# Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications

arXiv: https://arxiv.org/abs/2411.04717 · HTML: https://mlsanity.com/paper/2411.04717/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 37 references that are in the map.
- First posted on arXiv 7 Nov 2024.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [Non-Parametric Estimation of Manifolds from Noisy Data](https://mlsanity.com/paper/2105.04754/) (2021)
- [Manifold Approximation by Moving Least-Squares Projection (MMLS)](https://mlsanity.com/paper/1606.07104/) (2016)
- [Efficient Manifold and Subspace Approximations with Spherelets](https://mlsanity.com/paper/1706.08263/) (2017)
- [Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization](https://mlsanity.com/paper/1410.7220/) (2014)
- [Structure-adaptive manifold estimation](https://mlsanity.com/paper/1906.05014/) (2019)
- [Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications](https://mlsanity.com/paper/2209.12638/) (2022)
- [Quadratic Matrix Factorization with Applications to Manifold Learning](https://mlsanity.com/paper/2301.12965/) (2023)

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

- [Robust Subspace-Constrained Quadratic Models for Low-Dimensional Structure Learning](https://mlsanity.com/paper/2605.20300/) (2026)

## Abstract (arXiv)

Matrix Factorization has emerged as a widely adopted framework for modeling data exhibiting low-rank structures. To address challenges in manifold learning, this paper presents a subspace-constrained quadratic matrix factorization model. The model is designed to jointly learn key low-dimensional structures, including the tangent space, the normal subspace, and the quadratic form that links the tangent space to a low-dimensional representation. We solve the proposed factorization model using an alternating minimization method, involving an in-depth investigation of nonlinear regression and projection subproblems. Theoretical properties of the quadratic projection problem and convergence characteristics of the alternating strategy are also investigated. To validate our approach, we conduct numerical experiments on synthetic and real-world datasets. Results demonstrate that our model outperforms existing methods, highlighting its robustness and efficacy in capturing core low-dimensional structures.