Learning Bounds for Greedy Approximation with Explicit Feature Maps from Multiple Kernels

Nonlinear kernels can be approximated using finite-dimensional feature maps for efficient risk minimization. Due to the inherent trade-off between the dimension of the (mapped) feature space and the approximation accuracy, the key problem is to identify promising (explicit) features leading to a satisfactory out-of-sample performance. In this work, we tackle this problem by efficiently choosing such features from multiple kernels in a greedy fashion. Our method sequentially selects these explicit features from a set of candidate features using a correlation metric. We establish an out-of-sample error bound capturing the trade-off between the error in terms of explicit features (approximation error) and the error due to spectral properties of the best model in the Hilbert space associated to the combined kernel (spectral error). The result verifies that when the (best) underlying data model is sparse enough, i.e., the spectral error is negligible, one can control the test error with a small number of explicit features, that can scale poly-logarithmically with data. Our empirical results show that given a fixed number of explicit features, the method can achieve a lower test error with a smaller time cost, compared to the state-of-the-art in data-dependent random features.Comment: Proc. of 2018 Advances in Neural Information Processing Systems (NIPS 2018

Nonparametric Basis Pursuit via Sparse Kernel-based Learning

Signal processing tasks as fundamental as sampling, reconstruction, minimum mean-square error interpolation and prediction can be viewed under the prism of reproducing kernel Hilbert spaces. Endowing this vantage point with contemporary advances in sparsity-aware modeling and processing, promotes the nonparametric basis pursuit advocated in this paper as the overarching framework for the confluence of kernel-based learning (KBL) approaches leveraging sparse linear regression, nuclear-norm regularization, and dictionary learning. The novel sparse KBL toolbox goes beyond translating sparse parametric approaches to their nonparametric counterparts, to incorporate new possibilities such as multi-kernel selection and matrix smoothing. The impact of sparse KBL to signal processing applications is illustrated through test cases from cognitive radio sensing, microarray data imputation, and network traffic prediction.Comment: IEEE SIGNAL PROCESSING MAGAZINE, 2013 (TO APPEAR

Generalization Guarantees for Sparse Kernel Approximation with Entropic Optimal Features

Despite their success, kernel methods suffer from a massive computational cost in practice. In this paper, in lieu of commonly used kernel expansion with respect to $N$ inputs, we develop a novel optimal design maximizing the entropy among kernel features. This procedure results in a kernel expansion with respect to entropic optimal features (EOF), improving the data representation dramatically due to features dissimilarity. Under mild technical assumptions, our generalization bound shows that with only $O(N^{\frac{1}{4}})$ features (disregarding logarithmic factors), we can achieve the optimal statistical accuracy (i.e., $O(1/\sqrt{N})$). The salient feature of our design is its sparsity that significantly reduces the time and space cost. Our numerical experiments on benchmark datasets verify the superiority of EOF over the state-of-the-art in kernel approximation

Non-parametric Group Orthogonal Matching Pursuit for Sparse Learning with Multiple Kernels

We consider regularized risk minimization in a large dictionary of Reproducing kernel Hilbert Spaces (RKHSs) over which the target function has a sparse representation. This setting, commonly referred to as Sparse Multiple Kernel Learning (MKL), may be viewed as the non-parametric extension of group sparsity in linear models. While the two dominant algorithmic strands of sparse learning, namely convex relaxations using l1 norm (e.g., Lasso) and greedy methods (e.g., OMP), have both been rigorously extended for group sparsity, the sparse MKL literature has so far mainly adopted the former with mild empirical success. In this paper, we close this gap by proposing a Group-OMP based framework for sparse MKL. Unlike l1-MKL, our approach decouples the sparsity regularizer (via a direct l0 constraint) from the smoothness regularizer (via RKHS norms), which leads to better empirical performance and a simpler optimization procedure that only requires a black-box single-kernel solver. The algorithmic development and empirical studies are complemented by theoretical analyses in terms of Rademacher generalization bounds and sparse recovery conditions analogous to those for OMP [27] and Group-OMP [16].