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Found 60 papers, 19 papers with code
Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization
Simplex-structured matrix factorization (SSMF) is a generalization of nonnegative matrix factorization, a fundamental interpretable data analysis model, and has applications in hyperspectral unmixing and topic modeling.
Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software
However, this condition is NP-hard to check in general.
Block Majorization Minimization with Extrapolation and Application to $β$-NMF
We propose a Block Majorization Minimization method with Extrapolation (BMMe) for solving a class of multi-convex optimization problems.
Subtractive Mixture Models via Squaring: Representation and Learning
Mixture models are traditionally represented and learned by adding several distributions as components.
Deep Nonnegative Matrix Factorization with Beta Divergences
Deep Nonnegative Matrix Factorization (deep NMF) has recently emerged as a valuable technique for extracting multiple layers of features across different scales.
Algorithms for Boolean Matrix Factorization using Integer Programming
Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors.
Accelerated Algorithms for Nonlinear Matrix Decomposition with the ReLU function
In this paper, we study the following nonlinear matrix decomposition (NMD) problem: given a sparse nonnegative matrix $X$, find a low-rank matrix $\Theta$ such that $X \approx f(\Theta)$, where $f$ is an element-wise nonlinear function.
Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications
In this paper, we propose a new low-rank matrix factorization model dubbed bounded simplex-structured matrix factorization (BSSMF).
Least-squares methods for nonnegative matrix factorization over rational functions
When the data is made of samplings of continuous signals, the factors in NMF can be constrained to be samples of nonnegative rational functions, which allow fairly general models; this is referred to as NMF using rational functions (R-NMF).
Revisiting data augmentation for subspace clustering
Subspace clustering is the classical problem of clustering a collection of data samples that approximately lie around several low-dimensional subspaces.
A consistent and flexible framework for deep matrix factorizations
Deep matrix factorizations (deep MFs) are recent unsupervised data mining techniques inspired by constrained low-rank approximations.
Partial Identifiability for Nonnegative Matrix Factorization
In this paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of $C$ and $S$.
Smoothed Separable Nonnegative Matrix Factorization
More recently, Bhattacharyya and Kannan (ACM-SIAM Symposium on Discrete Algorithms, 2020) proposed an algorithm for learning a latent simplex (ALLS) that relies on the assumption that there is more than one nearby data point to each vertex.
Block Alternating Bregman Majorization Minimization with Extrapolation
In this paper, we consider a class of nonsmooth nonconvex optimization problems whose objective is the sum of a block relative smooth function and a proper and lower semicontinuous block separable function.
Beyond Linear Subspace Clustering: A Comparative Study of Nonlinear Manifold Clustering Algorithms
To overcome the restrictive linearity assumption, numerous nonlinear approaches were proposed to extend successful subspace clustering approaches to data on a union of nonlinear manifolds.
A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization
In this paper, we propose an algorithmic framework, dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints.
Successive Nonnegative Projection Algorithm for Linear Quadratic Mixtures
In this work, we tackle the problem of hyperspectral (HS) unmixing by departing from the usual linear model and focusing on a Linear-Quadratic (LQ) one.
Provably robust blind source separation of linear-quadratic near-separable mixtures
The BF is in turn shown to be robust to noise under easier-to-check and milder conditions than SNPALQ.
Matrix-wise $\ell_0$-constrained Sparse Nonnegative Least Squares
In this paper, as opposed to most previous works that enforce sparsity column- or row-wise, we first introduce a novel formulation for sparse MNNLS, with a matrix-wise sparsity constraint.
Multiplicative Updates for NMF with $β$-Divergences under Disjoint Equality Constraints
In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, and with penalty terms in the objective function.
An Inertial Block Majorization Minimization Framework for Nonsmooth Nonconvex Optimization
In this paper, we introduce TITAN, a novel inerTIal block majorizaTion minimizAtioN framework for non-smooth non-convex optimization problems.
Algorithms for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence
Nonnegative matrix factorization (NMF) is a standard linear dimensionality reduction technique for nonnegative data sets.
Deep matrix factorizations
Constrained low-rank matrix approximations have been known for decades as powerful linear dimensionality reduction techniques to be able to extract the information contained in large data sets in a relevant way.
Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms
In this paper, we provide novel algorithms with identifiability guarantees for simplex-structured matrix factorization (SSMF), a generalization of nonnegative matrix factorization.
Multi-Resolution Beta-Divergence NMF for Blind Spectral Unmixing
We show on numerical experiments that the MU are able to obtain high resolutions in both dimensions on two applications: (1) blind unmixing of audio spectrograms: to the best of our knowledge, this is the first time a coupled NMF model is used in this context, and (2) the fusion of hyperspectral and multispectral images: we show that the MU compete favorable with state-of-the-art algorithms in particular in the presence of non-Gaussian noise.
Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization.
Sparse Separable Nonnegative Matrix Factorization
We propose a new variant of nonnegative matrix factorization (NMF), combining separability and sparsity assumptions.
On a minimum enclosing ball of a collection of linear subspaces
By scaling the objective and penalizing the information lost by the rank-$k$ minimax center, we jointly recover an optimal dimension, $k^*$, and a central subspace, $U^* \in$ Gr$(k^*, n)$ at the center of the minimum enclosing ball, that best represents the data.
Accelerating Block Coordinate Descent for Nonnegative Tensor Factorization
This paper is concerned with improving the empirical convergence speed of block-coordinate descent algorithms for approximate nonnegative tensor factorization (NTF).
Explicit Group Sparse Projection with Applications to Deep Learning and NMF
Instead, in our approach we set the sparsity level for the whole set explicitly and simultaneously project a group of vectors with the sparsity level of each vector tuned automatically.
Near-Convex Archetypal Analysis
Archetypal analysis (AA), also referred to as convex NMF, is a well-known NMF variant imposing that the basis elements are themselves convex combinations of the data points.
Successive Projection Algorithm Robust to Outliers
In this paper, we propose a new SPA variant, dubbed Robust SPA (RSPA), that is robust to outliers while still being provably robust in low-noise settings, and that takes into account the reconstruction error for selecting the indices in $\mathcal{K}$.
Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization
We introduce and analyze BPALM and A-BPALM, two multi-block proximal alternating linearized minimization algorithms using Bregman distances for solving structured nonconvex problems.
Optimization and Control Numerical Analysis Numerical Analysis
Blind Audio Source Separation with Minimum-Volume Beta-Divergence NMF
Considering a mixed signal composed of various audio sources and recorded with a single microphone, we consider on this paper the blind audio source separation problem which consists in isolating and extracting each of the sources.
A Provably Correct and Robust Algorithm for Convolutive Nonnegative Matrix Factorization
We present an algorithm that takes advantage of the NMF model underlying CNMF and exploits existing algorithms for separable NMF to provably find a solution under certain conditions.
Generalized Separable Nonnegative Matrix Factorization
Given a data matrix $M$ and a factorization rank $r$, NMF looks for a nonnegative matrix $W$ with $r$ columns and a nonnegative matrix $H$ with $r$ rows such that $M \approx WH$.
Inertial Block Proximal Methods for Non-Convex Non-Smooth Optimization
We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems.
Distributionally Robust and Multi-Objective Nonnegative Matrix Factorization
We propose to use Lagrange duality to judiciously optimize for a set of weights to be used within the framework of the weighted-sum approach, that is, we minimize a single objective function which is a weighted sum of the all objective functions.
Identifiability of Complete Dictionary Learning
In this work, we provide new results in the deterministic scenario when the data has a low-rank structure, that is, when $D$ is (under)complete.
Improved SVD-based Initialization for Nonnegative Matrix Factorization using Low-Rank Correction
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.
Accelerating Nonnegative Matrix Factorization Algorithms using Extrapolation
In this paper, we propose a general framework to accelerate significantly the algorithms for nonnegative matrix factorization (NMF).
Scalable and Robust Sparse Subspace Clustering Using Randomized Clustering and Multilayer Graphs
To improve the scalability of SSC, we propose to select a few sets of anchor points using a randomized hierarchical clustering method, and, for each set of anchor points, solve the LASSO problems for each data point allowing only anchor points to have a non-zero weight (this reduces drastically the number of variables).
Low-Rank Matrix Approximation in the Infinity Norm
The low-rank matrix approximation problem with respect to the entry-wise $\ell_{\infty}$-norm is the following: given a matrix $M$ and a factorization rank $r$, find a matrix $X$ whose rank is at most $r$ and that minimizes $\max_{i, j} |M_{ij} - X_{ij}|$.
Dictionary-based Tensor Canonical Polyadic Decomposition
To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition which enforces one factor to belong exactly to a known dictionary.
Introduction to Nonnegative Matrix Factorization
In this paper, we introduce and provide a short overview of nonnegative matrix factorization (NMF).
UTA-poly and UTA-splines: additive value functions with polynomial marginals
In such models, a numerical value is associated to each alternative involved in the decision problem.
On the Complexity of Robust PCA and $\ell_1$-norm Low-Rank Matrix Approximation
The low-rank matrix approximation problem with respect to the component-wise $\ell_1$-norm ($\ell_1$-LRA), which is closely related to robust principal component analysis (PCA), has become a very popular tool in data mining and machine learning.
Coordinate Descent Methods for Symmetric Nonnegative Matrix Factorization
Given a symmetric nonnegative matrix $A$, symmetric nonnegative matrix factorization (symNMF) is the problem of finding a nonnegative matrix $H$, usually with much fewer columns than $A$, such that $A \approx HH^T$.
Sequential Dimensionality Reduction for Extracting Localized Features
In particular, nonnegative matrix factorization (NMF) has become very popular as it is able to extract sparse, localized and easily interpretable features by imposing an additive combination of nonnegative basis elements.
Heuristics for Exact Nonnegative Matrix Factorization
The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$.
Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization
Second, we propose an exact algorithm (that is, an algorithm that finds an optimal solution), also based on the SVD, for a certain class of matrices (including nonnegative irreducible matrices) from which we derive an initialization for matrices not belonging to that class.
Enhancing Pure-Pixel Identification Performance via Preconditioning
We analyze robustness of pre-whitening which allows us to characterize situations in which it performs competitively with the SDP-based preconditioning.
The Why and How of Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data vectors.
Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation
In this paper, we propose a new fast and robust recursive algorithm for near-separable nonnegative matrix factorization, a particular nonnegative blind source separation problem.
Semidefinite Programming Based Preconditioning for More Robust Near-Separable Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing.
Hierarchical Clustering of Hyperspectral Images using Rank-Two Nonnegative Matrix Factorization
The effectiveness of this approach is illustrated on several synthetic and real-world hyperspectral images, and shown to outperform standard clustering techniques such as k-means, spherical k-means and standard NMF.
Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization
Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns.
Robustness Analysis of Hottopixx, a Linear Programming Model for Factoring Nonnegative Matrices
Although nonnegative matrix factorization (NMF) is NP-hard in general, it has been shown very recently that it is tractable under the assumption that the input nonnegative data matrix is close to being separable (separability requires that all columns of the input matrix belongs to the cone spanned by a small subset of these columns).
Fast and Robust Recursive Algorithms for Separable Nonnegative Matrix Factorization
In this paper, we study the nonnegative matrix factorization problem under the separability assumption (that is, there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns), which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure-pixel assumption.
On the Geometric Interpretation of the Nonnegative Rank
We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity.
Optimization and Control Combinatorics
