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In recent years, the proximal gradient method and its variants have been generalized to Riemannian manifolds for solving optimization problems with an additively separable structure, i.e., $f + h$, where $f$ is continuously differentiable, and $h$ may be nonsmooth but convex with computationally reasonable proximal mapping. In this paper, we generalize the proximal Newton method to embedded submanifolds for solving the type of problem with $h(x) = \mu \|x\|_1$. The generalization relies on the Weingarten and semismooth analysis. It is shown that the Riemannian proximal Newton method has a local superlinear convergence rate under certain reasonable assumptions. Moreover, a hybrid version is given by concatenating a Riemannian proximal gradient method and the Riemannian proximal Newton method. It is shown that if the objective function satisfies the Riemannian KL property and the switch parameter is chosen appropriately, then the hybrid method converges globally and also has a local superlinear convergence rate. Numerical experiments on random and synthetic data are used to demonstrate the performance of the proposed methods.

We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on a stratified set and present a first-order algorithm designed to find a stationary point of that problem. Our assumptions on the stratified set are satisfied notably by the determinantal variety (i.e., matrices of bounded rank), its intersection with the cone of positive-semidefinite matrices, and the set of nonnegative sparse vectors. The iteration map of the proposed algorithm applies a step of projected-projected gradient descent with backtracking line search, as proposed by Schneider and Uschmajew (2015), to its input but also to a projection of the input onto each of the lower strata to which it is considered close, and outputs a point among those thereby produced that maximally reduces the cost function. Under our assumptions on the stratified set, we prove that this algorithm produces a sequence whose accumulation points are stationary, and therefore does not follow the so-called apocalypses described by Levin, Kileel, and Boumal (2022). We illustrate the apocalypse-free property of our method through a numerical experiment on the determinantal variety.

Orthogonality constraints naturally appear in many machine learning problems, from Principal Components Analysis to robust neural network training. They are usually solved using Riemannian optimization algorithms, which minimize the objective function while enforcing the constraint. However, enforcing the orthogonality constraint can be the most time-consuming operation in such algorithms. Recently, Ablin & Peyré (2022) proposed the Landing algorithm, a method with cheap iterations that does not enforce the orthogonality constraint but is attracted towards the manifold in a smooth manner. In this article, we provide new practical and theoretical developments for the landing algorithm. First, the method is extended to the Stiefel manifold, the set of rectangular orthogonal matrices. We also consider stochastic and variance reduction algorithms when the cost function is an average of many functions. We demonstrate that all these methods have the same rate of convergence as their Riemannian counterparts that exactly enforce the constraint. Finally, our experiments demonstrate the promise of our approach to an array of machine-learning problems that involve orthogonality constraints.

We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient over the real determinantal variety, and present a first-order algorithm designed to find stationary points of that problem. This algorithm applies steps of a retraction-free descent method proposed by Schneider and Uschmajew (2015), while taking the numerical rank into account to attempt rank reductions. We prove that this algorithm produces a sequence of iterates the accumulation points of which are stationary, and therefore does not follow the so-called apocalypses described by Levin, Kileel, and Boumal (2022). Moreover, the rank reduction mechanism of this algorithm requires at most one rank reduction attempt per iteration, in contrast with the one of the $\mathrm{P}^2\mathrm{GDR}$ algorithm introduced by Olikier, Gallivan, and Absil (2022) which can require a number of rank reduction attempts equal to the rank of the iterate in the worst-case scenario., arXiv admin note: text overlap with arXiv:2201.03962

We compare two first-order low-rank optimization algorithms, namely $\text{P}^2\text{GD}$ (Schneider and Uschmajew, 2015), which has been proven to be apocalypse-prone (Levin et al., 2021), and its apocalypse-free version $\text{P}^2\text{GDR}$ obtained by equipping $\text{P}^2\text{GD}$ with a suitable rank reduction mechanism (Olikier et al., 2022). Here an apocalypse refers to the situation where the stationarity measure goes to zero along a convergent sequence whereas it is nonzero at the limit. The comparison is conducted on two simple examples of apocalypses, the original one (Levin et al., 2021) and a new one. We also present a potential side effect of the rank reduction mechanism of $\text{P}^2\text{GDR}$ and discuss the choice of the rank reduction parameter.

We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the real determinantal variety, and present a first-order algorithm designed to find stationary points of that problem. This algorithm applies steps of steepest descent with backtracking line search on the variety, as proposed by Schneider and Uschmajew (2015), but by taking the numerical rank into account to perform suitable rank reductions. We prove that this algorithm produces sequences of iterates the accumulation points of which are stationary, and therefore does not follow the so-called apocalypses described by Levin, Kileel, and Boumal (2021).

Quadratic hypersurfaces are a natural generalization of affine subspaces, and projections are elementary blocks of algorithms in optimization and machine learning. It is therefore intriguing that no proper studies and tools have been developed to tackle this nonconvex optimization problem. The quadproj package is a user-friendly and documented software that is dedicated to project a point onto a non-cylindrical central quadratic hypersurface.

We study a continuous-time system that solves optimization problems over the set of orthonormal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but asymptotically lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical metric of the Stiefel manifold. We show that the vector field of the proposed flow can be interpreted as the sum of a Riemannian gradient on a generalized Stiefel manifold and a normal vector. Moreover, we prove that the proposed flow globally converges to the set of critical points, and any local minimum and isolated critical point is asymptotically stable.

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space $\mathbb{R}^{2n\times 2p}$. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expression of the Riemannian gradient with respect to the Euclidean metric, which then used in optimization algorithms. Numerical experiments on the nearest symplectic matrix problem and the symplectic eigenvalue problem illustrate the effectiveness of Euclidean-based algorithms.

This paper concerns the minimax center of a collection of linear subspaces. When the subspaces are $k$-dimensional subspaces of $\mathbb{R}^n$, this can be cast as finding the center of a minimum enclosing ball on a Grassmann manifold, Gr$(k,n)$. For subspaces of different dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the initial problem in this context leads to a candidate minimax center on each of the constituent manifolds, but does not inherently provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. We propose and solve an optimization problem parametrized by the rank of the minimax center. The solution is computed using a subgradient algorithm on the dual. 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., 26 pages

We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem. It is formulated as minimizing a trace cost function over the symplectic Stiefel manifold. We first investigate various theoretical aspects of this optimization problem such as characterizing the sets of critical points, saddle points, and global minimizers as well as proving that non-global local minimizers do not exist. Based on our recent results on constructing Riemannian structures on the symplectic Stiefel manifold and the associated optimization algorithms, we then propose solving the symplectic eigenvalue problem in the framework of Riemannian optimization. Moreover, a connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices is discussed. Numerical examples are presented., Comment: 24 pages, 2 figures

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point., 35 pages, 4 figures

We propose the SH model, a simplified version of the well-known SIR compartmental model of infectious diseases. With optimized parameters and initial conditions, this time-invariant two-parameter two-dimensional model is able to fit COVID-19 hospitalization data over several months with high accuracy (e.g., the root relative squared error is below 10% for Belgium over the period from 2020-03-15 to 2020-07-15). Moreover, we observed that, when the model is trained on a suitable three-week period around the first hospitalization peak for Belgium, it forecasts the subsequent two months with mean absolute percentage error (MAPE) under 4%. We repeated the experiment for each French department and found 14 of them where the MAPE was below 20%. However, when the model is trained in the increase phase, it is less successful at forecasting the subsequent evolution., Comment: Paper home page: https://sites.uclouvain.be/absil/2020.05

We show that H��lder continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the first-order Taylor approximation. We also relate this global upper bound to the H��lder constant of the gradient. This relation is expressed as an interval, depending on the H��lder constant, in which the error of the first-order Taylor approximation is guaranteed to be. We show that, for the Lipschitz continuous case, the interval cannot be reduced. An application to the norms of quadratic forms is proposed, which allows us to derive a novel characterization of Euclidean norms.

We consider a low-rank tensor completion (LRTC) problem which aims to recover a tensor from incomplete observations. LRTC plays an important role in many applications such as signal processing, computer vision, machine learning, and neuroscience. A widely used approach is to combine the tensor completion data fitting term with a regularizer based on a convex relaxation of the multilinear ranks of the tensor. For the data fitting function, we model the tensor variable by using the Canonical Polyadic (CP) decomposition and for the low-rank promoting regularization function, we consider a graph Laplacian-based function which exploits correlations between the rows of the matrix unfoldings. For solving our LRTC model, we propose an efficient alternating minimization algorithm. Furthermore, based on the Kurdyka-{\L}ojasiewicz property, we show that the sequence generated by the proposed algorithm globally converges to a critical point of the objective function. Besides, an alternating direction method of multipliers algorithm is also developed for the LRTC model. Extensive numerical experiments on synthetic and real data indicate that the proposed algorithms are effective and efficient.

The third edition of the "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) took place in Aalborg, the 4th largest city in Denmark situated beautifully in the northern part of the country, from the 24th to 26th of August 2016. The workshop venue was at the Aalborg University campus. One implicit objective of this biennial workshop is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For this third edition, iTWIST'16 gathered about 50 international participants and features 8 invited talks, 12 oral presentations, and 12 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing (e.g., optics, computer vision, genomics, biomedical, digital communication, channel estimation, astronomy); Application of sparse models in non-convex/non-linear inverse problems (e.g., phase retrieval, blind deconvolution, self calibration); Approximate probabilistic inference for sparse problems; Sparse machine learning and inference; "Blind" inverse problems and dictionary learning; Optimization for sparse modelling; Information theory, geometry and randomness; Sparsity? What's next? (Discrete-valued signals; Union of low-dimensional spaces, Cosparsity, mixed/group norm, model-based, low-complexity models, ...); Matrix/manifold sensing/processing (graph, low-rank approximation, ...); Complexity/accuracy tradeoffs in numerical methods/optimization; Electronic/optical compressive sensors (hardware)., Comment: 69 pages, 22 extended abstracts, iTWIST'16 website: http://www.itwist16.es.aau.dk

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference., 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist14

Optimal distribution of power among generating units to meet a specific demand subject to system constraints is an ongoing research topic in the power system community. The problem, even in a static setting, turns out to be hard to solve with conventional optimization methods owing to the consideration of valve-point effects which make the cost function nonsmooth and nonconvex. This difficulty gave rise to the proliferation of population-based global heuristics in order to address the multi-extremal and nonsmooth problem. In this paper, we address the economic load dispatch problem (ELDP) with valve-point effect in its classic formulation where the cost function for each generator is expressed as the sum of a quadratic term and a rectified sine term. We propose two methods that resort to piecewise-quadratic surrogate cost functions, yielding surrogate problems that can be handled by mixed-integer quadratic programming (MIQP) solvers. The first method shows that the global solution of the ELDP can often be found by using a fixed and very limited number of quadratic pieces in the surrogate cost function. The second method adaptively builds piecewise-quadratic surrogate under-estimations of the ELDP cost function, yielding a sequence of surrogate MIQP problems. It is shown that any limit point of the sequence of MIQP solutions is a global solution of the ELDP. Moreover, numerical experiments indicate that the proposed methods outclass the state-of-the-art algorithms in terms of minimization value and computation time on practical instances., Comment: 8 pages

We consider the problem of approximating a multiple-input multiple-output (MIMO) $p\times m$ rational transfer function $H(s)$ of high degree by another $p\times m$ rational transfer function $\hat H(s)$ of much smaller degree, so that the ${\cal H}_2$ norm of the approximation error is minimized. We characterize the stationary points of the ${\cal H}_2$ norm of the approximation error by tangential interpolation conditions and also extend these results to the discrete-time case. We analyze whether it is reasonable to assume that lower-order models can always be approximated arbitrarily closely by imposing only first-order interpolation conditions. Finally, we analyze the ${\cal H}_2$ norm of the approximation error for a simple case in order to illustrate the complexity of the minimization problem., Paper ID sheet: http://www.inma.ucl.ac.be/~absil/Publi/H2-modred-mimo.htm

We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem., Comment: submitted

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design effcient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. By dealing internally with most of the differential geometry, the package aims particularly at lowering the entrance barrier. © 2014 Nicolas Boumal.