# Structural Conditions for Projection-Cost Preservation via Randomized Matrix Multiplication

## TL;DR

This paper establishes structural conditions under which randomized matrix multiplication guarantees projection-cost preservation, enabling efficient low-rank approximations with theoretical assurances.

## Contribution

It provides a general structural framework with four sufficient conditions for projection-cost preservation using randomized linear algebra techniques.

## Key findings

- Four structural conditions for projection-cost preservation.
- Conditions can be satisfied with existing randomized linear algebra tools.
- Framework guides the design of efficient low-rank approximation algorithms.

## Abstract

Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank-$k$ projection can be preserved using a smaller sketch of the original data matrix. We present a general structural result outlining four sufficient conditions to achieve projection-cost preservation. These conditions can be satisfied using tools from the Randomized Linear Algebra literature.