int(Λ).

since E ǫ is continuously differentiable this claim follows from step 4.


For example, words like "neural" and "network" will tend to co-occur quite frequently together in NIPS papers.

we seek to relax the word independence assumption of the Multinomial.


However, co-training requires labeled data to train classifiers for each view, unlike monotonic feature classification which can operate without any labeled data.

there are cases where an MF-based algorithm like MFA is applicable, but co-training is not.


The task is therefore partially observable, since the agent must navigate the space and take in information through this viewport.

the agent must predict the dynamics of the ghosts in memory, and plan its navigation accordingly, also based on remembered information about which food has already been picked up.


These models effectively learn to sort the list.

by design, they cannot be used at test time without a list of candidates available.


For the boundary cases, we keep both the √ T and log T terms because the constant in the log T term is typically much larger than that in the √ T term.

the log T term may dominate the regret particularly when the number of context-action pairs is large for medium T . It is still an open problem if one can achieve regret lower than O( √ T ) in these cases.


However, previous work shows that the confidence score is unreliable in the adversarial setting [12,28].

we construct another metric which is more pertinent and helpful to our goal.


For 4 out of 5 subjects, with the corresponding False Positive rate at 1%, the proposed transform increased the system performance by 7.7%, 8.3%, 8.5% and 18.9% respectively in terms of True Positive rate.

the overall performance of the LF-ASD for these subjects was improved from 66.1%-82.7% to 85.0%-90.4%.


(Note that the k-cluster stopping condition yields a clustering function whose range is the antichain consisting of all partitions into k sets.)

to prove this result, we use a variant of the sum-of-pairs clustering function (see e.g.


It can be shown that L(q) = KL(q(f )||p(f |D)) up to an additive constant [5].

directly optimizing the possibly infinite-dimensional RKHS objects µ and Σ is computationally intractable except for the special case of a Gaussian likelihood and a small training set size N . in practice, we need to impose a certain sparse structure on µ and Σ.


The Hessian matrix of It is easy to show that if 2 σ (x − a i ) ≤ 1, for all i ∈ Π j , then H(f i ) is positive semi-definite, that is, f i (x) is convex.

f (x), the sum of convex functions is also convex.


Upon receiving x n , it learns two hypotheses, h ŷ = LEARN H (S ∪ {(x n , ŷ)}, T ) for ŷ ∈ {±1}, and then compares their empirical errors on S ∪ T . If the difference is large enough 3 , it is possible to infer how h * labels x n (as we show in Lemma 3).

in this case, the algorithm adds x n , with this inferred label, to S. Otherwise, the algorithm requests the label y n and adds (x n , y n ) to T . S contains examples with inferred labels consistent with h * , and T contains examples with their requested labels.


For all m 2 [M ], a product in a i 2 Gm induces a disagreement with all product a j 2 S such that j < i.

the sum of all these disagreements is equal to, Part 2.


The color of the points are the same as used for the pixels in the top part of the figure.

although only the projection onto the isoluminant plane is shown, the third dimension (i.e., luminance) can be inferred by the brightness of the points.


A very convenient and quite versatile modeling technique for analyzing sequences is the Hidden Markov Model (HMM).

the goal we set had been to introduce selection mechanisms for HMMs, capable of achieving useful risk-coverage trade-off in predicting next-day trends.


Before the starting of the experiments, we have standardized the input dataset.

filling zero value is the same as filling mean value.


We replace (11) with At a saddle point Pα(x β ) = P β (x β ) ∀ α⊃β and thus the difference between the logarithmic average (13) and the linear average (11) as well as its derivatives vanish.

(13) has the same local stability properties as (11).


5 and 6, are Markovian and thus we can update individual intensities and opinions in O(1) -let t i and t i+1 be two consecutive events, then, we can compute λ * (t i+1 ) as (λ * (t i ) − µ) exp(−ν(t i+1 − t i )) + µ and x * (t i+1 ) as (x * (t i ) − α) exp(−ω(t i+1 − t i )) + α, respectively; and, (ii) social networks are typically sparse and thus both A and B are also sparse, then, whenever a node expresses its opinion, only a small number of opinions and intensity functions in its local neighborhood will change.

we can reuse the majority of samples from the intensity functions and sentiment distributions for the next new sample.


Moreover, the assumption on the approximation error function is satisfied for β := s/m, whenever f * L,P ∈ W s (X) and s ∈ (d/2, m].

the resulting learning rate is n − 2sα 2s+d+2ds/m , which in the i.i.d.


Because C f and C are only used in the proof (they are not parameters of the OAL algorithm), we can choose them to be arbitrarily small.

oAL converges to an optimal Nash equilibrium w.p.1.


According to ( 16) and ( 17), we have ỹjii = where |R l i | denotes the number of elements in R l i , the subtree of T l rooted at v i .

from the nature of the maximal root-tree T l , l ∈ I 1 , we have we can construct from T l a new root-tree Tl by removing the subtree of T l rooted at v i , so that Tl achieves a larger value than T l . This contradicts with the argument that T l , l ∈ I 1 is the maximal root-tree of the working tree T . it follows from ( 18) that ( 14) holds.


The alternative assumption breaks this constancy, but would produce roughly equivalent results for place cell firing.


It is explained in [9] that small input perturbations can be amplified across layers and cause a hidden state to grow by ω T η, where ω means weight vector and η means perturbation.

even if a perturbation is small, the hidden states can be altered significantly and be easily detected by I-defender.


Let us also denote by φ φ φ T (x, y), the projection of φ φ φ(x, y) on the edges of T . Namely, (φ φ φ T (x, y)) i,j = φ φ φ i,j (x, y i , y j ) if (i, j) ∈ T , and (φ φ φ T (x, y)) i,j = 0 otherwise.

recall that φ φ φ i,j (x, for all (x, y) ∈ X × Y and for all T ∈ S(G), we have We now establish how F (w, x, y) can be written as an expectation over all the spanning trees of G. Let U(G) denote the uniform distribution on S(G).


It is usually infeasible to calculate (7) exactly due to expectation operator.

the empirical risk for ( 7) is used as a cost function as follows: where {(x (1) , y (1) ), • • • , (x (N ) , y (N ) )} are the N pairs of a training dataset, sampled from the joint distribution of x (j) ∼ p(x) and n (j) ∼ N 0,σ 2 .


Usually, we don't have a closed form expression for ŵ(θ) but we rather have an iterative procedure converging to ŵ(θ) that we arbitrarily stop after Q iterations.

we actually solve an approximate problem of the following form.


Our analysis differs from all of these, by focusing on the maximum likelihood situations noted above as convex optimization problems, and using SGD convergence techniques [29] for the sequence of iterates M Recall that the (negative) precision is a natural parameter of the normal distribution written in exponential family notation, and that the log-likelihood is a concave function of the natural parameters for this family; this makes max-likelihood a convex optimization problem over the precision, even in the presence of linear constraints [30,31].

this implies that the problems (3), ( 4) have a unique maximum, denoted by M * I above.


Each term corresponds to an independent M-step, just as in EM for maximizing O indep .

our product EM algorithm differs from independent EM only in the E-step, in which the submodels are multiplied together to produce one posterior over z rather than M separate ones.


Then from Lemma 3, the number of breakpoints is exactly x − y.

our goal is to compute E zt∈[0,1] [x − y].


Due to the nonlinearity and complexity of the model, the E step is intractable.

we considered a stochastic version of the EM algorithm, namely the Monte-Carlo Markov Chain Stochastic Approximation Expectation-Maximization (MCMC-SAEM) algorithm [2], based on [4].


However, this does not allow us to capture dependencies between words because there is only one word being drawn at a time.

we need to reformulate LDA in a way that the words from a topic are sampled jointly from a Multinomial.


From this, we can see that it is not possible to achieve a O(1/n) rate unless c ≥ 1/(2(λ 1 − λ 2 )).

we will assume this when stating our final results, although most of our analysis is in terms of general γ n . An interesting practical question, to which we do not have an answer, is how one would empirically set c without prior knowledge of the eigenvalue gap.


However, p(x ′ |x, u) is generally unknown, and it is a non-trivial problem to identify this distribution in large-scale applications.

it is desirable to find parameter θ in model-free ways, i.e., parameter is estimated from the sample sequences alone, instead of using p(x ′ |x, u).


Each product z e(i) j z j k (1−z k ) appearing in (3) results in a sum of O(2 D ) terms, each of which has (up to a sign) the form q M ≡ m∈M z m for some set M . It is still the case that q M ∈ {0, 1}.

objective function (3) can be interpreted as being linear on a set of binary variables {{z}, {q}}.


This is because the volume of feature space usually increases quickly with the feature dimensionality.

the same number of mixtures will result in a coarser partition resolution and imprecise modeling.


However, we also show that the gap between the elastic net and the k-support norm is at most a factor of √ 2, corresponding to a factor of two difference in the sample complexity.

our work can also be interpreted as justifying the use of the elastic net, viewing it as a fairly good approximation to the tightest possible convex relaxation of sparsity intersected with an 2 constraint.


which can be equivalently stated as The numerator of the second term above can be rewritten as where X i is the i th column vector of X.

the second term of (21) can be written as In ( 23), Since each matrix Z i is a symmetric projection matrix and X i 1 and X i 2 are orthogonal to each other when i 1 = i 2 , Z is a projection matrix of the form X(X T X) −1 X.


This can be seen by writing q( w q ) = M m=1 [q( w qm , s qm = 1) + q( w qm , s qm = 0)] and then by multiplying the terms a mixture of 2 M components is obtained.


The monotonicity of the objective function with respect to ε i implies that the constraints ) are saturated at the solution of (3) for t i = 0 or t i = 1 respectively.

the loss in (3) is the neg-log-likelihood of the lower or the upper bound on p(1|x i ; θ) respectively.


In particular, while the MDP dynamics assume that Mars is flat and every state can be reached, the safety constraint depends on the a priori unknown heights.

under the prior belief, it is unknown which transitions are safe.


Note that, in this new graphical model, the global parameters θ are independent of x tr and y tr and are independent of x test when y test is not observed.

we can now write down a variational lower bound for the logarithm of the approximate likelihood, which is given by In this bound, we essentially perform approximate inference via MAP on φ i to obtain p(φ i |x tr i , y tr i , θ), and use the variational distribution for θ only.


However, if the optimal value function is very complex, then learning an accurate low-dimensional representation can be challenging or even impossible.

when the optimal value function cannot easily be reduced to a low-dimensional representation, we argue to apply a complementary form of regularisation on the target side.


For LiNGAM, we combined the data from two environments as the input.

the distribution of the exogenous noise of variables in I 12 is not Guassian anymore.


As mentioned above, the affect of altering the position of a control point is to change the shape of the contour locally since the basis functions have limited support.

when evaluating p(x~lxt-t}p(ZtlxD for a proposed particle, we only need to reexamine measurement lines and evaluate p(zm,t Ix~,t) for lines in the effected interval and similarly for p(x~,t IXn,t-l).


Generally, we expect that correlations in data may result from several different factors.

we extend the optimization above to include Here we simultaneously search subsets of variables G j and over variables Y j that explain the correlations in each group.


Evidence from the cognitive science literature suggests humans find it difficult to predict the motion of objects when multiple information dimensions are involved [40].

we arrive at an alternative explanation for why humans may behave near-optimally with respect to a dynamics model that differs from the real dynamics: even if users have perfect knowledge of the real dynamics, they may not have the computational resources to plan under the real dynamics, and instead choose to plan using a simplified model.


Nonetheless, noisy labels are evenly spread across the whole space of examples.

the disagreement area includes a number of noisy labels, where the Decoupling approach cannot handle noisy labels explicitly.


On the other hand, F ( x T ) = f ( x T ) + γ ln 1 + exp λ 0 g( x T ) γ ≥ f ( x T ) + max (0, λ 0 g( x T )) .

with the value of γ = ln T /T , we have f ( x T ) + λ 0 g( x T ) ≤ f (x * ) + O ln T T . Applying the inequalities ( 13) and ( 14) to (16), and noting that γ = ln T /T , we have For λ 0 > G 1 /ρ, we have x T − x T 2 ≤ (1/(λ 0 ρ − G 1 ))O(ln T /T ).


Let l ij = log f1(eij ) f0(eij ) be the log-likelihood ratio score of tuple (i, j).

the likelihood function of Definition 1 can be written as c ij l ij .


Assuming that the average degree is ω(1) mismatches empirical social network data.

we use a bipartite model that introduces heterogeneity among nodes: By splitting the nodes into two classes, we achieve high estimation accuracy on the influencers and the degree distribution more closely matches real-world data.


), which is dominated by the quadratic term, and therefore diverges, rendering the model unnormalizeable.

this "Poisson equivalent" of the Ising model cannot model positive interactions between neurons, limiting its value.


The function C λ turns out to have intrinsic connections with statistical leverage scores, as the quantity 1/C λ corresponds precisely to a notion of leverage used in [6,2,28,7].

we uncover a variational formulation for leverage scores which helps elucidate connections with the RKHS and the density p on R d . Our main contribution is a precise asymptotic expansion of C λ as λ → 0 under restrictions on the RKHS.


Specifically, we suggest using a SGD-like low-regret update for the variable w, while updating the variables ξ and b via a PST-like step; the dual update of our method is similar to that of CHW.

our algorithm enjoys the benefits of both methods, each in its respective domain, and avoids the problem originating from the "size" of K. We defer the detailed description of the method to the following section.


The basic ingredients of the proof are: (i) In the limit σ → 0 the expansion coefficients approach α → 1/(1 + γ)1.


. . , Z be independent random variables that are uniformly distributed over S and let H = 1 i=1 Z i be their average.

e[H] = h * and E L S (H) = L * + Var[H] ≤ L * + .


However, this optimization problem has as many variables as the product of the numbers of positively and negatively labeled instances (n + n ), which makes it prohibitive to solve for large data sets within a runtime of a few days.

we experimented with the INFINITEPUSH algorithm only on the Ionosphere data set.


In contrast, the mean firing rates of the two cells during each stimulus were equal for s 1 and s 2 .

if the stimulus is 1 s long, we cannot discriminate two stimuli by using the independent model, that is, I * 1 = 0.


In this section, we focus on the application of GFM in the multi-label setting.

we consider the task of predicting a vector y = (y 1 , y 2 , .


Given two groups (G, • G ) and (H, • H ), we can construct the direct product group (G, Thus, for example, the product SO(2) × (R 2 , +) is again a group.

theorem 1 and 2 also apply for those combinations.


This set is convex and is a superset of the original non-convex set.

the above SDP is a relaxation of the QCQP in 3 with f SDP ≥ f QCQP , providing an upper bound on y (x,ȳ) that could serve as a certificate of robustness.


Note that the clustering game approach [12] utilizes a noninformative prior, that is, all vertices have equal probability.

it cannot obtain multiple clusters simultaneously.


That is, a network connection may be redundant due to the existence of some others, but it will soon become crucial once the others are removed.

it should be more appropriate to conduct a learning process and continually maintain the network structure.


In order to see the statement for the run-time, note that H t and A t−1 have both size O(n) × O(n).

computing H t A t−1 requires O(n 3 ) operations.


Formally, let T R→Φ be a |Φ (i−1) | by |S| matrix for which T R→Φ (φ, s) = |φ| −1 if s ∈ φ, and 0 otherwise, and let T Φ→R be a |S| by |Φ (i−1) | matrix for which T Φ→R (s, φ) = 1 if s ∈ φ, and 0 otherwise.

t Φ→R T R→Φ R is a vector where the reward in each state is the average over all rewards in the feature that state belongs to.


Most modern statistical problems are characterized by high dimensionality.

it is natural to ask the following meta question: We study this question via the problems of estimating parameters of distributions in the distributed setting.


In addition, the structures of sequences and trees are entirely included in that of DAG.

the HDAG kernel subsumes some of the discrete kernels, such as sequence, tree, and graph kernels.


In any case, the effect of this randomness in action-selection will differ across games (based, e.g., on whether a wrong action can be terminal).

we also present results for the UCTtoClassification-greedy agent in which we don't do any exploration.


Of course, L p -nested functions of L pnested functions are again L p -nested.

an L p -nested function f in its general form can be visualized by a tree in which each inner node corresponds to an L p -norm while the leaves stand for the coefficients of the vector y.


Now, consider the case that W > W is non-singular (note that N VOX N CELLS ).

this implies that there is a one-to-one mapping between the RNN state and predicted neural activity: we can rewrite equation 8 in terms of the effect changing r t1 has on the predicted neural activity u v t in each voxel at time t times what a change in u v t implies about a change in ⇡ t2 , operating implicitly via what the change in u v t tells us about a change in h t . We can write this as, Note that this is a correlational relationship -the direction of causality is from h t to u t . Nevertheless the individual terms in this sum: combine the influence that voxel u v t at time t receives from the reward at time t 1 (which is @u v t /@r t1 ), with the covariation between the voxel activity and the action at time t 2 (which is @⇡ t2 /@u v t ).


. . , x n , y n . Since the sampled variables alternate between the blocks, if the chain moves onto the bridge (necessarily by sampling a variable from the island it was previously on), it will always proceed to sample a variable from the other block, which will cause it to move onto the other island.


The derivation of ( 8) and a detailed description of the problem appear in [17].

h ∈ R[w, q, ξ] and X is a basic closed semialgebraic set.


Unlike RFs in classification and regression tasks [9,10,11], RFs are used in language modeling to deal with the data sparsenes problem [4].

the training data is not randomly sampled for each DT.


This is due to the flexibility of the GP framework that considers noisy measurements.

every state in the MDP represents a d ⇥ d square area with d = 1 m, as opposed to d = 20 m in [14].


This spiking condition can be written as E(no spike) > E(with spike).

using equation 6, this can be rewritten as x, x x x cancelling terms, and using equation 2, we can write our spiking condition as the spiking threshold for each neuron must be set to T k ≡ −Ω kk /2, though this condition can be relaxed considerably if our loss function has an additional linear cost term 1 .


Finally, a difficulty of the model proposed is that exact Bayesian inference is intractable.

expectation propagation (EP) is employed for efficient approximate inference.


In fact, for a reversible chain, we can choose L = 1 and α = |λ 2 |, the second largest magnitude eigenvalue of P . We consider an alteration of (1), For any baseline .


An analytic form of the true posterior distribution p(z n |x n ) is intractable for this problem.

we employ samples collected from MCMC as ground truth.


matching parts of the training captions in a word-by-word manner) but restrictive way.

captions generated for different images, especially those that contain objects of the same categories, are sometimes very similar [1], despite their differences in other aspects.


PCN-E-5), PCN slightly under-performs ResNet-50 but outperforms ResNet-34, while using fewer layers and parameters than both of them.

the classification performance of PCN compares favorably with other state-of-the-art models especially in terms of the performance-to-layer ratio.


The bandwidth between the PC host memory and the GPU device memory is ∼ 1.0 GB/s, which is higher than the computation bandwidth (size of data processed by the GPU per second) of the parallel CGS.

the speedup with or without data streaming is almost the same for the parallel CGS.


The weighted estimates P s w (σ 0:t ) for each context s depend on the ratio of the probability of the observed data at s itself, P s e (σ 0:t ), and that of the data observed at its children, P s w (σ 0:t ) at s = σs ∀σ ∈ Σ.

these estimates depend on the number of times each symbol followed a context, implying that c s,t = σ∈Σ c s ,t . the weighting in (3) assumes that each symbol that was observed to follow the non-leaf context s was also observed to follow exactly one of its children s . If this was not so and, e.g., s was created at time 0 but its children only at τ > 0, then, since P s w (σ τ :t ) ≥ P s w (σ 0:t ), the weighting would be biased towards the children, which would have been exposed to less data.


As previously noted, Lemma 1 and its proof remain valid for smooth loss functions such as the α -norm when α ∈ (1, ∞).

it makes sense to compare the results with the ones obtained for α ∈ (1, ∞) .


Note that those fake splits are all deterministically assigned, which means the corresponding conditional probabilities are all one.

the estimated probabilities of rooted trees only depend on their true splitting processes.


Indeed, consider multiple layers of the form Then their composition can be re-written This means that these operations can essentially be performed in Euclidean space.

it is the interposition between those with the bias translation of Eq.


In this domain, EM is competitive with fAfC as for this relatively smaller problem with 25 agents, the space of counts is much smaller than in the taxi domain.

eM's piecewise policy is able to provide a fine grained approximation over the count range.


However, conditioned on the labels of all but one mixture, we can compute the conditional distribution over the remaining label in O(N ) operations, and easily sample from it.

we may use a Gibbs sampler [9] to draw asymptotically unbiased samples, as illustrated in Figure 1.


For non-harmonic features, we use a radial basis function to define affinities.

if f a is the value of the feature for data point a, we use a basis affinity matrix defined as W ab = exp(−||f a − f b || β ), where β > 1.


To discourage abrupt changes in the number of inactive leaves, we introduce a variant of SGD, in which the assignments of data points to leaves are fixed for a number of gradient updates.

the bound is optimized with respect to a set of data point leaf assignment constraints.


is convex and continuous, the affine function ) is convex and continuous.


Hence, the expected number of iterations E[T ] for finding the desired solution is at most 2ek 2 n. Comparing with the approximation guarantee of FR, (1 − e −γ S F R ,k ) • OP T [3], it is easy to see that γ ∅,k ≥ γ S F R ,k from Definition 3.

pOSS with the simplest configuration of the isolation function can do at least as well as FR on any sparse regression problem, and achieves the best previous approximation guarantee.


Moreover, we can see that X T g Θ ⊆ Ξ g , g = 1, . . . , G. To develop the TLFre rule by (R1 * ) and (R2 * ), we need to solve the following optimization problems: t * gi (λ, λ; α) = sup θ {|x T gi θ| : θ ∈ Θ}, i = 1, . . . , n g , g = 1, . . . , G. (18) Solving problem (17) We consider the following equivalent problem of ( 17): We can see that the objective function of problem ( 19) is continuously differentiable and the feasible set is a ball.

( 19) is a nonconvex problem because we need to maximize a convex function subject to a convex set.


Note that this is not equivalent to adding λ k to the Fisher matrix.

nevertheless, it does become equivalent to a small SGD step in the limit of large damping factor λ k . at iteration k we update the synaptic weights in layer i according to We update λ k in each iteration to reflect the ratio ρ k between i) the actual decrease in the loss resulting from the latest damped NG parameter update, and ii) the decrease predicted by a quadratic approximation to the loss 1 .


The key idea of our method is that running multiple MCTS simulations generates a set of trajectories with more positive rewards (see Section 4 for more analysis), which can also be viewed as being generated by an improved policy ⇡ ✓ .

learning from these trajectories can further improve ⇡ ✓ .


In addition, performance criteria are different in the closed-loop and openloop settings (e.g., time per target vs. root mean squared error).

a SNN of a different size may be required to meet the desired specifications.


Since most of points (candidate matches) should not belong to any cluster, partition-based clustering method, such as clique aver-aging method, cannot be used.

we only compare our method with matching game approach and measure the performance of these two methods by counting how many matches agree with the ground truths.


Furthermore, on pos and cws, the best "gid" results with parameters ρ < 1 are competitive to the standard perceptron with exact inference (i.e., ρ = 1, u = 1).

it is possible for approximate inference to be both fast and good.


The Generic Viewpoint Assumption (GVA) states that the position of the viewer or the light in a scene is not special.

any estimated parameters from an observation should be stable under small perturbations such as object, viewpoint or light positions.


[26, Theorem 1] has proved the descent property, similar to that of (5), which simply follows from the linear majorization idea and therefore the descent property in condition (2) of Theorem 2 holds.

the result follows from Theorem 2 and Lemma 9.


23.2], which runs in time O(n 2 + n log n).

the approximation is computationally tractable.


This process reassembles the constructive clustering process used by [11] in transforming multiinstance examples into single-instance examples except that in [11] the clustering is executed at the instance level while here we execute it at the bag level.

the original MIML examples which corresponds to the Step 3 of MIMLSVM.


For Internet diagnosis, we do not know the structure of graphical model for a block of data ahead of time: the structure depends on the metadata for the requests in the log.

we cannot amortize the learning time of a predictive model.


( 1), we observe that applying multiplicative noise to weights is equivalent to applying it to activations, except that the noise mask is independently sampled for each hidden unit in the upper layer.

without loss of generality, we consider multiplicative noise of the form in Eq.


Furthermore, in a detection setup the number of region proposals for two algorithms are dependent on their underlying approaches.

the Recall@K is not comparable for two algorithms since the number of samples being compared are different.


It does not care which members are already met.

we define s i as the percentage of members we have not met in a community C i .


The decision function is of the form where x E ~d is the d-dimensional input vector of a test example, y E {-I, I} is a class label, Xi  O:S ai :S C Vi.

(4) The kernel K(X,Xi) can have different forms, such as the Radial Basis Function (RBF): K(Xi, Xj) = exp (-llxi(T~ Xj112) (5) with parameter (T. to train an SVM, we need to solve a quadratic optimization problem, where the number of parameters is N. This makes the use of SVMs for large datasets difficult: computing K(Xi' Xj) for every training pair would require O(N2) computation, and solving may take up to O(N3).


Theorem 1 establishes a smooth, algebraic characterization of acyclicity that is also computable.

the following equality-constrained program (ECP) is equivalent to (3): 3.


In particular, it is much more restrictive than the assumption of independent sampling that underlies most statistical language models.

we will focus on a different preferential attachment scheme, based upon the two-parameter species sampling model [4,5] known as the Pitman-Yor process [6].


Given v = [1, 1, −ǫ, −ǫ] T ∈ Null(Φ), this can be accomplished by computing ∂g (I) (x (2) + βv)/∂β = h ′ (|α 1 − α 2 + β|) + h ′ (|β|) + 2ǫh ′ (|ǫα 2 − ǫβ|).

in the limit as β → 0 (from the right or left), this expression will always be positive for ǫ < 0.5 based on the concavity of h. x (2) must be a minimum.


By direct analogy to FA, we defined the prior distribution of the neural state x :,t at each timepoint t to be N (0, I) by setting Furthermore, because we seek to extract smooth neural trajectories, we set σ 2 n,i to a small value (10 −3 ).

the timescale τ i is the only (hyper)parameter of the SE covariance that is learned.


The meta-algorithm tracks the state reached in the WFST and maintains a set of external regret minimizing algorithms that help the learner perform well at every state.

here, we need one external regret minimization algorithm A u,i , for each state u reached at time t after reading sequence x 1:t 1 and each i 2 ⌃ labeling an outgoing transition at u.


For the replication-based scheme, since the number of extra workers n−k < k, there is a non-zero probability (which does not change with T dl ) that the n − k extra workers do not replicate the computation in the slowest one among the first worker.

replication when n − k < k does not improve the error exponent, because the error is dominated by this slowest worker.


This is the classical coupon collector's problem (CCP), which is known to require Θ(K log K) samples [33,34].

under these assumptions the expected number of samples required to learn π * is Θ(K log K).


This result is based on recent approximation bounds dealing with the eigenvectors of the kernel matrix which show that if a function can be reconstructed using only a few kernel PCA components asymptotically, then the same already holds in a finite sample setting, even for small sample sizes.

the use of a kernel function not only greatly increases the expressive power of linear methods by non-linearly transforming the data, but it does so ensuring that the high dimensionality of the feature space will not become overwhelming: the relevant information for classification will stay confined within a comparably low dimensional subspace.


Note that (2) is a convex program and the time to solve it is negligible in Algorithm 1.

the running times with and without Warm-start are roughly the same.


Estimating Topic Matrices Z 1...n For LPMRF topic models, the estimation of the LPMRF parameters given the topic assignments requires solving another complex optimization problem.

we pursue an alternating EM-like scheme as in LPMRF mixtures.


The power of group bias: For this benchmark, good performance mostly depends on the quality of segmentation.

a simpler approach is to denoise with coarse-grained group bias, followed by classification.


. . , Z m be a sequence of random variables sampled uniformly without replacement from a fixed set of m + u elements Z, and let φ : Then, for all ǫ > 0, the following inequality holds: where α(m, u) We define the selection matrix corresponding to a sample of m columns as the matrix S ∈ R n×m defined by S ii = 1 if the ith column of K is among those sampled, S ij = 0 otherwise.

c = KS is the matrix formed by the columns sampled.


This problem is also found in differentiable variational Bayes and solved by using auxiliary MCMC strcuture [24].

we believe this problem is caused by MF-variational inference used in our model and we hope it can be solved if we try other inference methods.


There is no gold-standard list of topics to compare against for every corpus.

evaluating the latent space of topic models requires us to gather exogenous data.


The first component was grown around the point P in Figure 4 a.

the second component was grown around point Q in Figure 4 e. The symbolic signatures inside the region around P encode the geometric configuration of three of the four components learned by the first module (healthy heads), while the symbolic signatures around Q encode the geometric configuration of three of the remaining four components (malformed heads), Figure 4 b and f . the points of the output mesh of the second module will be set to "+1" if they belong to learned symbolic signatures associated with the healthy heads (Figure 4 c) , and "-1" otherwise (Figure 4 g).


Note that the trace is a linear function of , keeping the problem convex.

as long as dot products can be represented via kernels, the optimization can be carried out without explicitly using the high dimensional space.


When the And-Or fragment is added into the grammar, the size of the grammar is changed in two aspects: first, the size of the grammar is increased by the size of the And-Or fragment; second, the size of the grammar is decreased because of the reductions from configurations of multiple nodes to the new And-node.

the prior gain of learning from the And-Or fragment is:   where s a and s o are the number of bits needed to encode each node on the right-hand side of an And-rule and Or-rule respectively.


. . , x N ) be any multivariate distribution over real-valued random variables, then there exists a copula function such that The constructive converse which is of central interest from a modeling perspective is also true: since for any random variable the cumulative distribution F(x i ) is uniformly distributed on [0, 1], any copula function taking the marginal distributions {F(x i )} as its arguments, defines a valid joint distribution with marginals F(x i ).

copulas are "distribution-generating" functions that allow us to separate the choice of the univariate marginals and that of the dependence structure expressed in the copula function C, often resulting in an effective real-valued construction.


Note that exact inference is hard due to the global shape term.

we propose a spanning tree approximation to the graph to obtain multiple initial solutions for possible corner locations.


Let the optimal transport cost in (18) be the W 1 distance, and suppose F is a convex subset of 1-Lipschitz functions.

then, the minimax problem (18) will reduce to the moment-matching interpretation also holds for WGAN: for a convex set F of 1-Lipschitz functions WGAN finds the generative model with minimum W 1 distance to the distributions penalized to share the same moments over F with the data distribution.


This is different from the recent work [4] in which linear convergence for PSGM is obtained for sharp and weakly convex objective functions and convex constraint sets.

we believe the choice of piecewise geometrically diminishing step size is of independent interest and can be useful for other nonsmooth problems.


Other parameters describing the appearance of an object in an image can be added.

the image features most commonly used for describing local structures are the energy outputs of oriented band-pass filters, as they have been shown to be relevant for the task of object detection [9,10] and scene recognition [2,4,8,12]~the local image representation at the spatial location (x) is given by the vector VL(X) == {v(X,k)}k==l,N with: where i(x) is the input image and gk(X) are oriented band-pass filters defined by gk(i) == e-llxI12/u~e27fj<f~,x>.


Upon accepting a newly revealed sample, online RPCA updates the PCs estimation accordingly without re-exploring the previous samples.

online RPCA can deal with large amounts of data with low storage expense.


The bigger the value D ii (corresponding to y i ) is, the more "important" is y i .

we impose a constraint as follows: y T Dy = 1 ⇒ a T XDX T a = 1 Finally, the minimization problem reduces to finding: The transformation vector a that minimizes the objective function is given by the minimum eigenvalue solution to the generalized eigenvalue problem: It is easy to show that the matrices XLX T and XDX T are symmetric and positive semidefinite.


Further, one can use the conjugate gradient method to solve for (W t K −1 u,u W t +σ 2 I) −1 v which only involves computing the matrix-vector product In practice, the conjugate gradient method converges within only a few iterations.

approximating the posterior mean µ using SKI takes only O(n + d + m log m) time to compute.


Indeed, variational inference in this model follows more directly and with fewer separate bounding operations than the sparse auxiliary-variable approach used with the GP-LVM.

uncertainty in the low-dimensional coordinates and in the manifold shape (defined by the local maps) is captured naturally.


When used as a tool to assess an ANN index before actually putting it to work, it is also important that the tester actually reduces the total computation time compared to observing poor results at the end of the data processing pipeline (e.g., bad classification results) and starting over.

we seek to answer the following questions: Setup We implemented our property tester in C++ and integrated it into the Python framework ANN-Benchmarks [2,4].


£ ge £ ¦ 4 ne £ n 0 § fY "£ ny @ 2 # Now for each & (' v v Q£ ) , the term £ ge £ ¦ 2 4 u ge f£ n t 0 § fY "£ ¦y @ 2 is non-negative, and equal to zero only when y @ belongs to ¢ 0 ¤£ B£ ge f£ n t 2 .

the bound is met with equality if and only if y @ achieves the maximum defining £ ge £ ¦ 2 for all trees & (' v v Q£ ) .


sparse or Gaussian) can determine the type of structure captured in B, we discovered that it does not affect the nature of the lower-level representation.

for the results reported here, we assumed a Gaussian prior on v. as in other multi-variate Gaussian models, the precise directions of B are not important; the learned vectors only serve to collectively describe the volume of the space.


Similarly, when algorithm B runs at a lower rate of 1 − λ, it becomes a β 1−λ -approximation, and by monotonicity can only get better from concurrency with A.

both bounds hold simultaneously, and the overall guarantee is their minimum.


In the current, general model, these also depend on the isotropy of the optical blur (α 1 and α 2 ) and the variance of sensory noise (σ 2 ν ), and no simple meaning is attatched to the individual cases.)


The key to the utility of this technique is that the number of compound parameters can be polynomial in L even when the number of possible topologies is exponential in L. Previous analytic techniques [13] based on Kruskal's theorem [10] cannot be applied here because the possible topologies are too many and too varied.

note that the mixing equation µ = M Ψ holds for each sentence length L, but many compound parameters p appear in the equations of multiple L. we can combine the equations across all observed sentence lengths, yielding a more constrained system than if we considered the equations of each L separately.


Beginning with only the root ∅, AOBFS expands the search tree in a best-first manner.

more precisely, it maintains explicit AND/OR search tree of visited nodes, denoted S. For each node n in the AND/OR search tree, AOBFS maintains u n , an upper bound on v n , initialized via a pre-compiled heuristic v n ≤ h + n , and subsequently updated during search using information propagated from the frontier: the upper bound at the root, u ∅ , is an anytime deterministic upper bound of Z.


pLSI In pLSI, the topic mixture proportions θ d are a parameter for each document.

pLSI is not a fully generative model, and the number of parameters grows linearly with the number of documents.


According to lemma 2, a factorization model would at least require m latent components to recover this adjacency matrix exactly.

an algorithm with cubic runtime complexity in the rank would only be able to recover Y for this relation when the number of marriages is small, what limits its applicability to these relations.


However, current practice in population genetics collapses the data to a set of summary statistics before passing it through the machine learning models.

the performance still rests on the ability to laboriously hand-engineer informative statistics, and must be repeated from scratch for each new problem setting.


Two individual objects provided a single instance for training or testing and were never used again, neither in training nor in testing.

the training instances were independent of each other but they were not always informative (as defined in Section 3).


The coefficients of the quartic equation ( 5) are analytic, so γ h can also be obtained analytically 3 , e.g., by Ferrari's method [9] (we omit the details due to lack of space).

the global VB solution can be analytically computed.


is properly defined and meets requirements R1, R2, R3 iff [0,1] (y, H) = z + D φ (y||H) for some permissible φ.

φ maybe viewed as the "signature" of the loss.


Usually the optimal policy is defined as the policy that maximizes the value conditioned on the initial state, i.e., for any realization s of S 0 [10].

the policy that maximizes the conditional value given each realization of the initial state also maximizes the value averaged over the initial state, and it is the unique maximizer if the distribution of the initial state D is positive over S. when optimizing over all possible policies, the maximization of ( 1) and (2) are equivalent.


In contrast, by selecting an optimal subset of the clusters, our approach can still achieve superior performance as long as there are good clusters in the input clusterings.

our approach is much more robust, as confirmed by the results of this experiment.


The minimax problem ( 15) is the f-GAN problem introduced and discussed in [3].

theorem 2 reveals that f-GAN searches for the generative model minimizing the f-divergence to the discriminator moment matching models specified by discriminator set F. The following example shows the application of this result to the vanilla GAN introduced in the original GAN work [1].


The worst case setting corresponds to the situation where the adversary can choose any distribution over causal graphs.

our result implies a worst case −Θ(log log n) bound on the value of a game with multi-variable interventions and no latent variables.


The parameter updates ( 4)-( 7) imply µ This follows from the strong law of large numbers, as the updates are recursive implementations of the sample mean and sample covariance matrix.

the large-sample approximation to the conditional likelihood becomes: where we used lim u→∞ (1+ c u ) u = e c . The conditional likelihood (15) corresponds to the multivariate Gaussian distribution with mean µ h and covariance matrix Σ h . A similar asymptotic normality result was recently obtained in Tsiligkaridis & Forsythe (2015) for Gaussian observations with a von Mises prior.


The problem ( 8) is generally hard to solve because the model parameters and variational distributions are strongly coupled.

we develop a natural iterative procedure that estimates the parameters Θ and performs posterior inference alternatively.


We may think of w (i) v i : V (i−1) → R + as a modular function defined on set V (i−1) .

this matrix contains d i such modular functions.


Therefore, as V → ±∞, the function h(V ) goes to positive infinity and it thus has a global minimum at some value Ṽ ∈ R. At the global minimum, its derivative vanishes, .

at the global minimum of the polynomial h, all terms except the highest order term cancel, and we find  Avg h( Ṽ ) is strictly positive, and since Ṽ is a global minimum of h, we have h(V ) ≥ h( Ṽ ) > 0 for all V ∈ R. Inserting into the expression for L(λ * , V * 0 ) concludes the proof that the lower bound at the optimum is positive.


Here, the apparently small effect on the number of LPs for low ratio schedules actually arises because of the large amount of time spent eating.

according to the model DA is not really allowing animals to cope with higher work requirements, but rather is important for optimal choice of vigor at any work requirement, with the slowing effect of DA depletion more prominent (in the crude measure of LPs per session) when more time is spent leverpressing.


Namely for p far from uniform, Equation (1) will fail by a lot.

we can distinguish between the relevant cases by finding suitably close approximations to F 2 (p) and F 3 (p).


Such a direction w exists by the piecewise linearity of the generative model G. In fact, not only is the function piecewise linear, each of the pieces is the intersection of a finite number of half spaces.

with probability 1 any randomly chosen direction w moves strictly into one piece, allowing for differentiability at x + w for sufficiently small .


• We show in Section 3.3 that our choice to study the mean estimator here is not somehow "unlucky" (it is not a particularly bad linear smoother, nor is the upper bound on its MSE loose): the minimax linear risk over T d (C n ) is on the order C 2 n /n, for all d ≥ 2.

even the best linear smoothers have the same poor performance as the mean over T d (C n ).


To establish convergence of the off-policy update in the dual representation, first define the maxnorm for state-action visit distribution as Then one can simply reduce the dual to the primal case by appealing to the relationship MHr where the second equality holds since we assumed (1 − γ)q (sa) = [Hr] (sa) for all (sa).

given convergence of Mq to a fixed point Mq * = q * , the same must also hold for MH.


Although this can effectively reduce computational cost, we argue that early stage features are very important for vision tasks, and that premature reducing the feature map size can impair representational abilities.

we still maintain a four-stage structure.


4, is to show that this minimization can be framed as a minimization of a Difference of Convex functions (DC) [11].

a large literature on Difference of Convex functions Algorithms (DCA) [19,20](a rather standard approach to non-convex programming) is available to solve our problem.


The expected payoffs of the arms are given by the inner product of their associated vector and some fixed, but initially unknown parameter vector θ * .

the expected payoff of arm a is m ′ a θ * , which is linear in θ * .


Furthermore, one sees that lim µ→0 w µi = 1 if and only if Λ i > 1, and lim µ→0 w µi = 0 if and only if Λ i < 1.

lim µ→0 w µ = w * holds if and only if Λ i > 1 for all i with w * i = 1 and Λ i < 1 for all i with w * i = 0.


it will accept any price p t < v t and reject any price p t > v t ), since its decisions there do not affect any of its future prices.

the exploit rounds are the time at which the seller can exploit what it has learned from the buyer during exploration.


First, the pseudoinverse operation in (5) blows up variation in C V depending on the matrix condition number of this matrix.

c should be chosen such that the condition of C V gets close to 1.


As we have just shown, this objective function can be expressed in a form similar to (5), where the only difference lies in the norm that is used.

we have shown that the solution of the residual gradient algorithm can also be characterised in the general framework of quadratic error metrics IIq,w -V* liD.


defined on the same probability space) in such a way that G n → G and Q n → Q almost surely as n → ∞ for fixed T . Under assumption (5), it is easy to show that ( 11) holds for all n > n 0 with n 0 a sufficiently large constant (for a proof see the provided supplementary material).

by the proof of Theorem 3.1, the conditions in Proposition 4.1 hold for gradient G n and hessian Q n for any n ≥ n 0 , with probability larger than 1 − δ.


Note that the linear program in (21) is the same as the one in (18).

by checking the first-order optimality condition of x t , the variable v t is already computed, and we need to solve only one linear program per iteration.


By Corollary 1, it follows that w j f G ≤ 0.

by Assumption 1, w j f k > 0 and w j f j > 0, leading to contradiction.


For transcription, we model letters as coming from an n-gram language model, with no dependencies between words.

the probability of a letter depends on the k letters before it, where k = n unless this would cross a word boundary in which case the history terminates at this boundary.


If node i is a linear unit, it simply outputs the weighted sum We allow the networks to be heterogeneous, that is, they may contain all three types of computation nodes simultaneously.

this model encompasses a wide class of network types considered in research and applications.


In these situations not all variables are of equal relevance to the user's query; some variables become observed given the query, others become statistically independent given the query, and the remaining variables are typically marginalized.

a user-generated query provides a tremendous amount of information that can be exploited by an intelligent inference procedure.


These "perturb-max" models describe the robustness of the MAP prediction to random changes of its parameters.

one can draw unbiased samples from these distributions using MAP predictions.


Applying stable random projections on By the properties of Fourier transforms, the projected differences, v 1,j −v 2,j , j = 1, 2, ..., k, are i.i.d.

samples of the stable distribution S(α, d (α) ), i.e., the task is to estimate the scale parameter from k i.i.d.


It returns the state x(n) with highest evaluation.

note that Assumption 2 implies that the b-value of any cell containing x * upper bounds f * , i.e., for any cell a node (h, i) such that f (x h,i ) + δ(h) < f * will never be expanded (since at any time t, the b-value of such a node will be dominated by the b-value of the leaf containing x * ).


Note that a Gaussian with covariance matrix @7 has zero variance along directions that are orthogonal to the images of the data.

in order to compute the mutual information, we only need to compute the covariance matrix of the projection of ¥ onto the linear span of the data, that is, for all CB ED ¦ 0F : where © G denotes the § F ¥ vectors with only zeros except at position D , and I ¡ is the Gram matrix of the centered points, the so-called centered Gram matrix of the -th component, defined from the Gram matrix P ¡ of the original (non-centered) points as , where Q is a § F ( § matrix composed of ones [8].


p k is a continuous quantity whereas F is defined on at discrete 2D locations.

we use the differentiable bilinear sampling operation to sample from the discrete grid [25] to obtain the feature at X k w . Such an operation has the highly desirable property that features from pixels in multiple images that may correspond to the same 3D world point unproject to the same location in the 3D gridtrivially enforcing epipolar constraints.


For large £ , complexity of ¢¡ ¤£ ¥ § is a problem.

the rest of the paper explores possibilities of reducing the computation to make the method applicable to large databases.


Although the race model is elegant in its simplicity and captures key experimental data, it is descriptive in nature and does not address how the stopping latency and other elements of the model depend on various underlying cognitive factors.

it cannot explain why behavior and stopping latency varies systematically across different experimental conditions or across different subject populations.


The following corollary is straightforward.

corollary 1 For any s ∈ S * , we have: the homogeneity testing problem may be seen as closely related to the problem of estimating the optimal AUC * , since it may be re-formulated as follows: "H 0 : AUC * = 1/2" versus "H 1 : AUC * > 1/2".


Since the connection between rank and exact inference is obvious from Theorem 2, the more interesting question in Q2 is whether Boolean rank is indicative of the complexity of approximate lifted inference as well.

we investigate its influence on the Lifted MCMC algorithm (LMCMC) [29] with Rao-Blackwellized probability estimation [30].


For graph-based MDPs, the sizes of the full state and decision spaces are exponential in the number of variables.

the complexity of exact algorithms is exponentially large.


Algorithm 2 is based on a linear encoder and a linear decoder.

one may notice that the value function is also linear in the domain of the raw features, i.e., the value function can be represented as One may wonder, why it is not better to solve for w directly with regularization on w ?


In each iteration of our algorithm, we add an SPN structure exactly similar to HT-TMM.

the first iteration of our algorithm (i.e.


In our case, we define importance by separability-or a gap-rather than simply variance.

we distinguish panel (1) from ( 2) and (3), while [17] distinguishes between ( 2) and (3).


A graph Laplacian is not just a single observation-it is a positive semidefinite matrix with a very specific structure.

we will take L to be a random object with expectation L, where L is another normalized graph Laplacian.


2 and remain fairly constant, BP-spikes change their shapes along the dendrite.

figure 2: (A-F) STDP-curves obtained from Eqs.


predictions as compared with Caucasians (G 2 ) when automated risk tools are used in practice.


However, when the network is sparse, whenever we sample a new node (or link) event from the model, only a small number of intensity functions, in the local neighborhood of the node (or the link), will change.

we can reuse most of the samples from the intensity functions for the next new sample and find which intensity functions we need to change in O(log m) operations, using a heap.


As the parameter approaches zero, the expectation of each per-term probability becomes closer to its empirical estimate.

the expected distribution over terms becomes less smooth.


In other words, when the perfect-recall property is satisfied, it is possible to move between the sequence-form and the realization-form representations by means of a simple linear transformation.

the realization polytope of perfect-recall games can be described with a linear number (in the game size) of linear constraints.


In addition, in case of repetitive features, assigning the match to the first nearest neighbour might result in an incorrect match, in which case the hard assignment would lose valuable information about the subsequent closest neighbours.

in order to have an approach that is amenable to end-to-end training, all pairwise feature matches need to be computed and stored.


Note that this model captures both point-to-point communication as well as broadcast com-munication.

our lower bounds in this model apply to both the message passing setting and the broadcast setting.


The argument is that typical experimental setups make the reliability of each trial unpredictable for the participant.

it was argued, the brain can have no correct estimate of the reliability.


This stretching does multiplicatively attenuate any curvature in layer l − 1 by a factor 1/χ 1 (see the update equation for κl in ( 8)), but new curvature is added in due to a nonzero χ 2 , which originates from the curvature of the single neuron nonlinearity in (9).

unlike in linear expansion, extrinsic curvature is not lost, but maintained, and ultimately approaches a fixed point κ * .


In contrast, using stochastic search methods or descent methods to find the global minimum will generally incur a computational cost scaling which is exponential in the sample size times the parameter space dimensions.

when fitting high-dimensional mixture models, practitioners are forced to choose between the discretization errors inherent to NNLS, or the computational difficulties in the descent methods.


In such a case, the false negative rate is R 1 = 0 but the false positive rate is R −1 = 1.

the overall misclassification rate for the hinge loss will be 1 − π.


Second, the set of classifiers is restricted to differentiable parametric functions, which may not include the F-measure maximizer.

their proof of vanishing regret does in general not imply convergence to the optimal F-score.


. . , π t−N ∈ σ(u t−N ) and µ N t can be computed using where P a is the transition probability matrix when in every state action a is used and e x is the unit row vector corresponding to x (and we assumed that X = {1, . . . , |X |}).

moreover, a simple but tedious calculation shows that ( 3) and ( 4) ensure the conditional unbiasedness of our estimates, that is, It then follows that E[ ρt |u t−N ] = ρ t , and, hence, by the uniqueness of the solutions of the Bellman equations, we have, for all (x, a) ∈ X × A, we also have, for all (x, a) ∈ X × A, t ≥ N + 1, and The bandit algorithm that we propose is shown as Algorithm 1.


Third, a triangular matrix can be inverted in quadratic time.

we can efficiently compute A −1 t from A t when needed, instead of having two separate quadratic-time updates for A −1 t and A t , which requires more memory and is prone to numerical instabilities.


In some of such cases, ION will not discover any PAGs which entail the correct d-separations and d-connections.


This plot at least suggests that the constituent/non-constituent classification is less amenable to direct clustering.

it is important that an induction system be able to detect constituents, either implicitly or explicitly.


The task of learning is to find the model parameters Θ and the assignments Y which maximize the likelihood of the complete dataset {I, Y , A}.

the joint probability of {I, Y , A} given Θ from equation (1) can be written as  to minimize J , each latent variable Y i must belong to the set of possible assignments A i .


Even in the corrupted setting, the difference in leverage scores between corrupted and noncorrupted points is small (see §6).

the main contribution to the influence for each point will originate from the residual error, e 2 i . Consequently, we propose sampling with probability inversely proportional to the approximate residual, 1 ẽ2 i .


the lower bound for the log-likelihood is given by The maximum of F[q(z)] with respect to q(z j,i = t) = ϕ j,i,t and β is given by The updates of α are the same as Eq.


Notice that Fourier coefficients of p are the expectations p(z) = EX∼p [ψ z (X)].

p(z) := 1 n n j=1 ψ z (X j ) and q(z) := 1 n n j=1 ψ z (Y j ) are independent unbiased estimates of p and q, respectively.


Whereas, the corresponding feature map of videos where the action of interest is present is being decomposed into two components: a) the latent regions and b) context regions.

the scoring function is written: where K is the number of latent regions of the action model and b is the bias term.


Taking inverses, (B + δI) −1 (A + δI) −1 .

by Definition 11, the function h(t) = 1 δ+t is operator antitone.


Fazel [26] proposed using the Schatten 1-norm • S 1 , the sum of the matrix's singular values, as a convex proxy for rank.


We refer to these problems generically as "regression," which indicates our focus on the conditional probability density p Y |X (y|x).

our framework includes classification problems, where Y is discrete.


The wrapper method, on the other hand, is defined as a search through the space of feature subsets using the estimated accuracy from an induction algorithm as a measure of goodness of a particular feature subset.

one approximates T(O', a) by minimizing Twrap(O', a) = min Talg(O') IT (2) subject to 0' E {a, l}n where Talg is a learning algorithm trained on data preprocessed with fixed 0'.


permutation matrices since they corresponds to ordering K elements.

even though {q i,l } can be obviously decomposed into a linear combination of permutation matrices, it is not clear how to compute them without computing the set of all permutation matrices that are exponentially large in K. Algorithm 2 solves this problem: Let Ni,l be a temporal variable that is initialized by Ñi,l at the beginning.


This feature seems to weight the technical characteristics of a computer (RAM, CPU and CD-ROM) against its price.

in this application our algorithm is able to discern interesting patterns in people's decision process.


However, in practice, we observe that λ decreases quickly as δ increases.

our algorithm can finish in a reasonable time.


Yue and Joachims (2009) showed that a stochastic gradient descent algorithm can be applied to dueling bandit.

it is naturally expected that a stochastic mirror descent algorithm, which achieves the (near) optimal order in convex optimization with one/two-point bandit feedback, can be applied to dueling bandit setting and achieves good performance.


Third, large blocks may not be fit in the cache, leading to cache misses.

we use block cyclic data distribution which uses a small nonconsecutive blocks and thus can largely achieve load balance and scalability.


Using Lemma 4.2, it suffices to obtain a (1 − γ)-multiplicative approximation to the k-SVD of A t (i.e., .

we can directly apply the first running time in Theorem 4.3: .


In a POMDP, the agent reasons about the current state based on an observation.

pOMDP can be regarded as an MDP with observations, Z and an observation function O : Z × A × S → [0, 1], which determines P (z|a, s), the probability of observing z after performing action a in state s. In a POMDP, instead of knowing the current state, s t , the agent computes the belief state, b t , which is the posterior probability over states given all past observations and actions.


Learning/optimization is performed so that this sum (network plus prior) minimizes training loss.

with su cient network capacity and optimization, the ensemble members will agree at observed data.


The minimization of RCE is simple to implement using stochastic gradient descent methods, with little extra training cost, as compared to CE.

it can be easily applied to any deep networks and is as scalable as the CE training procedure.


The main difference is that MP solves a slightly different problem, where the 1 regularization is replaced with an 0 constraint.

the size of the support is a fixed parameter in MP, whereas it is controlled by the regularization parameter λ in our case.


We also show how to use the proposed architecture as an inference engine in more complex models which can handle multiple independent objects in the sensory input.

the proposed architecture is suitable for the type of inference required by rich models: those that can learn a hierarchy of abstractions, can handle temporal information and can model multiple objects in the input.


¡ 1 2 d ¢ 2C 8 3 ¨is sampled by executing action 3 in state ¨and thereafter following the policy.

in the episodic formulation, where the agent executes a maximum of 9 steps during each episode, at the end of each episode, ¡ 1 2 d ¢ 2¨6 8 6 ¨for step D can be evaluated as follows: ¡ 1 2 d ¢ ©¨6 8 6 ¨A £ $ ¥ @9 § 9 b 6 87 i ¨6 A ¡C 26 A § 8 given that the agent executes a complete episode ¢ ¢ 229 I9 ¨ ! "


Drug resistant viruses present a major cause of treatment failure and their occurrence renders many of the available drugs ineffective.

knowing the precise patterns of drug resistance is an important prerequisite for the choice of optimal drug combinations [9,10].


For each round t, we have that 120 ≥ err .

by Markov inequality, we have that Pr i∼p (t) (•) err Di (g (t) ) > 12 ≤ .1 .


Each of the parallel threads touches every data example in each pass.

the 10 thread run does 10x more gradient computations than the serial version.


We obtain similar results to AVI and API.

an interesting perspective would be to have a less naive setting for DCA by choosing different explicit decompositions and find a better convex solver for the intermediary convex minimizations problems.


. . ×N U N , the root mean square error (RMSE) is a frequently used measure of the difference between the recovered tensor and the true one: RMSE := 1 √ I N ∥D − A∥ F . [25] analyzes the statistical performance of the convex tensor Schatten 1-norm minimization problem with the general linear operator X : R I1×...×IN → R m . However, our model ( 6) is non-convex for the LRTC problem with the operator P Ω .

we follow the sketch of the proof in [26] to analyze the statistical performance of our model ( 6).


At iteration t, it requires two passes over the network: in the forward pass, the activations of all layers are computed from the bottom layer l = 1 to the top layer l = L following: ; in the backward pass, it applies the chain rule and propagates error gradients through the network from the top layer l = L to the bottom layer l = 1 following: According to (3), computing gradients for the weights w l of the layer l is dependent on the error gradient ∂f x t (w t ) ∂h t l from the layer l + 1, which is known as backward locking.

the backward locking prevents all layers from updating before receiving error gradients from dependent layers.


Our estimates can be further improved by a few steps of local search [6].

using the solution found by our hybrid algorithm to initialize some local search algorithms largely improves the performance on both accuracy and convergence speed, compared to the greedy stochastic search method described in [6].


Because of the nature of the problem, M i−1 + x i−1 ≤ M i ∀i.

these intervals are disjoint and sorted.


Before we can give eigenfunctions or state necessary and sufficient conditions we need some basic relations about Legendre Polynomials and spherical harmonics.

all we have to do in order to check whether a particular kernel may be used in a SV machine or a Gaussian Process is to look at its polynomial series expansion and check the coefficients.


The general dynamic regret in (2) includes the static regret in (1) and the restricted dynamic regret in (3) as special cases.

minimizing the general dynamic regret can automatically adapt to the nature of environments, stationary or dynamic.


For example, by fixing b = 1 and initializing a copositive network with some input, the permitted sets unequivocally determine the stable fixed points.

in this case, the notion of permitted sets is no different from fixed point attractors.


The world around us is structured in space and time, and the evolution over time of naturally occurring phenomena is related to the previous states.

changes in real-world scenes are sequential by nature and the human brain can perceive this sequentiality and use it to learn semantic relationships between the various events to analyze scenes; e.g.


This means that it, alone, tells us nothing about the model; for example, any distribution can be represented by a trivial factor graph with a single factor that contains all the variables.

in order to use hierarchy width to produce a result about the mixing time of Gibbs sampling, we constrain the maximum weight of the factors.


By Lemma 4, if a round t ends after I(⌧, t , N t ) queries are answered, then the total revenue collected from queries in that round is at least 6N t , so the "high price" at the end of the round is 0.

a query q uj from the non-adaptive user costs the low price 96/(⌧ 2 u j ) unless it causes an instantiation of VALIDATIONROUND to halt prematurely.


The constructive approach, however, has a drawback: Shallow networks are known not to handle non-linearities as effectively as deeper ones [Montufar et al., 2014].

the initial, shallow architectures may easily get trapped in bad optima, and thus provide poor initialization for the constructive steps.


We noted in our experiments that as α is large and our data lies between [0, 1], all units in the network tend to have small values that make the whole objective function relatively small.

a relatively large change in (w, u) might cause only small changes in the objective function but performance may vary significantly as the distance is large in the parameter space.


Regression with Brownian Sheets: When one uses Brownian sheets for regression with a target function f * = i α * i ϕ i that lies in the Cameron-Martin space K defined previously (i.e.

such that ||α * || < ∞), then the term ||α * || 2 sup x∈X ||ϕ(x)|| 2 that appears in Theorems 1 and 2 is bounded as: from Theorem 2, ordinary least-squares performed on random subspaces spanned by P Brownian sheets has an expected excess risk (and a similar bound holds in high probability).


We embedded PLSA into our model by treating the states of individual players as instances of PLSA topics (similar to [5]).

the PLSA model determines the emission probability in Eq.


For applications involving high-dimensional data, such as face recognition, σ is usually large (typically ranging from thousands to hundreds of thousands [13]), and the condition in Lemma 4.1 holds if we restrict our search space to the convex hull of each class in the original space.

the global minimum of the optimization problem in (7) can be found very efficiently using Newton's or gradient decent methods.


Here, -means is able to identify the inner circle as a cluster with ¢ §¤ .

the stability for this number of clusters is highest (figure 1  .


For any matrix norm II• II it holds that e(A) :S IIAII [5].

we know that ehIID~P) :S IbIID~PIID~' where the vector norm   Besides the iterative solution of (6) often a direct solution by matrix inversion is computed using equation ( 9) in the fixed point form AiFwiF + bIt = O.


Unfortunately, in general there is no closed form solution for evaluating the EMAX objective, even in our case of normally distributed vectors [20].

to implement the greedy algorithm, which requires many evaluations of the EMAX objective, we use Monte Carlo sampling, where for a given set B we sample the corresponding normally distributed vector and average the maximum values across the samples.


Each edge (i, t) is associated with exactly one ground set element i ∈ V, and selecting i (i ∈ T s ) is equivalent to cutting the edge (i, t).

the cost of edge (i, t) will model the cost g(i) of its element i ∈ V. Let E t be the set of such edges (i, t), and denote, for any subset C ⊆ E t the set of ground set elements adjacent to C by We define the cost of C to be the cost of its adjacent ground set elements, h g (C) g(V (C)); this implies h g (δ s (S ∩ E t )) = g(S).


The learned reward is then used for finding an optimal policy.

the expert's actions can be predicted in states that have not been encountered during the demonstration.


So far the segmentation S was assumed to be given, and we estimated GES(Q, H ref , S).

in order to obtain the global evidence score of Q, we marginalize over all possible segmentations S of Q: Namely, the likelihood P (S|H ref ) of a segmentation S can be interpreted as a weight for the likelihood ratio score of Q induced by S. we would like P (S|H ref ) to reflect the complexity of the segmentation S (e.g., its description length).


As we employ normalized kernels (where the diagonal is 1), it is sufficient to measure the pairwise kernel similarity.

for a test point x, the nearest neighbor classifier reduces to: The USPS hand written digits dataset Hull (1994) consists of n = 7291 training (and 2007 test) greyscale images of 10 handwritten digits from 0 to 9.


SGD explores the space of parameters by making small and noisy steps and that noise is averaged out by the stochastic gradient contributions accumulated in each weight.

it is important to keep sufficient resolution for these accumulators, which at first sight suggests that high precision is absolutely required.


For example, both convolutional and recurrent operations process a local neighborhood (several nearest neighboring neurons) in either space or time.

the long-range dependencies can only be captured when these operations are applied recursively, while those long-range dependencies are sometimes significant in practical learning problems, such as image or video classification, text summarization, and financial market analysis [1,6,22,27].


It is easy to see that X S X S is always invertible.

furthermore, one can show that if , we have |µ j | < 1, ∀j ∈ S c and sgn(β * i )γ i > 0, ∀i ∈ S. by our assumptions, the preconditions of Lemma 1 are satisfied for the original Lasso problem.


If all the neighbors of ¡ have been assigned, make sure ¡ @ 2' is a best response.

the first vertex chosen assigns both itself and all of its neighbors, but afterwards vertices assign only (some of) their neighbors, and receive their own values from a neighbor.


In our case, since each matrix multiplication is between a Toeplitz matrix K u,i and a matrix T (i) with n entries, it can be done in O(n log n) time using the FFT [29].

the overall running time is also O(n log n).


A classifier f is a function f : X → Y, and the true risk of f w.r. where : Y × Y → R + is a given loss function, for example the 0/1 error.

given a labeled set [5] is a pair (f, g), where f is a classifier, and g : X → {0, 1} is a selection function, which serves as a binary qualifier for f as follows, the selective classifier abstains from prediction at a point x iff g(x) = 0.


In this case, C is incorrectly classified by h P k , yielding a significant error.

in infinite-dimensional metric spaces, the shape of the Voronoi cells plays a fundamental role in the consistency of the partition.


Further, the narrower the peak of case-1, the lower is this probability.

bayesian optimisation algorithm converges to the narrower peak with lower probability.


The approximation error bound will not, in most cases, satisfy the sparsity assumption, as the errors are typically distributed almost uniformly over the state space and is received in every step as a result.

for sparse rewards, the discount error increase will typically be offset by the larger decrease in the approximation error.


Second, we note that because we initialized all weights to 0, only features with non-zero weights will contribute to the policy that Cobot uses.

we can determine that a feature is relevant for a user if that feature's weight vector is far from that user's bias feature weight vector, and from the all-zero vector.


The joint distribution, P (X, Y ), is valid provided P (Y |X) and P (X) are valid distributions.

a distributional assumption on P (X) does not restrict the set of covariate functions in any way.


Thus, we obtain that as {p k i } is bounded (Proposition 8 in Supplement F).

we conclude that, for all k ∈ K, k ≥ k = max( ǩ, k) and for all i ∈ {1, • • • , n}, at least one of the following three cases must happen: It follows that there exists a constant ᾱ > 0 such that Thus, considering According to Proposition 8 (in Supplement F), we know that the largest eigenvalue of H k is uniformly bounded from above by some positive constant M . Thus, we have which together with Eqs.


Moreover, the basis of such eigenfunctions can be learned using only the unlabeled data by constructing the Nystrom extension of the eigenvectors of an appropriate kernel matrix.

given unlabeled data we construct the basis of eigenfunctions and then apply L 1 penalized optimization procedure Lasso [Tib96] to fit a sparse linear combination of the basis elements to the labeled data.


We used a randomized tree approach similar to [24] to get features combining these modalities.

we first extracted the primary features in each pixel including the absolute differences with the previous frame and the background, the image intensity, and the absolute values xand y-derivatives.


At no step do we need to keep all of the dense matrices S, SΩ, ∇h 1 on a single machine.

cONCORD-ISTA and CONCORD-FISTA are scalable to any high dimensions restricted only by the number of machines.


However, G * (η) is difficult to deal with as it depends on the expectation parameters.

we approximate the model space using the tangent space at the center point of all training data [20]: and N is the number of training images.


In contrast, in convolutional networks the weight matrices are highly sparse with block structure due to striding filters, as in Figure 2(a).

[8]'s proof does not apply to convolutional networks, and the extension of concentration bounds for our case is not trivial.


It is easy to verify that Tit-for-Tat is flexible with respect to the experts AC and AD.

theorem 5.1 holds and the SEA achieves an average payoff at least as much as that of AC.


At test time, only the NTP is used for predictions.

the training loss for ComplEx can be seen as an auxiliary loss for the subsymbolic representations learned by the NTP.


Our model is easy to accommodate within a framework of temporal difference (TD) learning.

it naturally preserves the link between phasic DA signals and online learning of optimal values [24].