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P. -A. Absil

Publications and source records attributed to P. -A. Absil.

At least 19 recordsLinked to original sources
arXiv2026

MPFA: A Pareto Front Approximation Method for Riemannian Bi-objective Optimization

We propose a Pareto front approximation (MPFA) method for smooth bi-objective optimization problems on Riemannian manifolds based on a Hermite interpolation technique. Compared with the existing multiobjective optimization numerical algorithms, the proposed method can generate a continuous approximate Pareto front without multiple initial points. We establish convergence of the proposed method and analyze the approximation error of the resulting Pareto front. Numerical experiments on several test problems demonstrate that the proposed approach can effectively approximate the Pareto front with high accuracy and reasonable computational cost. Furthermore, the method is applied to a bi-objective formulation of sparse principal component analysis, illustrating its practical applicability in data analysis problems.

math.OC↗
arXiv2026

Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration

We consider optimization problems defined on product spaces of simplices. Examples of this class of problems include learning low-rank discrete multivariate probability distributions via simplex constrained tensor decomposition and performing functional data registration under the Square Root Velocity Function (SRVF) representation. In this work, we demonstrate the feasibility of replacing the product simplex with a smooth, elementwise strictly convex reparameterization, resulting in an unconstrained optimization problem on a manifold. We show that performing such a reparameterization results in the second order Karush-Kuhn-Tucker (KKT) points on the smooth manifold being mapped to the weak second order KKT points on the product simplex. This leads to a Riemannian Gradient Descent (RGD) algorithm for solving the reparameterized problem, which outperforms Projected Gradient Descent (PGD), and provides a more faithful representation of the original function shapes while performing curve registration.

cs.LG↗
arXiv2026

Graph-Regularized Low-Rank Matrix Completion by Variable Projection

We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.

cs.LG↗
arXiv2026

Diffeomorphic Logarithm of Special Orthogonal Matrices

The special orthogonal group $\mathbb{SO}_n$ is a Lie group whose geometry and local structure are encoded by the exponential map in its Lie algebra $\mathbf{Skew}_n$, the set of skew-symmetric matrices. The associated multi-valued inverse problem -- the matrix logarithm -- in $\mathbb{SO}_n$ exhibits a highly nontrivial local diffeomorphism structure, which differs from the matrix logarithm for invertible matrices. This work characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices where its derivative is invertible. We show that this set with an invertible derivative can be organized into diffeomorphic regions, using a canonical alignment of Schur decompositions. In particular, the region that contains the principal logarithm has a special multiplicity structure: each matrix in $\mathbb{SO}_n$ admits at most two skew-symmetric preimages in this region. Based on this geometric framework, we introduce the diffeomorphic logarithm of special orthogonal matrices together with an efficient and stable algorithm. Moreover, it is applied to the Karcher mean problem in $\mathbb{SO}_n$, demonstrating continuous behavior of the mean under perturbations of the data, which is not captured by the principal logarithm.

math.DG↗
arXiv2026

A Jacobi-like algorithm for normal matrices by the skew-symmetric part

We present a fast Jacobi-like algorithm for computing the eigenvalues, and optionally the eigenvectors, of a real normal matrix. The method gains a computational advantage by using Paardekooper's method for skew-symmetric matrices The method is most efficient for matrices where most eigenvalues are complex, such as random orthogonal matrices arising in the context of statistics on manifolds. In this case, the method is faster than the other Jacobi-like algorithms. In the last section of this paper, we also give explicit formulas for the nearest symmetric skew-Hamiltonian and the nearest ortho-symplectic matrix. These problems arise in the design and the analysis of the algorithm.

math.NA↗
arXiv2026

A second-order method landing on the Stiefel manifold via Newton$\unicode{x2013}$Schulz iteration

Retraction-free approaches offer attractive low-cost alternatives to Riemannian methods on the Stiefel manifold, but they are often first-order, which may limit the efficiency under high-accuracy requirements. To this end, we propose a second-order method landing on the Stiefel manifold without invoking retractions, which is proved to enjoy local quadratic (or superlinear for its inexact variant) convergence. The update consists of the sum of (i) a component tangent to the level set of the constraint-defining function that aims to reduce the objective and (ii) a component normal to the same level set that reduces the infeasibility. Specifically, we construct the normal component via Newton$\unicode{x2013}$Schulz, a fixed-point iteration for orthogonalization. Moreover, we establish a geometric connection between the Newton$\unicode{x2013}$Schulz iteration and Stiefel manifolds, in which Newton$\unicode{x2013}$Schulz moves along the normal space. For the tangent component, we formulate a modified Newton equation that incorporates Newton$\unicode{x2013}$Schulz. Numerical experiments on the orthogonal Procrustes problem, principal component analysis, and real-data independent component analysis illustrate that the proposed method performs better than the existing methods.

math.OC↗
arXiv2025

Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold

We determine the length of the shortest nontrivial geodesic loops on the Stiefel manifold endowed with any member of the one-parameter family of Riemannian metrics introduced by H\"uper et al. (2021). This family includes, in particular, the canonical and Euclidean metrics. By combining existing and new bounds on the sectional curvature, we determine the exact value of the injectivity radius of the Stiefel manifold under a wide range of members of the metric family.

math.DG↗
arXiv2025

The Exponential of Skew-Symmetric Matrices: A Nearby Inverse and Efficient Computation of Derivatives

The matrix exponential restricted to skew-symmetric matrices has numerous applications, notably in view of its interpretation as the Lie group exponential and Riemannian exponential for the special orthogonal group. We characterize the invertibility of the derivative of the skew-restricted exponential, thereby providing a simple expression of the tangent conjugate locus of the orthogonal group. In view of the skew restriction, this characterization differs from the classic result on the invertibility of the derivative of the exponential of real matrices. Based on this characterization, for every skew-symmetric matrix $A$ outside the (zero-measure) tangent conjugate locus, we explicitly construct the domain and image of a smooth inverse -- which we term \emph{nearby logarithm} -- of the skew-restricted exponential around $A$. This nearby logarithm reduces to the classic principal logarithm of special orthogonal matrices when $A=\mathbf{0}$. The symbolic formulae for the differentiation and its inverse are derived and implemented efficiently. The extensive numerical experiments show that the proposed formulae are up to $3.9$-times and $3.6$-times faster than the current state-of-the-art robust formulae for the differentiation and its inversion, respectively.

math.DG↗
arXiv2025

On the approximation of the Riemannian barycenter

We present a method for computing an approximate Riemannian barycenter of a collection of points lying on a Riemannian manifold. Our approach relies on the use of theoretically proven under- and over-approximations of the Riemannian distance function. We compare it to Riemannian steepest descent on the exact objective function of the Riemannian barycenter and to an approach that approximates the Riemannian logarithm using lifting maps. Experiments are conducted on the Stiefel manifold.

math.DG↗
arXiv2025

The tangent cone to the real determinantal variety: various expressions and a proof

The set of real matrices of upper-bounded rank is a real algebraic variety called the real generic determinantal variety. An explicit description of the tangent cone to that variety is given in Theorem 3.2 of Schneider and Uschmajew [SIAM J. Optim., 25 (2015), pp. 622-646]. The present paper shows that the proof therein is incomplete and provides a proof. It also reviews equivalent descriptions of the tangent cone to that variety. Moreover, it shows that the tangent cone and the algebraic tangent cone to that variety coincide, which is not true for all real algebraic varieties.

math.OC↗
arXiv2024

An Alternating Minimization Algorithm with Trajectory for Direct Exoplanet Detection -- The AMAT Algorithm

Effective image post-processing algorithms are vital for the successful direct imaging of exoplanets. Standard PSF subtraction methods use techniques based on a low-rank approximation to separate the rotating planet signal from the quasi-static speckles, and rely on signal-to-noise ratio maps to detect the planet. These steps do not interact or feed each other, leading to potential limitations in the accuracy and efficiency of exoplanet detection. We aim to develop a novel approach that iteratively finds the flux of the planet and the low-rank approximation of quasi-static signals, in an attempt to improve upon current PSF subtraction techniques. In this study, we extend the standard L2 norm minimization paradigm to an L1 norm minimization framework to better account for noise statistics in the high contrast images. Then, we propose a new method, referred to as Alternating Minimization Algorithm with Trajectory, that makes a more advanced use of estimating the low-rank approximation of the speckle field and the planet flux by alternating between them and utilizing both L1 and L2 norms. For the L1 norm minimization, we propose using L1 norm low-rank approximation, a low-rank approximation computed using an exact block-cyclic coordinate descent method, while we use randomized singular value decomposition for the L2 norm minimization. Additionally, we enhance the visibility of the planet signal using a likelihood ratio as a postprocessing step. Numerical experiments performed on a VLT/SPHERE-IRDIS dataset show the potential of AMAT to improve upon the existing approaches in terms of higher S/N, sensitivity limits, and ROC curves. Moreover, for a systematic comparison, we used datasets from the exoplanet data challenge to compare our algorithm to other algorithms in the challenge, and AMAT with likelihood ratio map performs better than most algorithms tested on the exoplanet data challenge.

astro-ph.IM↗
arXiv2024

Computing Bouligand stationary points efficiently in low-rank optimization

This paper considers the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the algebraic variety of all $m$-by-$n$ real matrices of rank at most $r$. Several definitions of stationarity exist for this nonconvex problem. Among them, Bouligand stationarity is the strongest necessary condition for local optimality. Only a handful of algorithms generate a sequence in the variety whose accumulation points are provably Bouligand stationary. Among them, the most parsimonious with (truncated) singular value decompositions (SVDs) or eigenvalue decompositions can still require a truncated SVD of a matrix whose rank can be as large as $\min\{m, n\}-r+1$ if the gradient does not have low rank, which is computationally prohibitive in the typical case where $r \ll \min\{m, n\}$. This paper proposes a first-order algorithm that generates a sequence in the variety whose accumulation points are Bouligand stationary while requiring SVDs of matrices whose smaller dimension is always at most $r$. A standard measure of Bouligand stationarity converges to zero along the bounded subsequences at a rate at least $O(1/\sqrt{i+1})$, where $i$ is the iteration counter. Furthermore, a rank-increasing scheme based on the proposed algorithm is presented, which can be of interest if the parameter $r$ is potentially overestimated.

math.OC↗
arXiv2024

Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics

We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by H\"uper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications.

math.DG↗
arXiv2024

Optimization without Retraction on the Random Generalized Stiefel Manifold

Optimization over the set of matrices $X$ that satisfy $X^\top B X = I_p$, referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as the canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods that require a fully formed $B$. We propose a cheap stochastic iterative method that solves the optimization problem while having access only to random estimates of $B$. Our method does not enforce the constraint in every iteration; instead, it produces iterations that converge to critical points on the generalized Stiefel manifold defined in expectation. The method has lower per-iteration cost, requires only matrix multiplications, and has the same convergence rates as its Riemannian optimization counterparts that require the full matrix $B$. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA, ICA, and the GEVP.

cs.LG↗
arXiv2024

A Riemannian Proximal Newton Method

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 $h(x) = μ\|x\|_1$. The generalization relies on the Weingarten and semismooth analysis. It is shown that the Riemannian proximal Newton method has a local quadratic 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 switch parameter is chosen appropriately, then the hybrid method converges globally and also has a local quadratic convergence rate. Numerical experiments on random and synthetic data are used to demonstrate the performance of the proposed methods.

math.OC↗
arXiv2024

The ultimate upper bound on the injectivity radius of the Stiefel manifold

We exhibit conjugate points on the Stiefel manifold endowed with any member of the family of Riemannian metrics introduced by H\"uper et al. (2021). This family contains the well-known canonical and Euclidean metrics. An upper bound on the injectivity radius of the Stiefel manifold in the considered metric is then obtained as the minimum between the length of the geodesic along which the points are conjugate and the length of certain geodesic loops. Numerical experiments support the conjecture that the obtained upper bound is in fact equal to the injectivity radius.

math.DG↗
arXiv2023

Alternating minimization algorithms for graph regularized tensor completion

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.

math.NA↗