Linear-time Learning on Distributions with Approximate Kernel Embeddings

TL;DR

This paper introduces a scalable method for kernel-based learning on distributions using approximate random features for non-Euclidean metrics, enabling large dataset processing without large Gram matrices.

Contribution

It develops the first random features for distribution kernels with non-Euclidean metrics, expanding scalable kernel methods to a broader class of similarity measures.

Findings

Effective approximation of kernels with non-Euclidean metrics

Scalable learning on large datasets without large Gram matrices

Empirical validation on real-world and synthetic data

Abstract

Many interesting machine learning problems are best posed by considering instances that are distributions, or sample sets drawn from distributions. Previous work devoted to machine learning tasks with distributional inputs has done so through pairwise kernel evaluations between pdfs (or sample sets). While such an approach is fine for smaller datasets, the computation of an $N \times N$ Gram matrix is prohibitive in large datasets. Recent scalable estimators that work over pdfs have done so only with kernels that use Euclidean metrics, like the $L_2$ distance. However, there are a myriad of other useful metrics available, such as total variation, Hellinger distance, and the Jensen-Shannon divergence. This work develops the first random features for pdfs whose dot product approximates kernels using these non-Euclidean metrics, allowing estimators using such kernels to scale to large datasets by working in a primal space, without computing large Gram matrices. We provide an analysis of the approximation error in using our proposed random features and show empirically the quality of our approximation both in estimating a Gram matrix and in solving learning tasks in real-world and synthetic data.