Unifying Conventional Matrix Factorization and Hadamard Decomposition

Matrix factorization (MF) is inherently constrained by a low-rank bottleneck, whereas the recently proposed Hadamard decomposition (HD) can represent high-rank matrices at the cost of a nearly rigid spectral profile. We first characterize when HD achieves rank amplification: the Hadamard product of two generic rank-$r$ matrices has rank $\min (M,N,r^2)$ almost surely. However, under generic orthonormal factors, its Frobenius energy scales as $\mathcal{O}(r^2/N^2)$, while its average nonzero singular value scales as $\mathcal{O}(1/N)$, revealing an energy-dilution effect that we also verify empirically. Motivated by this analysis, we propose Rada-MF, a rank-adaptive framework that, under a fixed parameter budget, selects among MF, HD, and their additive hybrid according to the entropy-based effective rank of the target spectrum. Rada-MF provides a unified higher-level formulation that subsumes both conventional low-rank MF and HD. On controlled spectra with effective ranks ranging from 8 to 96, Rada-MF-auto, which automatically selects the decomposition regime, closely tracks the best fixed model, achieving a mean oracle gap of 0.012. Truncated SVD performs best at low effective ranks, the SVD-core+HD-residual hybrid is strongest in the intermediate regime, and HD dominates at high effective ranks. Experiments on binary matrices provide further support for the proposed framework.