Estimation of embedding vectors in high dimensions

TL;DR

This paper investigates the theoretical limits of learning embeddings in high-dimensional settings using a probabilistic model and AMP algorithms, providing precise accuracy predictions and insights into key parameters affecting embedding quality.

Contribution

It introduces a probabilistic model for embedding estimation and applies AMP methods to predict estimation accuracy in high dimensions, validated by simulations.

Findings

AMP can accurately predict embedding estimation accuracy

Key parameters like sample size and correlation strength influence learning success

Theoretical results align with experiments on synthetic and real data

Abstract

Embeddings are a basic initial feature extraction step in many machine learning models, particularly in natural language processing. An embedding attempts to map data tokens to a low-dimensional space where similar tokens are mapped to vectors that are close to one another by some metric in the embedding space. A basic question is how well can such embedding be learned? To study this problem, we consider a simple probability model for discrete data where there is some "true" but unknown embedding where the correlation of random variables is related to the similarity of the embeddings. Under this model, it is shown that the embeddings can be learned by a variant of low-rank approximate message passing (AMP) method. The AMP approach enables precise predictions of the accuracy of the estimation in certain high-dimensional limits. In particular, the methodology provides insight on the relations of key parameters such as the number of samples per value, the frequency of the terms, and the strength of the embedding correlation on the probability distribution. Our theoretical findings are validated by simulations on both synthetic data and real text data.