# A Quantum-inspired Classical Algorithm for Separable Non-negative Matrix Factorization

## TL;DR

This paper introduces a new classical algorithm for separable Non-negative Matrix Factorization that is inspired by quantum techniques, achieving exponential speedup for large-scale, low-rank datasets.

## Contribution

It presents a polynomial-time classical algorithm for separable NMF inspired by quantum dequantization methods, enabling efficient processing of large datasets.

## Key findings

- Runs in polynomial time in rank and logarithmic in input size
- Achieves exponential speedup in low-rank scenarios
- Applicable to large-scale text and image data

## Abstract

Non-negative Matrix Factorization (NMF) asks to decompose a (entry-wise) non-negative matrix into the product of two smaller-sized nonnegative matrices, which has been shown intractable in general. In order to overcome this issue, the separability assumption is introduced which assumes all data points are in a conical hull. This assumption makes NMF tractable and is widely used in text analysis and image processing, but still impractical for huge-scale datasets. In this paper, inspired by recent development on dequantizing techniques, we propose a new classical algorithm for separable NMF problem. Our new algorithm runs in polynomial time in the rank and logarithmic in the size of input matrices, which achieves an exponential speedup in the low-rank setting.