We introduce a Gaussian process model of functions which are additive. An additive function is one which decomposes into a sum of low-dimensional functions, each depending on only a subset of the input variables. Additive GPs generalize both Generalized Additive Models, and the standard GP models which use squared-exponential kernels. Hyperparameter learning in this model can be seen as Bayesian Hierarchical Kernel Learning (HKL). We introduce an expressive but tractable parameterization of the kernel function, which allows efficient evaluation of all input interaction terms, whose number is exponential in the input dimension. The additional structure discoverable by this model results in increased interpretability, as well as state-of-the-art predictive power in regression tasks.
Bach @cite_10 uses a regularized optimization framework to learn a weighted sum over an exponential number of kernels which can be computed in polynomial time. The subsets of kernels considered by this method are restricted to be a of kernels. In the setting we are considering in this paper, a hull can be defined as a subset of all terms such that if term @math is included in the subset, then so are all terms @math , for all @math . For details, see @cite_10 . Given each dimension's kernel, and a pre-defined weighting over all terms, HKL performs model selection by searching over hulls of interaction terms. In @cite_10 , Bach also fixes the relative weighting between orders of interaction with a single term @math , computing the sum over all orders by: which has computational complexity @math . However, this formulation forces the weight of all @math th order terms to be weighted by @math .
@cite_10
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We present a geometric formulation of the Multiple Kernel Learning (MKL) problem. To do so, we reinterpret the problem of learning kernel weights as searching for a kernel that maximizes the minimum (kernel) distance between two convex polytopes. This interpretation combined with novel structural insights from our geometric formulation allows us to reduce the MKL problem to a simple optimization routine that yields provable convergence as well as quality guarantees. As a result our method scales efficiently to much larger data sets than most prior methods can handle. Empirical evaluation on eleven datasets shows that we are significantly faster and even compare favorably with a uniform unweighted combination of kernels. We analyze mixture density approximation and estimation. We form a convex set of density functions by taking the convex hull of a parametric family, e.g. mixtures of the Gaussian location family. A sequence of finite mixture densities is formulated to provide a parsimonious approximation for the target density. If the target density itself is in the convex hull, we show that the approximation error goes to zero with a rate of 1 k, where k is the number of components in the approximation. If the target density is outside of the convex hull, the approximation error is equal to the best achievable error plus a term that goes to zero with a rate of 1 k. A greedy algorithm that introduces one component at each step is shown to achieve such an error rate. Similarly, a greedy estimation algorithm is provided to find such approximation for data from an arbitrary density. This algorithm estimates one mixture component at one time. We prove that such an algorithm achieves a likelihood nearly as good as the MLE (maximum likelihood estimate) over the whole convex hull. And we identify the difference as being bounded by order O(1 k), where k is the number of components in the estimate. Risks of such estimators are shown to be bounded by a sum of approximation error and estimation error. The error terms are identified. An optimal choice of k can be derived by minimizing the risk bound. Acting as a similar role as the bandwidth in non-parainetric density estimation, k controls two error terms in opposite directions. A large k reduces approximation error and increases estimation error. A MDL (minimum description length) principle is derived to provide an estimation method for k. And the estimated k is shown to achieve the risk bound as if we know the best k in advance. A new information projection theory is derived to expand the approximating class to include its information closure. We prove the existence and uniqueness of a f* in the closure of the convex hull C (in a sense we identify), such that D ( fpf* ) = infg∈CD fpg , where Dfpg is the Kullback-Leibler divergence. And log(fk) → log(f*) in L1 (f) for any sequence fk in C with Dfpfk→ infg∈CD fpg. Other characterizing properties of f* are also given. Several object categorization algorithms use kernel methods over multiple cues, as they offer a principled approach to combine multiple cues, and to obtain state-of-the-art performance. A general drawback of these strategies is the high computational cost during training, that prevents their application to large-scale problems. They also do not provide theoretical guarantees on their convergence rate. Here we present a Multiclass Multi Kernel Learning (MKL) algorithm that obtains state-of-the-art performance in a considerably lower training time. We generalize the standard MKL formulation to introduce a parameter that allows us to decide the level of sparsity of the solution. Thanks to this new setting, we can directly solve the problem in the primal formulation. We prove theoretically and experimentally that 1) our algorithm has a faster convergence rate as the number of kernels grow; 2) the training complexity is linear in the number of training examples; 3) very few iterations are enough to reach good solutions. Experiments on three standard benchmark databases support our claims. Abstract In this paper we report on our early experience on porting, optimizing and benchmarking a Lattice Boltzmann (LB) code on the Xeon-Phi co-processor, the first generally available version of the new Many Integrated Core (MIC) architecture, developed by Intel. We consider as a test-bed a state-of-the-art LB model, that accurately reproduces the thermo-hydrodynamics of a 2D- fluid obeying the equations of state of a perfect gas. The regular structure of LB algorithms makes it relatively easy to identify a large degree of available parallelism. However, mapping a large fraction of this parallelism onto this new class of processors is not straightforward. The D2Q37 LB algorithm considered in this paper is an appropriate test-bed for this architecture since the critical computing kernels require high performances both in terms of memory bandwidth for sparse memory access patterns and number crunching capability. We describe our implementation of the code, that builds on previous experience made on other (simpler) many-core processors and GPUs, present benchmark results and measure performances, and finally compare with the results obtained by previous implementations developed on state-of-the-art classic multi-core CPUs and GP-GPUs.