arXiv subjects

Nicolas Gillis

Publications and source records attributed to Nicolas Gillis.

At least 91 records · Page 5Linked to original sources

arXiv2019

Near-Convex Archetypal Analysis

Nonnegative matrix factorization (NMF) is a widely used linear dimensionality reduction technique for nonnegative data. NMF requires that each data point is approximated by a convex combination of basis elements. 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. AA has the advantage to be more interpretable than NMF because the basis elements are directly constructed from the data points. However, it usually suffers from a high data fitting error because the basis elements are constrained to be contained in the convex cone of the data points. In this letter, we introduce near-convex archetypal analysis (NCAA) which combines the advantages of both AA and NMF. As for AA, the basis vectors are required to be linear combinations of the data points and hence are easily interpretable. As for NMF, the additional flexibility in choosing the basis elements allows NCAA to have a low data fitting error. We show that NCAA compares favorably with a state-of-the-art minimum-volume NMF method on synthetic datasets and on a real-world hyperspectral image.

eess.SP↗

arXiv2019

Successive Projection Algorithm Robust to Outliers

The successive projection algorithm (SPA) is a fast algorithm to tackle separable nonnegative matrix factorization (NMF). Given a nonnegative data matrix $X$, SPA identifies an index set $\mathcal{K}$ such that there exists a nonnegative matrix $H$ with $X \approx X(:,\mathcal{K})H$. SPA has been successfully used as a pure-pixel search algorithm in hyperspectral unmixing and for anchor word selection in document classification. Moreover, SPA is provably robust in low-noise settings. The main drawbacks of SPA are that it is not robust to outliers and does not take the data fitting term into account when selecting the indices in $\mathcal{K}$. 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 $\mathcal{K}$. We illustrate the effectiveness of RSPA on synthetic data sets and hyperspectral images.

eess.SP↗

arXiv2019

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to tackle this problem. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library.

math.OC↗

arXiv2019

Algorithms and Comparisons of Non-negative Matrix Factorization with Volume Regularization for Hyperspectral Unmixing

In this work, we consider nonnegative matrix factorization (NMF) with a regularization that promotes small volume of the convex hull spanned by the basis matrix. We present highly efficient algorithms for three different volume regularizers, and compare them on endmember recovery in hyperspectral unmixing. The NMF algorithms developed in this work are shown to outperform the state-of-the-art volume-regularized NMF methods, and produce meaningful decompositions on real-world hyperspectral images in situations where endmembers are highly mixed (no pure pixels). Furthermore, our extensive numerical experiments show that when the data is highly separable, meaning that there are data points close to the true endmembers, and there are a few endmembers, the regularizer based on the determinant of the Gramian produces the best results in most cases. For data that is less separable and/or contains more endmembers, the regularizer based on the logarithm of the determinant of the Gramian performs best in general.

math.NA↗

arXiv2019

Identifiability of Complete Dictionary Learning

Sparse component analysis (SCA), also known as complete dictionary learning, is the following problem: Given an input matrix $M$ and an integer $r$, find a dictionary $D$ with $r$ columns and a matrix $B$ with $k$-sparse columns (that is, each column of $B$ has at most $k$ non-zero entries) such that $M \approx DB$. A key issue in SCA is identifiability, that is, characterizing the conditions under which $D$ and $B$ are essentially unique (that is, they are unique up to permutation and scaling of the columns of $D$ and rows of $B$). Although SCA has been vastly investigated in the last two decades, only a few works have tackled this issue in the deterministic scenario, and no work provides reasonable bounds in the minimum number of samples (that is, columns of $M$) that leads to identifiability. 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. While previous bounds feature a combinatorial term $r \choose k$, we exhibit a sufficient condition involving $\mathcal{O}(r^3/(r-k)^2)$ samples that yields an essentially unique decomposition, as long as these data points are well spread among the subspaces spanned by $r-1$ columns of $D$. We also exhibit a necessary lower bound on the number of samples that contradicts previous results in the literature when $k$ equals $r-1$. Our bounds provide a drastic improvement compared to the state of the art, and imply for example that for a fixed proportion of zeros (constant and independent of $r$, e.g., 10\% of zero entries in $B$), one only requires $\mathcal{O}(r)$ data points to guarantee identifiability.

stat.ML↗

arXiv2019

Approximating the nearest stable discrete-time system

In this paper, we consider the problem of stabilizing discrete-time linear systems by computing a nearby stable matrix to an unstable one. To do so, we provide a new characterization for the set of stable matrices. We show that a matrix $A$ is stable if and only if it can be written as $A=S^{-1}UBS$, where $S$ is positive definite, $U$ is orthogonal, and $B$ is a positive semidefinite contraction (that is, the singular values of $B$ are less or equal to 1). This characterization results in an equivalent non-convex optimization problem with a feasible set on which it is easy to project. We propose a very efficient fast projected gradient method to tackle the problem in variables $(S,U,B)$ and generate locally optimal solutions. We show the effectiveness of the proposed method compared to other approaches.

math.OC↗

arXiv2019

On approximating the nearest Ω-stable matrix

In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω, within the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips and disks. We refer to this problem as the nearest Ω-stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time-invariant systems. In order to achieve this goal, we parametrize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.

math.OC↗

arXiv2018

A note on approximating the nearest stable discrete-time descriptor system with fixed rank

Consider a discrete-time linear time-invariant descriptor system $Ex(k+1)=Ax(k)$ for $k \in \mathbb Z_{+}$. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system $\hat E x(k+1)= \hat A x(k)$ with $\text{rank}(\hat E)=r$. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.

math.OC↗

arXiv2018

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). This framework is inspired from the extrapolation scheme used to accelerate gradient methods in convex optimization and from the method of parallel tangents. However, the use of extrapolation in the context of the two-block exact coordinate descent algorithms tackling the non-convex NMF problems is novel. We illustrate the performance of this approach on two state-of-the-art NMF algorithms, namely, accelerated hierarchical alternating least squares (A-HALS) and alternating nonnegative least squares (ANLS), using synthetic, image and document data sets.

math.NA↗

arXiv2018

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. Many initialization schemes have been proposed for NMF, among which one of the most popular class of methods are based on the singular value decomposition (SVD). However, these SVD-based initializations do not satisfy a rather natural condition, namely that the error should decrease as the rank of factorization increases. In this paper, we propose a novel SVD-based \textsc{NMF} initialization to specifically address this shortcoming by taking into account the SVD factors that were discarded to obtain a nonnegative initialization. This method, referred to as nonnegative SVD with low-rank correction (NNSVD-LRC), allows us to significantly reduce the initial error at a negligible additional computational cost using the low-rank structure of the discarded SVD factors. NNSVD-LRC has two other advantages compared to previous SVD-based initializations: (1) it provably generates sparse initial factors, and (2) it is faster as it only requires to compute a truncated SVD of rank $\lceil r/2 + 1 \rceil$ where $r$ is the factorization rank of the sought NMF decomposition (as opposed to a rank-$r$ truncated SVD for other methods). We show on several standard dense and sparse data sets that our new method competes favorably with state-of-the-art SVD-based initializations for NMF.

math.NA↗

arXiv2018

Scalable and Robust Sparse Subspace Clustering Using Randomized Clustering and Multilayer Graphs

Sparse subspace clustering (SSC) is one of the current state-of-the-art methods for partitioning data points into the union of subspaces, with strong theoretical guarantees. However, it is not practical for large data sets as it requires solving a LASSO problem for each data point, where the number of variables in each LASSO problem is the number of data points. 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). This generates a multilayer graph where each layer corresponds to a different set of anchor points. Using the Grassmann manifold of orthogonal matrices, the shared connectivity among the layers is summarized within a single subspace. Finally, we use $k$-means clustering within that subspace to cluster the data points, similarly as done by spectral clustering in SSC. We show on both synthetic and real-world data sets that the proposed method not only allows SSC to scale to large-scale data sets, but that it is also much more robust as it performs significantly better on noisy data and on data with close susbspaces and outliers, while it is not prone to oversegmentation.

cs.CV↗

arXiv2017

Multiplicative Updates for Polynomial Root Finding

Let $f(x)=p(x)-q(x)$ be a polynomial with real coefficients whose roots have nonnegative real part, where $p$ and $q$ are polynomials with nonnegative coefficients. In this paper, we prove the following: Given an initial point $x_0 > 0$, the multiplicative update $x_{t+1} = x_t \, p(x_t)/q(x_t)$ ($t=0,1,\dots$) monotonically and linearly converges to the largest (resp. smallest) real roots of $f$ smaller (resp. larger) than $x_0$ if $p(x_0) < q(x_0)$ (resp. $q(x_0) < p(x_0)$). The motivation to study this algorithm comes from the multiplicative updates proposed in the literature to solve optimization problems with nonnegativity constraints; in particular many variants of nonnegative matrix factorization.

math.NA↗

arXiv2017

Finding the nearest positive-real system

The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive-real (PR) system to a non PR system: given an LTI control system defined by $E \dot{x}=Ax+Bu$ and $y=Cx+Du$, minimize the Frobenius norm of $(Δ_E,Δ_A,Δ_B,Δ_C,Δ_D)$ such that $(E+Δ_E,A+Δ_A,B+Δ_B,C+Δ_C,D+Δ_D)$ is a PR system. We first show that a system is extended strictly PR if and only if it can be written as a strict port-Hamiltonian system. This allows us to reformulate the nearest PR system problem into an optimization problem with a simple convex feasible set. We then use a fast gradient method to obtain a nearby PR system to a given non PR system, and illustrate the behavior of our algorithm on several examples. This is, to the best of our knowledge, the first algorithm that computes a nearby PR system to a given non PR system that (i) is not based on the spectral properties of related Hamiltonian matrices or pencils, (ii) allows to perturb all matrices $(E,A,B,C,D)$ describing the system, and (iii) does not make any assumption on the original given system.

math.OC↗

arXiv2017

Spectral Unmixing with Multiple Dictionaries

Spectral unmixing aims at recovering the spectral signatures of materials, called endmembers, mixed in a hyperspectral or multispectral image, along with their abundances. A typical assumption is that the image contains one pure pixel per endmember, in which case spectral unmixing reduces to identifying these pixels. Many fully automated methods have been proposed in recent years, but little work has been done to allow users to select areas where pure pixels are present manually or using a segmentation algorithm. Additionally, in a non-blind approach, several spectral libraries may be available rather than a single one, with a fixed number (or an upper or lower bound) of endmembers to chose from each. In this paper, we propose a multiple-dictionary constrained low-rank matrix approximation model that address these two problems. We propose an algorithm to compute this model, dubbed M2PALS, and its performance is discussed on both synthetic and real hyperspectral images.

math.OC↗

arXiv2017

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. A new formulation of sparse coding is proposed which enables high dimensional tensors dictionary-based canonical polyadic decomposition. The benefits of using a dictionary in tensor decomposition models are explored both in terms of parameter identifiability and estimation accuracy. Performances of the proposed algorithms are evaluated on the decomposition of simulated data and the unmixing of hyperspectral images.

stat.ML↗

arXiv2017

A Fast Gradient Method for Nonnegative Sparse Regression with Self Dictionary

A nonnegative matrix factorization (NMF) can be computed efficiently under the separability assumption, which asserts that all the columns of the given input data matrix belong to the cone generated by a (small) subset of them. The provably most robust methods to identify these conic basis columns are based on nonnegative sparse regression and self dictionaries, and require the solution of large-scale convex optimization problems. In this paper we study a particular nonnegative sparse regression model with self dictionary. As opposed to previously proposed models, this model yields a smooth optimization problem where the sparsity is enforced through linear constraints. We show that the Euclidean projection on the polyhedron defined by these constraints can be computed efficiently, and propose a fast gradient method to solve our model. We compare our algorithm with several state-of-the-art methods on synthetic data sets and real-world hyperspectral images.

math.OC↗

arXiv2017

Algorithms for Positive Semidefinite Factorization

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization problem consists in finding, if possible, symmetric $k$-by-$k$ positive semidefinite matrices $\{A^1,...,A^m\}$ and $\{B^1,...,B^n\}$ such that $X_{i,j}=\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size $k=1+ \lceil \log_2(n) \rceil$ for the regular $n$-gons when $n=5$, $8$ and $10$. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices $A^i$ and $B^j$ have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality $A^i=B^i$ is required for all $i$).

math.OC↗

arXiv2017

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 $\max_{i,j} |M_{ij} - X_{ij}|$. In this paper, we prove that the decision variant of this problem for $r=1$ is NP-complete using a reduction from the problem `not all equal 3SAT'. We also analyze several cases when the problem can be solved in polynomial time, and propose a simple practical heuristic algorithm which we apply on the problem of the recovery of a quantized low-rank matrix.

cs.CC↗