Hadamard Decomposition: A Denser Alternative to Standard Matrix Factorization
New manifold-based algorithms enable efficient computation of element-wise matrix products that represent higher-rank data with fewer parameters than truncated SVD.
Source: arxiv/cs.LG · Nicolas Gillis, Subhayan Saha, Stefano Sicilia, Arnaud Vandaele · open original ↗
Hadamard decomposition factors matrices into element-wise products of low-rank factors, enabling denser representations than standard truncated SVD.
- — Hadamard decomposition (HD) approximates matrix X as product of two low-rank matrices via element-wise multiplication.
- — HD achieves rank r₁r₂ with fewer parameters than standard low-rank methods, improving expressiveness.
- — Three new algorithms proposed: direct HD optimization, block projected gradient, and projection-free manifold descent.
- — Manifold reformulation X ≈ WH⊤ enables efficient handling of large sparse datasets.
- — New initialization strategies improve convergence accuracy compared to existing approaches.
- — Methods outperform truncated SVD and prior state-of-the-art on synthetic and real-world benchmarks.
- — Projection-free variant eliminates computational overhead of feasible-set constraints.
Astrobobo tool mapping
- Knowledge Capture Record the three algorithm variants (direct HD, block projected gradient, projection-free manifold descent) with pseudocode and computational complexity notes for future reference.
- Focus Brief Summarize when to use Hadamard decomposition (sparse, high-rank data) versus truncated SVD (dense, low-rank data) as a decision tree for matrix factorization selection.
- Reading Queue Queue related papers on manifold optimization (Manopt toolbox docs) and recent work on structured matrix factorization to deepen understanding of the theoretical foundation.
Frequently asked
- Hadamard decomposition approximates a matrix X as the element-wise (Hadamard) product of two low-rank matrices X₁ and X₂. Unlike SVD, which decomposes X into orthogonal factors, Hadamard decomposition can represent higher-rank matrices with fewer total parameters. For example, two rank-10 matrices multiplied element-wise produce rank-100 data, whereas SVD would need rank-100 factors directly.
cite ▸
APA
Nicolas Gillis, Subhayan Saha, Stefano Sicilia, Arnaud Vandaele. (2026, May 29). Hadamard Decomposition: A Denser Alternative to Standard Matrix Factorization. Astrobobo Content Engine (rewrite of arxiv/cs.LG). https://astrobobo-content-engine.vercel.app/article/hadamard-decomposition-a-denser-alternative-to-standard-matrix-factorization-dc1569
MLA
Nicolas Gillis, Subhayan Saha, Stefano Sicilia, Arnaud Vandaele. "Hadamard Decomposition: A Denser Alternative to Standard Matrix Factorization." Astrobobo Content Engine, 29 May 2026, https://astrobobo-content-engine.vercel.app/article/hadamard-decomposition-a-denser-alternative-to-standard-matrix-factorization-dc1569. Based on "arxiv/cs.LG", https://arxiv.org/abs/2605.28980.
BibTeX
@misc{astrobobo_hadamard-decomposition-a-denser-alternative-to-standard-matrix-factorization-dc1569_2026,
author = {Nicolas Gillis, Subhayan Saha, Stefano Sicilia, Arnaud Vandaele},
title = {Hadamard Decomposition: A Denser Alternative to Standard Matrix Factorization},
year = {2026},
url = {https://astrobobo-content-engine.vercel.app/article/hadamard-decomposition-a-denser-alternative-to-standard-matrix-factorization-dc1569},
note = {Astrobobo rewrite of arxiv/cs.LG, https://arxiv.org/abs/2605.28980},
}