# Multi-Scale Vector Quantization with Reconstruction Trees

## TL;DR

This paper introduces a multi-scale vector quantization method called reconstruction trees, which efficiently reconstructs data by leveraging a family of partitions, with theoretical analysis under various data distribution assumptions.

## Contribution

It presents a novel multi-scale vector quantization algorithm inspired by decision trees, with theoretical analysis of its expected distortion for data from unknown distributions.

## Key findings

- Achieves low expected distortion under regularity assumptions.
- Provides asymptotic and finite sample bounds.
- Applicable to data on Riemannian manifolds.

## Abstract

We propose and study a multi-scale approach to vector quantization. We develop an algorithm, dubbed reconstruction trees, inspired by decision trees. Here the objective is parsimonious reconstruction of unsupervised data, rather than classification. Contrasted to more standard vector quantization methods, such as K-means, the proposed approach leverages a family of given partitions, to quickly explore the data in a coarse to fine-- multi-scale-- fashion. Our main technical contribution is an analysis of the expected distortion achieved by the proposed algorithm, when the data are assumed to be sampled from a fixed unknown distribution. In this context, we derive both asymptotic and finite sample results under suitable regularity assumptions on the distribution. As a special case, we consider the setting where the data generating distribution is supported on a compact Riemannian sub-manifold. Tools from differential geometry and concentration of measure are useful in our analysis.