# Regularized Residual Quantization: a multi-layer sparse dictionary learning approach

## TL;DR

This paper introduces Regularized Residual Quantization (RRQ), a multi-layer sparse dictionary learning method that improves high-dimensional data quantization and super-resolution tasks by incorporating variance regularization inspired by rate-distortion theory.

## Contribution

The paper proposes a novel regularization technique for residual quantization, enabling scalable, sparse, multi-layer dictionaries suitable for high-dimensional data and natural images.

## Key findings

- Efficient quantization of high-dimensional variance-decaying data.
- Improved super-resolution results with quantized facial images.
- Effective extension of residual quantization to many layers without overfitting.

## Abstract

The Residual Quantization (RQ) framework is revisited where the quantization distortion is being successively reduced in multi-layers. Inspired by the reverse-water-filling paradigm in rate-distortion theory, an efficient regularization on the variances of the codewords is introduced which allows to extend the RQ for very large numbers of layers and also for high dimensional data, without getting over-trained. The proposed Regularized Residual Quantization (RRQ) results in multi-layer dictionaries which are additionally sparse, thanks to the soft-thresholding nature of the regularization when applied to variance-decaying data which can arise from de-correlating transformations applied to correlated data. Furthermore, we also propose a general-purpose pre-processing for natural images which makes them suitable for such quantization. The RRQ framework is first tested on synthetic variance-decaying data to show its efficiency in quantization of high-dimensional data. Next, we use the RRQ in super-resolution of a database of facial images where it is shown that low-resolution facial images from the test set quantized with codebooks trained on high-resolution images from the training set show relevant high-frequency content when reconstructed with those codebooks.