Dimensionality-Dependent Generalization Bounds for $k$-Dimensional Coding Schemes

TL;DR

This paper derives a new dimensionality-dependent generalization bound for $k$-dimensional coding schemes, improving upon previous bounds by providing tighter estimates for finite-dimensional data features.

Contribution

The paper introduces a novel dimensionality-dependent generalization bound for $k$-dimensional coding schemes, which is tighter than existing bounds for finite-dimensional data.

Findings

The new bound is of order $ig(rac{mk ext{ln}(mkn)}{n}ig)^{ ext{lambda}_n}$.

The bound converges faster and avoids worst-case upper bounds on $k$.

It is applicable to various coding schemes, complementing existing bounds.

Abstract

The $k$-dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative $k$-dimensional vectors, and include non-negative matrix factorization, dictionary learning, sparse coding, $k$-means clustering and vector quantization as special cases. Previous generalization bounds for the reconstruction error of the $k$-dimensional coding schemes are mainly dimensionality independent. A major advantage of these bounds is that they can be used to analyze the generalization error when data is mapped into an infinite- or high-dimensional feature space. However, many applications use finite-dimensional data features. Can we obtain dimensionality-dependent generalization bounds for $k$-dimensional coding schemes that are tighter than dimensionality-independent bounds when data is in a finite-dimensional feature space? The answer is positive. In this paper, we address this problem and derive a dimensionality-dependent generalization bound for $k$-dimensional coding schemes by bounding the covering number of the loss function class induced by the reconstruction error. The bound is of order $\mathcal{O}\left(\left(mk\ln(mkn)/n\right)^{\lambda_n}\right)$, where $m$ is the dimension of features, $k$ is the number of the columns in the linear implementation of coding schemes, $n$ is the size of sample, $\lambda_n>0.5$ when $n$ is finite and $\lambda_n=0.5$ when $n$ is infinite. We show that our bound can be tighter than previous results, because it avoids inducing the worst-case upper bound on $k$ of the loss function and converges faster. The proposed generalization bound is also applied to some specific coding schemes to demonstrate that the dimensionality-dependent bound is an indispensable complement to these dimensionality-independent generalization bounds.