# Identifiability in Two-Layer Sparse Matrix Factorization

arXiv: https://arxiv.org/abs/2110.01235 · HTML: https://mlsanity.com/paper/2110.01235/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 49 references that are in the map.
- First posted on arXiv 4 Oct 2021.

## Builds on (most cited first, 4 of 4)

- [Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations](https://mlsanity.com/paper/1903.05895/) (2019)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [Flexible Multi-layer Sparse Approximations of Matrices and Applications](https://mlsanity.com/paper/1506.07300/) (2015)
- [On the stable recovery of deep structured linear networks under sparsity constraints](https://mlsanity.com/paper/1706.00342/) (2017)

## Built on by (most cited first, 1 of 1)

- [Pruning at Initialization -- A Sketching Perspective](https://mlsanity.com/paper/2305.17559/) (2023)

## Abstract (arXiv)

Sparse matrix factorization is the problem of approximating a matrix Z by a product of J sparse factors X(J) X(J-1) ldots X(1). This paper focuses on identifiability issues that appear in this problem, in view of better understanding under which sparsity constraints the problem is well-posed. We give conditions under which the problem of factorizing a matrix into two sparse factors admits a unique solution, up to unavoidable permutation and scaling equivalences. Our general framework considers an arbitrary family of prescribed sparsity patterns, allowing us to capture more structured notions of sparsity than simply the count of nonzero entries. These conditions are shown to be related to essential uniqueness of exact matrix decomposition into a sum of rank-one matrices, with structured sparsity constraints. In particular, in the case of fixed-support sparse matrix factorization, we give a general sufficient condition for identifiability based on rank-one matrix completability, and we derive from it a completion algorithm that can verify if this sufficient condition is satisfied, and recover the entries in the two sparse factors if this is the case. A companion paper further exploits these conditions to derive identifiability properties and theoretically sound factorization methods for multi-layer sparse matrix factorization with support constraints associated to some well-known fast transforms such as the Hadamard or the Discrete Fourier Transforms.