Laplacian Eigenmaps from Sparse, Noisy Similarity Measurements

TL;DR

This paper investigates how Laplacian eigenmaps, a dimensionality reduction technique, can be effectively approximated from noisy and incomplete similarity data, providing theoretical guarantees and experimental validation.

Contribution

It offers the first theoretical analysis of Laplacian eigenmaps under noise and occlusion, demonstrating conditions for accurate approximation and the benefits of regularization.

Findings

High-probability recovery of embeddings under modest noise and occlusion

Regularization improves embedding quality in noisy conditions

Experimental validation on real-world and synthetic data sets

Abstract

Manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering, but can be computationally expensive procedures when applied to large data sets or when similarities are expensive to compute. To date, little work has been done to investigate the tradeoff between computational resources and the quality of learned representations. We present both theoretical and experimental explorations of this question. In particular, we consider Laplacian eigenmaps embeddings based on a kernel matrix, and explore how the embeddings behave when this kernel matrix is corrupted by occlusion and noise. Our main theoretical result shows that under modest noise and occlusion assumptions, we can (with high probability) recover a good approximation to the Laplacian eigenmaps embedding based on the uncorrupted kernel matrix. Our results also show how regularization can aid this approximation. Experimentally, we explore the effects of noise and occlusion on Laplacian eigenmaps embeddings of two real-world data sets, one from speech processing and one from neuroscience, as well as a synthetic data set.