Contextual bandits with linear payoffs, which are also known as linear bandits, provide a powerful alternative for solving practical problems of sequential decisions, e.g., online advertisements. In the era of big data, contextual data usually tend to be high-dimensional, which leads to new challenges for traditional linear bandits mostly designed for the setting of low-dimensional contextual data. Due to the curse of dimensionality, there are two challenges in most of the current bandit algorithms: the first is high time-complexity; and the second is extreme large upper regret bounds with high-dimensional data. In this paper, in order to attack the above two challenges effectively, we develop an algorithm of Contextual Bandits via RAndom Projection ( CBRAP ) in the setting of linear payoffs, which works especially for high-dimensional contextual data. The proposed CBRAP algorithm is time-efficient and flexible, because it enables players to choose an arm in a low-dimensional space, and relaxes the sparsity assumption of constant number of non-zero components in previous work. Besides, we provide a linear upper regret bound for the proposed algorithm, which is associated with reduced dimensions.
@cite_5 , the authors proposed new methods for constructing sparse random sign matrix for dimensionality reduction. In the ensuing section, we name our algorithm of CBRAP with Random Sign (RS) matrix (abbreviated as CBRAP.RS ).
@cite_5
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Erasure codes, such as Reed-Solomon (RS) codes, are increasingly being deployed as an alternative to data-replication for fault tolerance in distributed storage systems. While RS codes provide significant savings in storage space, they can impose a huge burden on the I O and network resources when reconstructing failed or otherwise unavailable data. A recent class of erasure codes, called minimum-storage-regeneration (MSR) codes, has emerged as a superior alternative to the popular RS codes, in that it minimizes network transfers during reconstruction while also being optimal with respect to storage and reliability. However, existing practical MSR codes do not address the increasingly important problem of I O overhead incurred during reconstructions, and are, in general, inferior to RS codes in this regard. In this paper, we design erasure codes that are simultaneously optimal in terms of I O, storage, and network bandwidth. Our design builds on top of a class of powerful practical codes, called the product-matrix-MSR codes. Evaluations show that our proposed design results in a significant reduction the number of I Os consumed during reconstructions (a 5× reduction for typical parameters), while retaining optimality with respect to storage, reliability, and network bandwidth. Consider an @math matrix @math with the restricted isometry property of order k and level @math ; that is, the norm of any k-sparse vector in @math is preserved to within a multiplicative factor of @math under application of @math . We show that by randomizing the column signs of such a matrix @math , the resulting map with high probability embeds any fixed set of @math points in @math into @math without distorting the norm of any point in the set by more than a factor of @math . Consequently, matrices with the restricted isometry property and with randomized column signs provide optimal Johnson–Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound @math given by Ailon and Liberty to $m... Petabyte-scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency. Classical codes like Reed-Solomon are highly sub-optimal for distributed environments due to their high overhead in single-failure events. Locally Repairable Codes (LRCs) form a new family of codes that are repair efficient. In particular, LRCs minimize the number of nodes participating in single node repairs during which they generate small network traffic. Two large-scale distributed storage systems have already implemented different types of LRCs: Windows Azure Storage and the Hadoop Distributed File System RAID used by Facebook. The fundamental bounds for LRCs, namely the best possible distance for a given code locality, were recently discovered, but few explicit constructions exist. In this work, we present an explicit and optimal LRCs that are simple to construct. Our construction is based on grouping Reed-Solomon (RS) coded symbols to obtain RS coded symbols over a larger finite field. We then partition these RS symbols in small groups, and re-encode them using a simple local code that offers low repair locality. For the analysis of the optimality of the code, we derive a new result on the matroid represented by the code generator matrix. Random projections have recently emerged as a powerful method for dimensionality reduction. Theoretical results indicate that the method preserves distances quite nicely; however, empirical results are sparse. We present experimental results on using random projection as a dimensionality reduction tool in a number of cases, where the high dimensionality of the data would otherwise lead to burden-some computations. Our application areas are the processing of both noisy and noiseless images, and information retrieval in text documents. We show that projecting the data onto a random lower-dimensional subspace yields results comparable to conventional dimensionality reduction methods such as principal component analysis: the similarity of data vectors is preserved well under random projection. However, using random projections is computationally significantly less expensive than using, e.g., principal component analysis. We also show experimentally that using a sparse random matrix gives additional computational savings in random projection. This paper analyzes circulant Johnson-Lindenstrauss (JL) embeddings which, as an important class of structured random JL embeddings, are formed by randomizing the column signs of a circulant matrix generated by a random vector. With the help of recent decoupling techniques and matrix-valued Bernstein inequalities, we obtain a new bound @math for Gaussian circulant JL embeddings. Moreover, by using the Laplace transform technique (also called Bernstein's trick), we extend the result to subgaussian case. The bounds in this paper offer a small improvement over the current best bounds for Gaussian circulant JL embeddings for certain parameter regimes and are derived using more direct methods. We propose a new method for linear dimensionality reduction of manifold-modeled data. Given a training set X of Q points belonging to a manifoldM ⊂ R , we construct a linear operator P : R → R that approximately preserves the norms of all Q 2 pairwise difference vectors (or secants) of X . We design the matrix P via a trace-norm minimization that can be efficiently solved as a semi-definite program (SDP). When X comprises a sufficiently dense sampling of M, we prove that the optimal matrix P preserves all pairs of secants over M. We numerically demonstrate the considerable gains using our SDP-based approach over existing linear dimensionality reduction methods, such as principal components analysis (PCA) and random projections. Recently, great progress has been made in sign language recognition. Most approaches are based on the Hidden Markov Model (HMM) with various features, such as motion trajectory. Recognition for sign sentences is obtained from optimal path by Viterbi algorithm, however, some wrong jumps are usually caused by transitional movements between signs. To address the problem, in this paper, we propose an approach consisting of two stages: offline training and online recognition. In the offline training stage, we propose a threshold matrix and rate thresholds. Each element of the threshold matrix describes the minimal probability when a segment belongs to a sign, and rate thresholds are defined as the average probability for signs. So, if certain segment's evaluation is smaller than all the thresholds, it is regarded as a transitional movement and then it should be removed. In the online recognition stage, coarse segmentation, based on the threshold matrix, records the time interval for fine segmentation, and fine segmentation, based on Dynamic Time Warping(DTW) and Length-Root method, determines the endpoint for each candidate sign and selects the most possible one. The final recognition is obtained by concatenating the most possible signs. We evaluate our approach with Kinect-based dataset and the experiments demonstrate the effectiveness of our approach. In this work, we propose a linear base station (BS) precoding strategy based on switched relaying (SR) processing for downlink multiuser multiple-input multiple-output (MIMO) relay systems. The BS and the MIMO relay station (RS) are both equipped with a codebook of interleaving matrices. For a given channel state information (CSI) the selection function at the BS chooses the optimum interleaving matrix from the codebook based on the maximum sum received signal-to-interference-plus-noise ratio (SINR) or sum rate to design the linear BS precoder. Prior to the payload transmission the BS sends the index corresponding to the selected interleaving matrix to the RS, where the optimum interleaving matrix is selected as the optimum relay processing matrix. The entries of the codebook are randomly generated unitary matrices. Simulation results show that the performance of the proposed techniques is significantly better than prior art in the case of perfect CSI.1 We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = FGT, we focus on algorithms in which G is restricted to containing nonnegative entries, but allowing the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods. In this letter, we propose a new decoding algorithm based on a soft-input soft-output run-length limited (RLL) decoder in visible light communication (VLC) with on–off keying (OOK) modulation and Reed–Solomon (RS) codes. Conventional RLL codes are used for dimming adjustments in VLC; however, in our receiver model, the posterior probability matrix is produced by the new RLL decoder, and it can be matched as the input of the soft RS decoder to enhance the throughput in the VLC system. A significant advantage of our proposed method is that it leads to the enhancement of bit error rate (BER) and frame error rate (FER) performances without any change in the transmitter in VLC systems. The simulation results show that the majority of the BER and FER performances of our proposed method are better than those of hard-input hard-output RLL decoding and soft-input hard-output RLL decoding methods. Trust networks, where people leave trust and distrust feedback, are becoming increasingly common. These networks may be regarded as signed graphs, where a positive edge weight captures the degree of trust while a negative edge weight captures the degree of distrust. Analysis of such signed networks has become an increasingly important research topic. One important analysis task is that of sign inference, i.e., infer unknown (or future) trust or distrust relationships given a partially observed signed network. Most state-of-the-art approaches consider the notion of structural balance in signed networks, building inference algorithms based on information about links, triads, and cycles in the network. In this paper, we first show that the notion of weak structural balance in signed networks naturally leads to a global low-rank model for the network. Under such a model, the sign inference problem can be formulated as a low-rank matrix completion problem. We show that we can perfectly recover missing relationships, under certain conditions, using state-of-the-art matrix completion algorithms. We also propose the use of a low-rank matrix factorization approach with generalized loss functions as a practical method for sign inference - this approach yields high accuracy while being scalable to large signed networks, for instance, we show that this analysis can be performed on a synthetic graph with 1.1 million nodes and 120 million edges in 10 minutes. We further show that the low-rank model can be used for other analysis tasks on signed networks, such as user segmentation through signed graph clustering, with theoretical guarantees. Experiments on synthetic as well as real data show that our low rank model substantially improves accuracy of sign inference as well as clustering. As an example, on the largest real dataset available to us (Epinions data with 130K nodes and 840K edges), our matrix factorization approach yields 94.6 accuracy on the sign inference task as compared to 90.8 accuracy using a state-of-the-art cycle-based method - moreover, our method runs in 40 seconds as compared to 10,000 seconds for the cycle-based method. In this paper, we study the problem of one-bit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm such that a k-sparse unit vector x* can be efficiently recovered from the sign of its linear measurements, i.e., b = sign(Ax*). This is an important problem for signal acquisition and has several learning applications as well, e.g., multi-label classification (, 2009). We study this problem in two settings: a) support recovery: recover the support of x*, b) approximate vector recovery: recover a unit vector x such that ||x - x*||2 = e. For support recovery, we propose two novel and efficient solutions based on two combinatorial structures: union free families of sets and expanders. In contrast to existing methods for support recovery, our methods are universal i.e. a single measurement matrix A can recover all the signals. For approximate recovery, we propose the first method to recover a sparse vector using a near optimal number of measurements. We also empirically validate our algorithms and demonstrate that our algorithms recover the true signal using fewer measurements than the existing methods. In this paper we present an algorithm which, for a given (sparse) matrix, constructs a partition of its set of row-indices, such that each subset of this partition (except the last one obtained) contains indices which correspond to mutually orthogonal rows. We then use such decompositions in some classes of block-projections methods, previously extended by the author to general inconsistent linear least-squares problems. Numerical experiments on an inconsistent and rank-deficient least-squares model problem are described in the last section of the paper. We present a novel method for elastic animation editing with space-time constraints. In a sharp departure from previous approaches, we not only optimize control forces added to a linearized dynamic model, but also optimize material properties to better match user constraints and provide plausible and consistent motion. Our approach achieves efficiency and scalability by performing all computations in a reduced rotation-strain (RS) space constructed with both cubature and geometric reduction, leading to two orders of magnitude improvement over the original RS method. We demonstrate the utility and versatility of our method in various applications, including motion editing, pose interpolation, and estimation of material parameters from existing animation sequences. In this thesis, I study the spectral characteristics of large dynamic networks and formulate the spectral evolution model. The spectral evolution model applies to networks that evolve over time, and describes their spectral decompositions such as the eigenvalue and singular value decomposition. The spectral evolution model states that over time, the eigenvalues of a network change while its eigenvectors stay approximately constant. I validate the spectral evolution model empirically on over a hundred network datasets, and theoretically by showing that it generalizes arncertain number of known link prediction functions, including graph kernels, path counting methods, rank reduction and triangle closing. The collection of datasets I use contains 118 distinct network datasets. One dataset, the signed social network of the Slashdot Zoo, was specifically extracted during work on this thesis. I also show that the spectral evolution model can be understood as a generalization of the preferential attachment model, if we consider growth in latent dimensions of a network individually. As applications of the spectral evolution model, I introduce two new link prediction algorithms that can be used for recommender systems, search engines, collaborative filtering, rating prediction, link sign prediction and more. The first link prediction algorithm reduces to a one-dimensional curve fitting problem from which a spectral transformation is learned. The second method uses extrapolation of eigenvalues to predict future eigenvalues. As special cases, I show that the spectral evolution model applies to directed, undirected, weighted, unweighted, signed and bipartite networks. For signed graphs, I introduce new applications of the Laplacian matrix for graph drawing, spectral clustering, and describe new Laplacian graph kernels. I also define the algebraic conflict, a measure of the conflict present in a signed graph based on the signed graph Laplacian. I describe the problem of link sign prediction spectrally, and introduce the signed resistance distance. For bipartite and directed graphs, I introduce the hyperbolic sine and odd Neumann kernels, which generalize the exponential and Neumann kernels for undirected unipartite graphs. I show that the problem of directed and bipartite link prediction are related by the fact that both can be solved by considering spectral evolution in the singular value decomposition. The robust estimation of multivariate location and shape is one of the most challenging problems in statistics and crucial in many application areas. The objective is to find highly efficient, robust, computable and affine equivariant location and covariance matrix estimates. In this paper, three different concepts of multivariate sign and rank are considered and their ability to carry information about the geometry of the underlying distribution (or data cloud) are discussed. New techniques for robust covariance matrix estimation based on different sign and rank concepts are proposed and algorithms for computing them outlined. In addition, new tools for evaluating the qualitative and quantitative robustness of a covariance estimator are proposed. The use of these tools is demonstrated on two rank-based covariance matrix estimates. Finally, to illustrate the practical importance of the problem, a signal processing example where robust covariance matrix estimates are needed is given. In this paper, we consider the design of joint network coding (NC) and relay selection (RS) in two-way relay channels. In the proposed schemes, two users first sequentially broadcast their respective information to all the relays. We propose two RS schemes: 1) a single RS with NC and 2) a dual RS with NC. For both schemes, the selected relays perform NC on the received signals sent from the two users and forward them to both users. The proposed schemes are analyzed, and the exact bit-error-rate (BER) expressions are derived and verified through Monte Carlo simulations. It is shown that the dual RS with NC outperforms other considered RS schemes in two-way relay channels. The results also reveal that the proposed RS-NC schemes provide a selection gain compared with an NC scheme with no RS and an NC gain relative to a conventional RS scheme with no NC. The relationship between running speed (RS) and heart rate (HR) was determined in 210 runners. On a 400-m track the athletes ran continuously from an initial velocity of 12–14 km h to submaximal velocities varying according to the athlete's capability. The HRs were determined through ECG. In all athletes examined, a deflection from the expected linearity of the RS-HR relationship was observed at submaximal RS. The test-retest correlation for the velocities at which this deflection from linearity occurred (Vd) determined in 26 athletes was 0.99. The velocity at the anaerobic threshold (AT), established by means of blood lactate measurements, and Vd were coincident in 10 runners. The correlation between Vd and average running speed (mean RS) in competition was 0.93 in the 5,000 m (mean Vd = 19.13 + - 1.08 km h; mean RS = 20.25 + - 1.15 km h), 0.95 in the marathon (mean Vd = 18.85 + - 1.15 km h; mean RS = 17.40 + - 1.14 km h), and 0.99 in the 1-h race (mean Vd = 18.70 + - 0.98 km h; mean RS = 18.65 + - 0.92 km h), thus showing that AT is critical in determining the running pace in aerobic competitive events. Along with data on the web increasing dramatically, hashing is becoming more and more popular as a method of approximate nearest neighbor search. Previous supervised hashing methods utilized similarity dissimilarity matrix to get semantic information. But the matrix is not easy to construct for a new dataset. Rather than to reconstruct the matrix, we proposed a straightforward CNN-based hashing method, i.e. binarilizing the activations of a fully connected layer with threshold 0 and taking the binary result as hash codes. This method achieved the best performance on CIFAR-10 and was comparable with the state-of-the-art on MNIST. And our experiments on CIFAR-10 suggested that the signs of activations may carry more information than the relative values of activations between samples, and that the co-adaption between feature extractor and hash functions is important for hashing. In this paper we present a practical solution with performance guarantees to the problem of dimensionality reduction for very large scale sparse matrices. We show applications of our approach to computing the Principle Component Analysis (PCA) of any @math matrix, using one pass over the stream of its rows. Our solution uses coresets: a scaled subset of the @math rows that approximates their sum of squared distances to @math -dimensional subspace. An open theoretical problem has been to compute such a coreset that is independent of both @math and @math . An open practical problem has been to compute a non-trivial approximation to the PCA of very large but sparse databases such as the Wikipedia document-term matrix in a reasonable time. We answer both of these questions affirmatively. Our main technical result is a new framework for deterministic coreset constructions based on a reduction to the problem of counting items in a stream.