François Glineur

Nicolas Gillis, François Glineur

Weighted low-rank approximation (WLRA), a dimensionality reduction technique for data analysis, has been successfully used in several applications, such as in collaborative filtering to design recommender systems or in computer vision to recover structure from motion. In this paper, we study the computational complexity of WLRA and prove that it is NP-hard to find an approximate solution, even when a rank-one approximation is sought. Our proofs are based on a reduction from the maximum-edge biclique problem, and apply to strictly positive weights as well as binary weights (the latter corresponding to low-rank matrix approximation with missing data).

Nicolas Gillis, François Glineur

Nonnegative Matrix Factorization (NMF) is a data analysis technique which allows compression and interpretation of nonnegative data. NMF became widely studied after the publication of the seminal paper by Lee and Seung (Learning the Parts of Objects by Nonnegative Matrix Factorization, Nature, 1999, vol. 401, pp. 788--791), which introduced an algorithm based on Multiplicative Updates (MU). More recently, another class of methods called Hierarchical Alternating Least Squares (HALS) was introduced that seems to be much more efficient in practice. In this paper, we consider the problem of approximating a not necessarily nonnegative matrix with the product of two nonnegative matrices, which we refer to as Nonnegative Factorization (NF); this is the subproblem that HALS methods implicitly try to solve at each iteration. We prove that NF is NP-hard for any fixed factorization rank, using a reduction to the maximum edge biclique problem. We also generalize the multiplicative updates to NF, which allows us to shed some light on the differences between the MU and HALS algorithms for NMF and give an explanation for the better performance of HALS. Finally, we link stationary points of NF with feasible solutions of the biclique problem to obtain a new type of biclique finding algorithm (based on MU) whose iterations have an algorithmic complexity proportional to the number of edges in the graph, and show that it performs better than comparable existing methods.

Cécile Hautecoeur, Lieven De Lathauwer, Nicolas Gillis et al.

Nonnegative Matrix Factorization (NMF) models are widely used to recover linearly mixed nonnegative data. 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). We first show that, under mild assumptions, R-NMF has an essentially unique factorization unlike NMF, which is crucial in applications where ground-truth factors need to be recovered such as blind source separation problems. Then we present different approaches to solve R-NMF: the R-HANLS, R-ANLS and R-NLS methods. From our tests, no method significantly outperforms the others, and a trade-off should be done between time and accuracy. Indeed, R-HANLS is fast and accurate for large problems, while R-ANLS is more accurate, but also more resources demanding, both in time and memory. R-NLS is very accurate but only for small problems. Moreover, we show that R-NMF outperforms NMF in various tasks including the recovery of semi-synthetic continuous signals, and a classification problem of real hyperspectral signals.

Philémon Beghin, Anne-Emmanuelle Ceulemans, Paul Fisette et al.

Some early violins have been reduced during their history to fit imposed morphological standards, while more recent ones have been built directly to these standards. We propose an objective photogrammetric approach to differentiate between a reduced and an unreduced instrument, whereby a three-dimensional mesh is studied geometrically by examining 2D slices. Our contribution is twofold. First, we validate the quality of the photogrammetric mesh through a comparison with reference images obtained by medical imaging, and conclude that a sub-millimetre accuracy is achieved. Then, we show how quantitative and qualitative features such as contour lines, channel of minima and a measure of asymmetry between the upper and lower surfaces of a violin can be automatically extracted from the validated photogrammetric meshes, allowing to successfully highlight differences between instruments.

Sofiane Tanji, Andrea Della Vecchia, François Glineur et al.

Kernel methods provide a powerful framework for non parametric learning. They are based on kernel functions and allow learning in a rich functional space while applying linear statistical learning tools, such as Ridge Regression or Support Vector Machines. However, standard kernel methods suffer from a quadratic time and memory complexity in the number of data points and thus have limited applications in large-scale learning. In this paper, we propose Snacks, a new large-scale solver for Kernel Support Vector Machines. Specifically, Snacks relies on a Nyström approximation of the kernel matrix and an accelerated variant of the stochastic subgradient method. We demonstrate formally through a detailed empirical evaluation, that it competes with other SVM solvers on a variety of benchmark datasets.

Valentin Hamaide, Denis Joassin, Lauriane Castin et al.

Predicting incoming failures and scheduling maintenance based on sensors information in industrial machines is increasingly important to avoid downtime and machine failure. Different machine learning formulations can be used to solve the predictive maintenance problem. However, many of the approaches studied in the literature are not directly applicable to real-life scenarios. Indeed, many of those approaches usually either rely on labelled machine malfunctions in the case of classification and fault detection, or rely on finding a monotonic health indicator on which a prediction can be made in the case of regression and remaining useful life estimation, which is not always feasible. Moreover, the decision-making part of the problem is not always studied in conjunction with the prediction phase. This paper aims to design and compare different formulations for predictive maintenance in a two-level framework and design metrics that quantify both the failure detection performance as well as the timing of the maintenance decision. The first level is responsible for building a health indicator by aggregating features using a learning algorithm. The second level consists of a decision-making system that can trigger an alarm based on this health indicator. Three degrees of refinements are compared in the first level of the framework, from simple threshold-based univariate predictive technique to supervised learning methods based on the remaining time before failure. We choose to use the Support Vector Machine (SVM) and its variations as the common algorithm used in all the formulations. We apply and compare the different strategies on a real-world rotating machine case study and observe that while a simple model can already perform well, more sophisticated refinements enhance the predictions for well-chosen parameters.

Pierre Vernimmen, François Glineur

This work assesses both empirically and theoretically, using the performance estimation methodology, how robust different first-order optimization methods are when subject to relative inexactness in their gradient computations. Relative inexactness occurs, for example, when compressing the gradient using fewer bits of information, which happens when dealing with large-scale problems on GPUs. Three major families of methods are analyzed: constant step gradient descent, long-step methods, and accelerated methods. The latter two are first shown to be theoretically not robust to inexactness. Then, a semi-heuristic shortening factor is introduced to improve their theoretical guarantees. All methods are subsequently tested on a concrete inexact problem, with two different types of relative inexactness, and it is observed that both accelerated methods are much more robust than expected, and that the shortening factor significantly helps the long-step methods. In the end, all shortened methods appear to be promising, even in this inexact setting.

Philémon Beghin, Anne-Emmanuelle Ceulemans, François Glineur

Some early violins have been reduced during their history to fit imposed morphological standards, while more recent ones have been built directly to these standards. We can observe differences between reduced and unreduced instruments, particularly in their contour lines and channel of minima. In a recent preliminary work, we computed and highlighted those two features for two instruments using triangular 3D meshes acquired by photogrammetry, whose fidelity has been assessed and validated with sub-millimetre accuracy. We propose here an extension to a corpus of 38 violins, violas and cellos, and introduce improved procedures, leading to a stronger discussion of the geometric analysis. We first recall the material we are working with. We then discuss how to derive the best reference plane for the violin alignment, which is crucial for the computation of contour lines and channel of minima. Finally, we show how to compute efficiently both characteristics and we illustrate our results with a few examples.

Philémon Beghin, Anne-Emmanuelle Ceulemans, François Glineur

The first violins appeared in late 16th-century Italy. Over the next 200 years, they spread across Europe and luthiers of various royal courts, eager to experiment with new techniques, created a highly diverse family of instruments. Around 1750, size standards were introduced to unify violin making for orchestras and conservatories. Instruments that fell between two standards were then reduced to a smaller size by luthiers. These reductions have an impact on several characteristics of violins, in particular on the contour lines, i.e. lines of constant altitude, which look more like a U for non reduced instruments and a V for reduced ones. While such differences are observed by experts, they have not been studied quantitatively. This paper presents a method for classifying violins as reduced or non-reduced based on their contour lines. We study a corpus of 25 instruments whose 3D geometric meshes were acquired via photogrammetry. For each instrument, we extract 10-20 contour lines regularly spaced every millimetre. Each line is fitted with a parabola-like curve (with an equation of the type y = alpha*abs(x)**beta) depending on two parameters, describing how open (beta) and how vertically stretched (alpha) the curve is. We compute additional features from those parameters, using regressions and counting how many values fall under some threshold. We also deal with outliers and non equal numbers of levels, and eventually obtain a numerical profile for each instrument. We then apply classification methods to assess whether geometry alone can predict size reduction. We find that distinguishing between reduced and non reduced instruments is feasible to some degree, taking into account that a whole spectrum of more or less transformed violins exists, for which it is more difficult to quantify the reduction. We also find the opening parameter beta to be the most predictive.

Baptiste Goujaud, Céline Moucer, François Glineur et al.

PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. To do that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modeling parts, and the worst-case analysis is performed numerically via a standard solver.

Shuyu Dong, Bin Gao, Yu Guan et al.

We propose new Riemannian preconditioned algorithms for low-rank tensor completion via the polyadic decomposition of a tensor. These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. This new metric is designed using an approximation of the diagonal blocks of the Hessian of the tensor completion cost function, thus has a preconditioning effect on these algorithms. We prove that the proposed Riemannian gradient descent algorithm globally converges to a stationary point of the tensor completion problem, with convergence rate estimates using the $Ł$ojasiewicz property. Numerical results on synthetic and real-world data suggest that the proposed algorithms are more efficient in memory and time compared to state-of-the-art algorithms. Moreover, the proposed algorithms display a greater tolerance for overestimated rank parameters in terms of the tensor recovery performance, thus enable a flexible choice of the rank parameter.

Yu Guan, Shuyu Dong, Bin Gao et al.

We consider a Canonical Polyadic (CP) decomposition approach to low-rank tensor completion (LRTC) by incorporating external pairwise similarity relations through graph Laplacian regularization on the CP factor matrices. The usage of graph regularization entails benefits in the learning accuracy of LRTC, but at the same time, induces coupling graph Laplacian terms that hinder the optimization of the tensor completion model. In order to solve graph-regularized LRTC, we propose efficient alternating minimization algorithms by leveraging the block structure of the underlying CP decomposition-based model. For the subproblems of alternating minimization, a linear conjugate gradient subroutine is specifically adapted to graph-regularized LRTC. Alternatively, we circumvent the complicating coupling effects of graph Laplacian terms by using an alternating directions method of multipliers. Based on the Kurdyka-Łojasiewicz property, we show that the sequence generated by the proposed algorithms globally converges to a critical point of the objective function. Moreover, the complexity and convergence rate are also derived. In addition, numerical experiments including synthetic data and real data show that the graph regularized tensor completion model has improved recovery results compared to those without graph regularization, and that the proposed algorithms achieve gains in time efficiency over existing algorithms.

Arnaud Vandaele, Nicolas Gillis, François Glineur et al.

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 $X = WH$. In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure. We show that these two heuristics are able to compute exact nonnegative factorizations for several classes of nonnegative matrices (namely, linear Euclidean distance matrices, slack matrices, unique-disjointness matrices, and randomly generated matrices) and as such demonstrate their superiority over standard multi-start strategies. We also consider a hybridization between these two heuristics that allows us to combine the advantages of both methods. Finally, we discuss the use of these heuristics to gain insight on the behavior of the nonnegative rank, i.e., the minimum factorization rank such that an exact NMF exists. In particular, we disprove a conjecture on the nonnegative rank of a Kronecker product, propose a new upper bound on the extension complexity of generic $n$-gons and conjecture the exact value of (i) the extension complexity of regular $n$-gons and (ii) the nonnegative rank of a submatrix of the slack matrix of the correlation polytope.

Filippo Pompili, Nicolas Gillis, P. -A. Absil et al.

Approximate matrix factorization techniques with both nonnegativity and orthogonality constraints, referred to as orthogonal nonnegative matrix factorization (ONMF), have been recently introduced and shown to work remarkably well for clustering tasks such as document classification. In this paper, we introduce two new methods to solve ONMF. First, we show athematical equivalence between ONMF and a weighted variant of spherical k-means, from which we derive our first method, a simple EM-like algorithm. This also allows us to determine when ONMF should be preferred to k-means and spherical k-means. Our second method is based on an augmented Lagrangian approach. Standard ONMF algorithms typically enforce nonnegativity for their iterates while trying to achieve orthogonality at the limit (e.g., using a proper penalization term or a suitably chosen search direction). Our method works the opposite way: orthogonality is strictly imposed at each step while nonnegativity is asymptotically obtained, using a quadratic penalty. Finally, we show that the two proposed approaches compare favorably with standard ONMF algorithms on synthetic, text and image data sets.