arXiv subjects

Valentin Leplat

Publications and source records attributed to Valentin Leplat.

At least 19 recordsLinked to original sources

arXiv2026

Iterative Methods for Computing the Moore--Penrose Inverse of Split-Quaternion Matrices with Applications

We study iterative methods for computing the Moore--Penrose inverse of split-quaternion matrices. We first establish a consistent framework based on a $2\times2$ real representation and the associated $i$-conjugate transpose. This representation gives a direct definition of the Moore--Penrose inverse and clarifies the treatment of nonzero zero divisors. We then analyze Newton--Schulz iterations for rectangular and rank-deficient matrices. Using a thin singular value decomposition of the real representative, we derive the convergence conditions, the evolution of the projector residuals, and an exact relation between the residual and the error. The resulting analysis applies both to an embedded real implementation and to a native split-quaternion iteration, which are shown to be equivalent under the representation. We also propose a low-degree polynomial initialization inspired by Souriau's inverse recursion. The polynomial is obtained by a least-squares approximation of the inverse Gram factor and is combined with a spectral acceptance test and a safe fallback initialization. Finally, we apply the proposed methods to cross and CUR approximations of split-quaternion matrices. We characterize the optimal middle factor for fixed sampled rows and columns and give conditions under which the cheaper cross factor gives an exact reconstruction. Numerical experiments illustrate the accuracy of the methods and the practical benefit of the polynomial initialization.

math.NA↗

arXiv2026

Adaptive Randomized Pivoting for Tensor Cross Approximation in the T-Product Framework

This paper studies extensions of adaptive randomized pivoting (ARP), recently introduced for matrix column subset selection, to tensors in the t-product framework. We propose two constructions. ARP-T-CUR applies matrix ARP-cross at the nonredundant Fourier frequencies and uses conjugate symmetry to preserve real-valued tensors. Assuming that the selected intersections are nonsingular almost surely, the matrix theory gives a direct expected-error bound for this Fourier-slicewise approximation. T-ARP instead selects common lateral and horizontal slices, with the same indices used at every Fourier frequency. This common-index constraint requires a new analysis. Under frequency-alignment and frequency-wise rank-growth assumptions, we prove an expected-error bound and recover the matrix ARP factor $r+1$ when the frequency-wise sampling distributions are aligned. We also derive bounds for tensor cross approximation and t-DEIM. The numerical experiments show that both proposed methods give accurate tensor approximations and perform favorably against the considered slice-selection baselines. A separate full-enumeration experiment verifies the assumptions and the expected-error bound for common-index T-ARP.

math.NA↗

arXiv2026

Quaternion Maximum-Volume Submatrix Selection with Applications to Multichannel Imaging and Visual Data

Low-rank approximation based on selected rows and columns is a useful alternative to singular value decompositions when the goal is an interpretable and compact matrix representation. A standard way to choose these rows and columns is the maximum-volume principle: it selects submatrices with large volume, which usually leads to stable interpolation coefficients and accurate CUR-type approximations. In this paper, we study this idea for quaternion matrices. This setting is natural for color images, three-dimensional motion data, and multi-channel signals, but requires care because quaternion multiplication is noncommutative. We define quaternion maximum-volume submatrix selection using quaternion singular values and the Study determinant. We then derive quaternion rank-one update formulas and use them to build two selection procedures: a greedy square-core method for row and column replacement, and a rectangular method that enlarges a selected row set until the interpolation coefficients are controlled. We prove that successful row and column swaps increase the quaternion volume of the selected square core when the exact quaternion inverse is used. We also connect the stopping criterion with quasi-dominance, prove an exact quaternion CUR identity in the full-rank case, and derive an interpolation stability bound. For the rectangular case, we derive an append-row pseudoinverse update and show how it gives a natural right preconditioner for overdetermined quaternion least-squares problems. Finally, we illustrate the methods on three applications: quaternion CUR approximation of RGB images, RectMaxVol-based preconditioning for ill-conditioned quaternion least-squares systems, and row selection in quaternion motion-capture data. The experiments show that the proposed quaternion MaxVol and RectMaxVol methods provide stable and efficient selection routines.

math.NA↗

arXiv2026

Prox-NAG-GS: A Semi-Implicit Proximal Method for Composite Optimization

Composite optimization problems, where a smooth loss is combined with a nonsmooth regularizer, are common in machine learning and inverse problems. In this work, we study a proximal extension of NAG-GS, a semi-implicit accelerated method obtained from a Gauss-Seidel discretization of an inertial dynamics. The proposed method, called Prox-NAG-GS, keeps the coupled structure of NAG-GS for the smooth part and replaces the second update by a proximal step. It therefore applies to objectives of the form $F=f+r$, where $f$ is smooth and $r$ is convex and proximable. We derive deterministic convergence guarantees for this method. The analysis has to account for a specific feature of the scheme. Prox-NAG-GS keeps two coupled sequences: an $x$-sequence, on which the gradient of the smooth term is evaluated, and a $v$-sequence, produced by the proximal update. The gradient is evaluated at $x_{k+1}$, whereas the proximal step returns $v_{k+1}$, which creates a mismatch absent from the standard proximal-gradient analysis. Under the sufficient condition that the proximal quadratic parameter is at least as large as the smoothness constant of $f$, we control this mismatch through an augmented Lyapunov function involving both sequences. This gives a linear convergence result in the strongly convex composite case. In the convex case, the same Lyapunov structure yields an $O(1/k)$ rate for the best iterate and for the averaged iterate. We test the method on deterministic Elastic Net and Group Lasso problems, and on stochastic sparse softmax-regression benchmarks. In the deterministic tests, Prox-NAG-GS reaches the same solutions as the baselines with substantially fewer iterations; for Group Lasso this also gives the best wall-clock time. In the stochastic tests, Prox-NAG-GS compares favorably with Prox-SGD in terms of data-fit reduction and gives similar test accuracies.

math.OC↗

arXiv2026

IRON: Implicit Resolvent Optimization under Noise

We study stochastic optimization from a joint continuous-discrete point of view. Starting from a second-order stochastic differential equation interpreted as a noisy accelerated gradient flow, we discretize the dynamics by a fully implicit Backward-Euler scheme. This leads to a resolvent, or proximal-type, update, computed in practice through Levenberg-Marquardt, Newton, or trust-region-type inner solves. The resulting method, denoted by $\text{IRON}_{\text{FI}}$, admits a Lyapunov mean-square recursion. The main conclusion is that increasing the implicit stepsize $\alpha$ improves the contraction factor and decreases the stationary mean-square error bound. Under sufficiently accurate inner solves, this bound scales as $O(1/\alpha)$; in particular, for large enough $\alpha$, the recursion is contractive and the stationary error bound vanishes as $\alpha\to\infty$. We establish the theory for smooth strongly convex objectives and provide a sharper quadratic analysis with an explicit stationary constant. The numerical experiments support the theory in the strongly convex case and illustrate the same qualitative behavior in nonconvex and learning settings. To the best of our knowledge, this fully implicit inertial-resolvent discretization, where the noise acts as an additive perturbation of the resolvent center and yields an $O(1/\alpha)$ stationary-MSE law, has not been isolated in this form before. The broader message is that discretization is not only an implementation choice: in stochastic optimization, the numerical integration rule can directly affect the long-time stability of the iterates under noise.

math.OC↗

arXiv2026

QuatIca: Advanced Numerical Linear Algebra and Optimization for Quaternionic Matrices in Python

Quaternion-valued representations provide a convenient way to model coupled multi-channel signals (e.g., RGB imagery, polarization data, vector fields, and multi-detector time series). Yet practical and numerically reliable software support remains far less mature than those based on the real/complex setting. Here, we present QuatIca, an open-source Python library for quaternion numerical linear algebra and optimization, designed for both research prototyping and reproducible experimentation. QuatIca provides core quaternion matrix operations and norms; dense decompositions and reductions (QR, LU, Q-SVD, eigendecomposition, Hessenberg/tridiagonal reduction, Cholesky decomposition, and Schur helpers); iterative solvers including quaternion GMRES (with preconditioning) and Newton-Schulz pseudoinverse schemes; and domain-focused routines for signal and image processing such as quaternion Tikhonov restoration. The library also includes OptiQ, which solves quaternion Hermitian semidefinite programs using log-det barrier Newton methods with $\mu$-continuation. We highlight design choices that preserve quaternion structure, and we provide end-to-end demonstrations including quaternion image deblurring, Lorenz-attractor filtering, and quaternion image completion. QuatIca is distributed via PyPI and accompanied by open-source development on GitHub and continuously deployed documentation with runnable tutorials.

math.NA↗

arXiv2026

A New Tensor Network: Tubal Tensor Train and Its Applications

We introduce the tubal tensor train (TTT) decomposition, a tensor-network model that combines the t-product algebra of the tensor singular value decomposition (T-SVD) with the low-order core structure of the tensor train (TT) format. For an order-$(N+1)$ tensor with a distinguished tube mode, the proposed representation consists of two third-order boundary cores and $N-2$ fourth-order interior cores linked through the t-product. As a result, for bounded tubal ranks, the storage scales linearly with the number of modes, in contrast to direct high-order extensions of T-SVD. We present two computational strategies: a sequential fixed-rank construction, called TTT-SVD, and a Fourier-slice alternating scheme based on the alternating two-cores update (ATCU). We also state a TT-SVD-type error bound for TTT-SVD and illustrate the practical performance of the proposed model on image compression, video compression, tensor completion, and hyperspectral imaging.

math.NA↗

arXiv2026

Joint Majorization-Minimization for Nonnegative CP and Tucker Decompositions under $\beta$-Divergences: Unfolding-Free Updates

We study majorization-minimization methods for nonnegative tensor decompositions under the $\beta$-divergence family, focusing on nonnegative CP and Tucker models. Our aim is to avoid explicit mode unfoldings and large auxiliary matrices by deriving separable surrogates whose multiplicative updates can be implemented using only tensor contractions (einsum-style operations). We present both classical block-MM updates in contraction-only form and a joint majorization strategy, inspired by joint MM for matrix $\beta$-NMF, that reuses cached reference quantities across inexpensive inner updates. We prove tightness of the proposed majorizers, establish monotonic decrease of the objective, and show convergence of the sequence of objective values. For block-MM, we discuss how BSUM theory applies to the analysis of stationary accumulation points. For J-CoMM, we further establish, under a set of standard regularity assumptions and for one inner sweep per outer iteration, convergence of the iterates to a critical point through a KL-based analysis. Finally, experiments on synthetic tensors and the Uber spatiotemporal count tensor demonstrate substantial speedups over unfolding-based baselines and competitive runtime relative to a recent einsum-factorization framework.

math.OC↗

arXiv2025

A Provably-Correct and Robust Convex Model for Smooth Separable NMF

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.

math.NA↗

arXiv2025

Market-Driven Subset Selection for Budgeted Training

Training large language models on massive datasets is computationally expensive, yet empirical evidence suggests that substantial portions of training examples contribute minimally to final performance. Data subset selection addresses this inefficiency by identifying small, high-utility subsets under resource constraints. However, example utility is inherently multi-faceted, encompassing uncertainty, distributional rarity, and diversity signals that are heterogeneous and typically combined through ad hoc weighted sums lacking theoretical grounding. We propose a market-based framework that treats each training example as a tradeable contract and employs the Logarithmic Market Scoring Rule to aggregate multiple utility signals into coherent prices. Heterogeneous signals act as traders, a single liquidity parameter controls concentration versus smoothing, and topic-wise normalization ensures calibrated aggregation. Token budgets are handled explicitly through a price-per-token decision rule with an interpretable length-bias parameter. We establish theoretical connections to maximum-entropy aggregation and provide utility recovery guarantees under noisy but monotone signals. On GSM8K mathematical reasoning under strict 60k-token budgets, our selector achieves parity with strong single-signal baselines while exhibiting lower variance and incurring less than 0.1 GPU-hour overhead. On AGNews classification at 5-25\% retention rates, the market formulation delivers competitive accuracy with improved stability. Our framework unifies multi-signal data curation under fixed computational budgets for prompt-level reasoning and classification tasks.

cs.LG↗

arXiv2025

Constrained Optimization via Constraint-Induced Geometry: Implicit Feasible Dynamics and Optimality from Stationarity

We introduce Gravidy, a geometry-aware framework for constrained optimization in which constraints are encoded directly into the dynamics, so the motion remains feasible by construction. The geometric mechanism depends on the feasible set: componentwise reparameterizations and induced Hessian geometries for the nonnegative orthant and box constraints, Fisher-Shahshahani and KL geometry for the simplex, and canonical Riemannian geometry for the Stiefel manifold. We derive feasible continuous-time flows and implicit discretizations adapted to each geometry. On the vector domains, the implicit updates admit exact Bregman-proximal interpretations, yielding monotone descent and convergence guarantees for convex objectives, linear contraction under relative strong convexity, and a Kurdyka-Lojasiewicz analysis for nonconvex problems under compact-interiority, decrease, and relative-error assumptions. We also show that convergent trajectories generated from the interior recover the Karush-Kuhn-Tucker conditions on the orthant, simplex, and box. The same holds for convergent implicit sequences generated from the interior when their stepsizes are bounded away from zero. On the Stiefel manifold, stationarity of the canonical Riemannian gradient is equivalent to the usual first-order optimality condition. The algorithms combine large implicit outer steps with problem-adapted Newton, modified Gauss-Newton, Newton-KKT, and Newton-Krylov inner solvers. Numerical experiments on nonnegative, simplex-constrained, box-constrained, and orthogonality-constrained problems show rapid convergence to high accuracy in a small number of outer iterations while preserving feasibility of accepted iterates. A sparse elastic-obstacle experiment further shows that the orthant construction can exploit large structured systems directly. This illustrates their accuracy and ability to exploit sparsity in sparse settings.

math.OC↗

arXiv2025

Norm-Constrained Flows and Sign-Based Optimization: Theory and Algorithms

Sign Gradient Descent (SignGD) uses only the coordinate-wise sign of the gradient. We study this method through norm-constrained continuous-time dynamics: at each point, the velocity is chosen to minimize the directional derivative over a unit norm ball. This recovers the sign flow for the $\ell_\infty$ constraint, normalized gradient flow for $\ell_2$, and greedy coordinate directions for $\ell_1$. We formulate the resulting dynamics as a set-valued differential inclusion, prove existence of solutions, and derive an exact energy identity. This identity also gives finite-time convergence under a Polyak-Lojasiewicz inequality expressed in the dual norm. We then connect the canonical set-valued flow with classical Filippov regularization of discontinuous selectors, which gives a precise description of crossing and sliding near switching sets. Motivated by this behavior, we introduce two face-aware SignGD variants, one-hit freeze and two-hit sliding-track. Both methods modify the usual sign direction by damping selected coordinates when a crossing or persistent switching is detected. We derive descent certificates for these damped sign directions and introduce a safeguard that preserves a global linear convergence rate under $\ell_\infty$-smoothness and a Polyak-Lojasiewicz inequality. For the $\ell_1$ flow, we also introduce convex-combination updates on active faces and prove a corresponding linear convergence guarantee. Finally, we analyze mass-aware face restrictions and an inertial SignGD method with restart. Numerical experiments illustrate the certified step rules, the face-aware variants, and the inertial scheme.

math.OC↗

arXiv2025

Iterative Methods for Computing the Moore-Penrose Pseudoinverse of Quaternion Matrices, with Applications

We develop quaternion--native iterative methods for computing the Moore--Penrose (MP) pseudoinverse of quaternion matrices and analyze their convergence. Our starting point is a damped Newton--Schulz (NS) iteration tailored to noncommutativity: we enforce the appropriate left/right identities for rectangular inputs and prove convergence directly in $\mathbb{H}$ under a simple spectral scaling. We then derive higher--order (\emph{hyperpower}) NS schemes with exact residual recurrences that yield order-$p$ local convergence, together with factorizations that reduce the number of $s\times s$ quaternion products per iteration. Beyond NS, we introduce a randomized sketch--and--project method (RSP--Q), a hybrid RSP+NS scheme that interleaves inexpensive randomized projections with an exact hyperpower step, and a matrix--form conjugate gradient on the normal equations (CGNE--Q). All algorithms operate directly in $\mathbb{H}$ (no real or complex embeddings) and are matrix--free.

math.NA↗

arXiv2025

GRAFT: Gradient-Aware Fast MaxVol Technique for Dynamic Data Sampling

Training modern neural networks on large datasets is computationally and environmentally costly. We introduce GRAFT, a scalable in-training subset selection method that (i) extracts a low-rank feature representation for each batch, (ii) applies a Fast MaxVol sampler to select a small, diverse subset that spans the batch's dominant subspace, and (iii) dynamically adjusts the subset size using a gradient-approximation criterion. By operating in low-rank subspaces and training on carefully chosen examples instead of full batches, GRAFT preserves the training trajectory while reducing wall-clock time, energy consumption, and $\mathrm{CO}_2$ emissions. Across multiple benchmarks, GRAFT matches or exceeds recent selection baselines in both accuracy and efficiency, providing a favorable trade-off between accuracy, efficiency, and emissions.

cs.LG↗

arXiv2025

Pass-efficient Randomized Algorithms for Low-rank Approximation of Quaternion Matrices

Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input matrix-which is critical in modern computing environments dominated by communication costs. We address this gap by proposing a suite of pass-efficient randomized algorithms that let users directly trade pass budget for approximation accuracy. Our contributions include: (i) a family of arbitrary-pass randomized algorithms for low-rank approximation of quaternion matrices that operate under a user-specified number of matrix views, and (ii) a pass-efficient extension of block Krylov subspace methods that accelerates convergence for matrices with slowly decaying spectra. Furthermore, we establish spectral norm error bounds showing that the expected approximation error decays exponentially with the number of passes. Finally, we validate our framework through extensive numerical experiments and demonstrate its practical relevance across multiple applications, including quaternionic data compression, matrix completion, image super-resolution, and deep learning.

math.NA↗

arXiv2024

Nonnegative Tensor Decomposition Via Collaborative Neurodynamic Optimization

This paper introduces a novel collaborative neurodynamic model for computing nonnegative Canonical Polyadic Decomposition (CPD). The model relies on a system of recurrent neural networks to solve the underlying nonconvex optimization problem associated with nonnegative CPD. Additionally, a discrete-time version of the continuous neural network is developed. To enhance the chances of reaching a potential global minimum, the recurrent neural networks are allowed to communicate and exchange information through particle swarm optimization (PSO). Convergence and stability analyses of both the continuous and discrete neurodynamic models are thoroughly examined. Experimental evaluations are conducted on random and real-world datasets to demonstrate the effectiveness of the proposed approach.

math.NA↗

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

Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models

Regularized nonnegative low-rank approximations, such as sparse Nonnegative Matrix Factorization or sparse Nonnegative Tucker Decomposition, form an important branch of dimensionality reduction models known for their enhanced interpretability. From a practical perspective, however, selecting appropriate regularizers and regularization coefficients, as well as designing efficient algorithms, remains challenging due to the multifactor nature of these models and the limited theoretical guidance available. This paper addresses these challenges by studying a more general model, the Homogeneous Regularized Scale-Invariant model. We prove that the scale-invariance inherent to low-rank approximation models induces an implicit regularization effect that balances solutions. This insight provides a deeper understanding of the role of regularization functions in low-rank approximation models, informs the selection of regularization hyperparameters, and enables the design of balancing strategies to accelerate the empirical convergence of optimization algorithms. Additionally, we propose a generic Majorization-Minimization (MM) algorithm capable of handling $\ell_p^p$-regularized nonnegative low-rank approximations with non-Euclidean loss functions, with convergence guarantees. Our contributions are demonstrated on sparse Nonnegative Matrix Factorization, ridge-regularized Nonnegative Canonical Polyadic Decomposition, and sparse Nonnegative Tucker Decomposition.

cs.LG↗