Recovering the Optimal Solution by Dual Random Projection

TL;DR

This paper introduces Dual Random Projection, an algorithm that accurately recovers the optimal high-dimensional solution from low-dimensional random projections, especially effective for low-rank data matrices.

Contribution

The paper proposes a novel dual random projection method for recovering the original optimal solution from low-dimensional projections, with theoretical guarantees.

Findings

High probability of accurate recovery for low-rank data

Effective solution recovery from low-dimensional projections

Theoretical analysis supports the method's reliability

Abstract

Random projection has been widely used in data classification. It maps high-dimensional data into a low-dimensional subspace in order to reduce the computational cost in solving the related optimization problem. While previous studies are focused on analyzing the classification performance of using random projection, in this work, we consider the recovery problem, i.e., how to accurately recover the optimal solution to the original optimization problem in the high-dimensional space based on the solution learned from the subspace spanned by random projections. We present a simple algorithm, termed Dual Random Projection, that uses the dual solution of the low-dimensional optimization problem to recover the optimal solution to the original problem. Our theoretical analysis shows that with a high probability, the proposed algorithm is able to accurately recover the optimal solution to the original problem, provided that the data matrix is of low rank or can be well approximated by a low rank matrix.