Toward a Unified Theory of Sparse Dimensionality Reduction in Euclidean Space
📜 Abstract
We aim to take steps toward a unified theory of sparse dimensionality reduction in Euclidean spaces, especially concerning whether nearly optimal dimension reduction can be achieved when aiming for a 1+ε distortion embedding. Our focus is on the relationship between the number of non-zero entries in the projection matrix, known as the sparsity of the embedding. We show that several classes of sparse matrices are indeed sufficient for nearly optimal dimension reduction, provided that certain structural properties are satisfied.
✨ Summary
The paper “Toward a Unified Theory of Sparse Dimensionality Reduction in Euclidean Space” by David P. Woodruff, published in April 2013, seeks to advance understanding in dimensionality reduction, particularly the use of sparse matrices. The work explores whether nearly optimal dimension reduction with a 1+ε distortion embedding can be achieved through specific sparse matrices in Euclidean spaces. This involves examining the relationship between sparsity and dimensionality reduction, focusing on structural properties. This paper advances the field by proposing general conditions under which sparse matrices achieve optimal embeddings.
The work is especially relevant in contexts dealing with high-dimensional data, as seen in applications of sketching, random projections, and compressed sensing. After conducting a search, there are no explicit references showing significant direct influence of this paper on subsequent specific research or industry applications, suggesting its impact may be more implicit, contributing foundational insights in the mathematical and theoretical understanding of sparse dimensionality reduction.
No direct citations of this paper were found impacting further research or industry applications. However, the concepts discussed are foundational and have likely contributed to further work in related areas such as sketching and high-dimensional data analysis.