相关论文: Nonconvex Factorization and Manifold Formulations …

Many high-dimensional optimisation problems exhibit rich geometric structures in their set of minimisers, often forming smooth manifolds due to over-parametrisation or symmetries. When this structure is known, at least locally, it can be…

This article presents a novel approach to solving the sparsity-constrained Orthogonal Nonnegative Matrix Factorization (SCONMF) problem, which requires decomposing a non-negative data matrix into the product of two lower-rank non-negative…

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult;…

Symmetry is the essential element of lifted inference that has recently demon- strated the possibility to perform very efficient inference in highly-connected, but symmetric probabilistic models models. This raises the question, whether…

Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local…

This paper considers the minimization of a general objective function $f(X)$ over the set of rectangular $n\times m$ matrices that have rank at most $r$. To reduce the computational burden, we factorize the variable $X$ into a product of…

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing:…

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…

The remarkable generalization ability of neural networks is usually attributed to the implicit bias of SGD, which often yields models with lower complexity using simpler (e.g. linear) and low-rank features. Recent works have provided…

Robust principal component analysis is an important representative method in data analysis. It is usually viewed as an optimization problem involving the rank and $\ell_0$-norm of matrices. In this paper, we study the rank and $\ell_0$…

Due to the iterative nature of most nonnegative matrix factorization (\textsc{NMF}) algorithms, initialization is a key aspect as it significantly influences both the convergence and the final solution obtained. Many initialization schemes…

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued…

With the explosive deployment of non-terrestrial networks (NTNs), the computational complexity of network performance analysis is rapidly escalating. As one of the most suitable mathematical tools for analyzing large-scale network…

A new concept for the higher-order accurate approximation of partial differential equations on manifolds is proposed where a surface mesh composed by higher-order elements is automatically generated based on level-set data. Thereby, it…

We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one…

We propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the metric on our search space to the…

We consider problems of dimensionality reduction and learning data representations for continuous spaces with two or more independent degrees of freedom. Such problems occur, for example, when observing shapes with several components that…

We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the…

In modern deep learning, highly subsampled stochastic approximation (SA) methods are preferred to sample average approximation (SAA) methods because of large data sets as well as generalization properties. Additionally, due to perceived…

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best…