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On Unbiased Low-Rank Approximation with Minimum Distortion

Leighton Pate Barnes, Stephen Cameron, Benjamin Howard

TL;DR

The paper addresses unbiased low-rank approximation of a complex matrix $P$ by a random matrix $Q$ with $\mathbb{E}[Q]=P$ and $\operatorname{rank}(Q)\le r$, aiming to minimize the expected Frobenius error $\mathbb{E}\|P-Q\|_F^2$. It introduces a simple, basis-independent sampling procedure that operates on the singular components of $P$, reducing the problem to sampling diagonal entries after an SVD, and proves that this procedure achieves the minimum possible distortion. The key contribution is showing that this sampling strategy matches a known dual lower bound for all $r$, thus providing an optimal, unbiased, low-rank approximation method whose performance matches theoretical limits. The approach emphasizes a direct spectral (diagonal) construction, with the dominant cost arising from the initial singular value decomposition, and it extends prior co-rank-one results to arbitrary $r$ while highlighting the richness of optimal unbiased approximations beyond diagonal-only solutions.

Abstract

We describe an algorithm for sampling a low-rank random matrix $Q$ that best approximates a fixed target matrix $P\in\mathbb{C}^{n\times m}$ in the following sense: $Q$ is unbiased, i.e., $\mathbb{E}[Q] = P$; $\mathsf{rank}(Q)\leq r$; and $Q$ minimizes the expected Frobenius norm error $\mathbb{E}\|P-Q\|_F^2$. Our algorithm mirrors the solution to the efficient unbiased sparsification problem for vectors, except applied to the singular components of the matrix $P$. Optimality is proven by showing that our algorithm matches the error from an existing lower bound.

On Unbiased Low-Rank Approximation with Minimum Distortion

TL;DR

The paper addresses unbiased low-rank approximation of a complex matrix by a random matrix with and , aiming to minimize the expected Frobenius error . It introduces a simple, basis-independent sampling procedure that operates on the singular components of , reducing the problem to sampling diagonal entries after an SVD, and proves that this procedure achieves the minimum possible distortion. The key contribution is showing that this sampling strategy matches a known dual lower bound for all , thus providing an optimal, unbiased, low-rank approximation method whose performance matches theoretical limits. The approach emphasizes a direct spectral (diagonal) construction, with the dominant cost arising from the initial singular value decomposition, and it extends prior co-rank-one results to arbitrary while highlighting the richness of optimal unbiased approximations beyond diagonal-only solutions.

Abstract

We describe an algorithm for sampling a low-rank random matrix that best approximates a fixed target matrix in the following sense: is unbiased, i.e., ; ; and minimizes the expected Frobenius norm error . Our algorithm mirrors the solution to the efficient unbiased sparsification problem for vectors, except applied to the singular components of the matrix . Optimality is proven by showing that our algorithm matches the error from an existing lower bound.
Paper Structure (3 sections, 15 equations, 2 figures, 1 algorithm)

This paper contains 3 sections, 15 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Sampling Procedure
  3. Matching Lower Bound

Figures (2)

  • Figure 1: Upper-left is an image of the third author's pet cat "Nutmeg", of rank $288$, and the lower-right is the average of 16 unbiased low-rank approximations, all instances of Algorithm \ref{['fig1']} with rank $r=30$. The images in the middle are the unbiased approximations.
  • Figure 2: Upper-left is an image of rank $\approx 700$, and the lower-right is the average of 16 unbiased low-rank approximations, all instances of Algorithm \ref{['fig1']} with rank $r=3$. The images in the middle are the unbiased approximations.