Time for dithering: fast and quantized random embeddings via the restricted isometry property

TL;DR

This paper demonstrates that RIP matrices with fast multiplication algorithms can be used to create efficient, quantized random embeddings with controllable distortions, enhanced by a novel bi-dithered quantization scheme.

Contribution

It introduces a method to generate fast, quantized embeddings from RIP matrices using dithering, with bounds on distortion and a new bi-dithered scheme for improved accuracy.

Findings

RIP matrices induce quantized embeddings with small distortions.

Distortion decreases as embedding dimension increases or pairwise distances decrease.

The bi-dithered scheme further reduces distortion independently of vector pairs.

Abstract

Recently, many works have focused on the characterization of non-linear dimensionality reduction methods obtained by quantizing linear embeddings, e.g., to reach fast processing time, efficient data compression procedures, novel geometry-preserving embeddings or to estimate the information/bits stored in this reduced data representation. In this work, we prove that many linear maps known to respect the restricted isometry property (RIP) can induce a quantized random embedding with controllable multiplicative and additive distortions with respect to the pairwise distances of the data points beings considered. In other words, linear matrices having fast matrix-vector multiplication algorithms (e.g., based on partial Fourier ensembles or on the adjacency matrix of unbalanced expanders) can be readily used in the definition of fast quantized embeddings with small distortions. This implication is made possible by applying right after the linear map an additive and random "dither" that stabilizes the impact of the uniform scalar quantization operator applied afterwards. For different categories of RIP matrices, i.e., for different linear embeddings of a metric space $(\mathcal K \subset \mathbb R^n, \ell_q)$ in $(\mathbb R^m, \ell_p)$ with $p,q \geq 1$, we derive upper bounds on the additive distortion induced by quantization, showing that it decays either when the embedding dimension $m$ increases or when the distance of a pair of embedded vectors in $\mathcal K$ decreases. Finally, we develop a novel "bi-dithered" quantization scheme, which allows for a reduced distortion that decreases when the embedding dimension grows and independently of the considered pair of vectors.