# Quadratic Matrix Factorization with Applications to Manifold Learning

arXiv: https://arxiv.org/abs/2301.12965 · HTML: https://mlsanity.com/paper/2301.12965/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 50 references that are in the map.
- First posted on arXiv 30 Jan 2023.

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

- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [Variational Autoencoders with Riemannian Brownian Motion Priors](https://mlsanity.com/paper/2002.05227/) (2020)
- [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)
- [Asymptotic theory for density ridges](https://mlsanity.com/paper/1406.5663/) (2014)
- [Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization](https://mlsanity.com/paper/1410.7220/) (2014)
- [Inferring manifolds using Gaussian processes](https://mlsanity.com/paper/2110.07478/) (2021)
- [Structure-adaptive manifold estimation](https://mlsanity.com/paper/1906.05014/) (2019)

## Built on by (most cited first, 5 of 5)

- [Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications](https://mlsanity.com/paper/2411.04717/) (2024)
- [Ridge Estimation with Nonlinear Transformations](https://mlsanity.com/paper/2306.05722/) (2023)
- [Regularized Projection Matrix Approximation with Applications to Community Detection](https://mlsanity.com/paper/2405.16598/) (2024)
- [Robust Subspace-Constrained Quadratic Models for Low-Dimensional Structure Learning](https://mlsanity.com/paper/2605.20300/) (2026)
- [A Generalized Deep Non-negative Matrix Factorization Approach for SAR Automatic Target Recognition](https://mlsanity.com/paper/2607.09779/) (2026)

## Abstract (arXiv)

Matrix factorization is a popular framework for modeling low-rank data matrices. Motivated by manifold learning problems, this paper proposes a quadratic matrix factorization (QMF) framework to learn the curved manifold on which the dataset lies. Unlike local linear methods such as the local principal component analysis, QMF can better exploit the curved structure of the underlying manifold. Algorithmically, we propose an alternating minimization algorithm to optimize QMF and establish its theoretical convergence properties. Moreover, to avoid possible over-fitting, we then propose a regularized QMF algorithm and discuss how to tune its regularization parameter. Finally, we elaborate how to apply the regularized QMF to manifold learning problems. Experiments on a synthetic manifold learning dataset and two real datasets, including the MNIST handwritten dataset and a cryogenic electron microscopy dataset, demonstrate the superiority of the proposed method over its competitors.