Reduced Basis Decomposition: a Certified and Fast Lossy Data Compression Algorithm

TL;DR

This paper introduces Reduced Basis Decomposition (RBD), a fast and certified lossy data compression algorithm inspired by RBM, offering efficient dimension reduction with out-of-sample mapping and error certification.

Contribution

The paper presents a novel RBD method that is faster than SVD, provides out-of-sample mapping, and includes an error indicator for certified compression.

Findings

RBD is significantly faster than SVD for dimension reduction.

RBD achieves comparable accuracy to traditional methods.

The method provides an error indicator to certify compression quality.

Abstract

Dimension reduction is often needed in the area of data mining. The goal of these methods is to map the given high-dimensional data into a low-dimensional space preserving certain properties of the initial data. There are two kinds of techniques for this purpose. The first, projective methods, builds an explicit linear projection from the high-dimensional space to the low-dimensional one. On the other hand, the nonlinear methods utilizes nonlinear and implicit mapping between the two spaces. In both cases, the methods considered in literature have usually relied on computationally very intensive matrix factorizations, frequently the Singular Value Decomposition (SVD). The computational burden of SVD quickly renders these dimension reduction methods infeasible thanks to the ever-increasing sizes of the practical datasets. In this paper, we present a new decomposition strategy, Reduced Basis Decomposition (RBD), which is inspired by the Reduced Basis Method (RBM). Given $X$ the high-dimensional data, the method approximates it by $Y \, T (\approx X)$ with $Y$ being the low-dimensional surrogate and $T$ the transformation matrix. $Y$ is obtained through a greedy algorithm thus extremely efficient. In fact, it is significantly faster than SVD with comparable accuracy. $T$ can be computed on the fly. Moreover, unlike many compression algorithms, it easily finds the mapping for an arbitrary ``out-of-sample'' vector and it comes with an ``error indicator'' certifying the accuracy of the compression. Numerical results are shown validating these claims.