# Structured Variational D-Decomposition for Accurate and Stable Low-Rank Approximation

arXiv: https://arxiv.org/abs/2506.08535 · HTML: https://mlsanity.com/paper/2506.08535/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 27 references that are in the map.
- First posted on arXiv 10 Jun 2025.

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

- [A Validation Approach to Over-parameterized Matrix and Image Recovery](https://mlsanity.com/paper/2209.10675/) (2022)
- [Accelerating nuclear-norm regularized low-rank matrix optimization through Burer-Monteiro decomposition](https://mlsanity.com/paper/2204.14067/) (2022)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models](https://mlsanity.com/paper/2403.18517/) (2024)

## Abstract (arXiv)

We introduce the D-decomposition, a non-orthogonal matrix factorization of the form A ≈ P D Q, where P ∈ ℝn × k, D ∈ ℝk × k, and Q ∈ ℝk × n. The decomposition is defined variationally by minimizing a regularized Frobenius loss, allowing control over rank, sparsity, and conditioning. Unlike algebraic factorizations such as LU or SVD, it is computed by alternating minimization. We establish existence and perturbation stability of the solution and show that each update has complexity O(n2k). Benchmarks against truncated SVD, CUR, and nonnegative matrix factorization show improved reconstruction accuracy on MovieLens, MNIST, Olivetti Faces, and gene expression matrices, particularly under sparsity and noise.