arXiv subjects

Christian Kümmerle

Publications and source records attributed to Christian Kümmerle.

At least 19 recordsLinked to original sources

arXiv2026

Support Discovery With Iteratively Reweighted Least Squares for Fixed-Charge Network Flow

The fixed-charge network flow problem (FCNFP) couples continuous flow allocation with discrete arc-activation decisions, making it a canonical but computationally challenging model for a variety of network design and resource allocation problems. Exact mixed-integer linear programming formulations capture the fixed-charge structure faithfully, but often become difficult to solve on large networks. We propose a scalable continuous-optimization algorithm for large-scale single-commodity FCNFP based on an iteratively reweighted least-squares (IRLS) framework. The method replaces the discontinuous fixed-charge and linear arc cost objective with a smooth nonconvex Lasry--Lions surrogate and solves a sequence of weighted quadratic flow subproblems. Each subproblem is solved by a warm-started dual semismooth Newton method whose Newton systems have weighted graph-Laplacian structure, enabling the use of modern Laplacian solvers. To further improve the discovered arc supports of the challenging underlying combinatorial problem, we also develop an algorithmic variant that incorporates objective-driven perturbation restarts and an anchor-union restricted search that jointly leverages supports discovered by IRLS and by complementary FCNFP heuristics. Computational experiments on 410 benchmark, synthetic, and large-scale instances show that our method obtains the best objective quality among the evaluated scalable FCNFP algorithms, with a mean gap of $1.316\%$ to a time-limited MILP reference and a win-or-tie rate of $90.0\%$ among the non-MILP methods. The results indicate that combining smooth continuous optimization with support-level search is an effective strategy for producing high-quality feasible solutions to large-scale FCNFP.

math.OC↗

arXiv2026

Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS

Iteratively reweighted least squares (IRLS) methods constitute a natural approach to nuclear norm minimization, but their convergence rates and the role of the weight operator have remained poorly understood. This paper establishes sharp convergence rates for IRLS methods for constrained nuclear norm minimization in low-rank recovery. A central ingredient is a new majorization analysis for the smoothed nuclear norm: we prove that the harmonic-mean weight operator defines a valid global quadratic majorizer. Furthermore, we show that this weight operator is optimal within the family of power-mean weights, clarifying why it improves over classical one-sided reweighting schemes that use only row- or column-space information. Under a Schatten-1 null space property, we prove global linear convergence of IRLS algorithms using a variety of weight operators, including the harmonic-mean weights. For IRLS with harmonic-mean weights, we prove a dimension-independent, locally linear convergence rate. We provide a counterexample showing that this dimension-independent local rate cannot in general be obtained for IRLS algorithms using one-sided weight operators, which predominate in the literature. Numerical experiments corroborate the theoretical results and illustrate the practical advantage of harmonic-mean reweighting across square, rectangular, and adversarially initialized recovery problems.

cs.LG↗

arXiv2026

Hankel low-rank matrix approximation for gravitational-wave data analysis

Next-generation gravitational-wave (GW) detectors, such as the Laser Interferometer Space Antenna (LISA), will observe vast numbers of overlapping signals. Disentangling these signals from instrumental noise and from one another constitutes a significant data analysis challenge. We explore a denoising technique based on embedding time series into Hankel matrices: a superposition of $n$ (damped) sinusoids corresponds to a matrix of rank $2n$. Thus, the problem of signal extraction is reduced to a structured low-rank approximation problem. Using synthetic data tailored to GW applications, we benchmark three Hankel-based algorithms: ESPRIT, Cadzow iterations, and iteratively reweighted least squares (IRLS). Our test scenarios include isolated and multi-component monochromatic signals, the resolution of sources with closely spaced frequencies, and the recovery of black hole quasinormal modes (QNM). All three algorithms achieve near-optimal performance consistent with Fisher matrix bounds, evidenced by an inverse-square scaling of the mismatch with the signal-to-noise ratio. Furthermore, a proof-of-concept application to numerical relativity waveforms validates the ability of these algorithms to extract QNM frequencies from ringdown signals. Hankel low-rank approximation therefore offers a transparent, computationally efficient avenue for preprocessing GW time series.

astro-ph.HE↗

arXiv2026

Q3R: Quadratic Reweighted Rank Regularizer for Effective Low-Rank Training

Parameter-efficient training based on low-rank optimization has become a highly successful tool for fine-tuning large deep learning models. However, these methods often fail for low-rank pre-training, where simultaneously maintaining low-rank weight structure and optimizing the task objective remains challenging. We propose the $\textit{Quadratic Reweighted Rank Regularizer}$ ($\texttt{Q3R}$), which leads to a novel low-rank-inducing training strategy inspired by the Iteratively Reweighted Least Squares (IRLS) framework. $\texttt{Q3R}$ is based on a quadratic regularizer term that majorizes a smoothed log-determinant rank surrogate. Unlike other low-rank training techniques, $\texttt{Q3R}$ can train weight matrices to prescribed low target ranks while achieving predictive performance comparable to dense models, with small computational overhead and full compatibility with existing architectures. For example, we demonstrate a $\texttt{Q3R}$-regularized ViT-Tiny experiment where truncating the model to $60\%$ and $80\%$ of its parameters results in only minor absolute accuracy drops of $1.3\%$ and $4\%$, respectively, on CIFAR-10. We confirm the efficacy of $\texttt{Q3R}$ on Transformers across both vision and language tasks, including low-rank fine-tuning.

cs.LG↗

arXiv2026

Fibottention: Inceptive Visual Representation Learning with Diverse Attention Across Heads

Vision Transformers and their variants have achieved remarkable success in diverse visual perception tasks. Despite their effectiveness, they suffer from two significant limitations. First, the quadratic computational complexity of multi-head self-attention (MHSA), which restricts scalability to large token counts, and second, a high dependency on large-scale training data to attain competitive performance. In this paper, to address these challenges, we propose a novel sparse self-attention mechanism named Fibottention. Fibottention employs structured sparsity patterns derived from the Wythoff array, enabling an $\mathcal{O}(N \log N)$ computational complexity in self-attention. By design, its sparsity patterns vary across attention heads, which provably reduces redundant pairwise interactions while ensuring sufficient and diverse coverage. This leads to an \emph{inception-like functional diversity} in the attention heads, and promotes more informative and disentangled representations. We integrate Fibottention into standard Transformer architectures and conduct extensive experiments across multiple domains, including image classification, video understanding, and robot learning. Results demonstrate that models equipped with Fibottention either significantly outperform or achieve on-par performance with their dense MHSA counterparts, while leveraging only $2\%$ of all pairwise interactions across self-attention heads in typical settings, $2-6\%$ of the pairwise interactions in self-attention heads, resulting in substantial computational savings. Moreover, when compared to existing sparse attention mechanisms, Fibottention consistently achieves superior results on a FLOP-equivalency basis. Finally, we provide an in-depth analysis of the enhanced feature diversity resulting from our attention design and discuss its implications for efficient representation learning.

cs.CV↗

arXiv2026

Approximate Message Passing for Quantum State Tomography

Quantum state tomography (QST) is an indispensable tool for characterizing many-body quantum systems. However, due to the exponential scaling of the cost of the protocol with system size, many approaches have been developed for quantum states with specific structure, such as low-rank states. In this paper, we show how approximate message passing (AMP), an algorithmic framework for sparse signal recovery, can be used to perform low-rank QST. AMP provides asymptotically optimal performance guarantees for large sparse recovery problems, which suggests its utility for QST. We discuss the design challenges that come with applying AMP to QST, and show that by properly designing the AMP algorithm, we can reduce the reconstruction error by over an order of magnitude compared to existing approaches to low-rank QST. We also performed tomographic experiments on IBM Kingston and considered the effect of device noise on the reliability of the predicted fidelity of state preparation. Our work advances the state of low-rank QST and may be applicable to other quantum tomography protocols.

quant-ph↗

arXiv2025

Procrustes Problems on Random Matrices

Meaningful comparison between sets of observations often necessitates alignment or registration between them, and the resulting optimization problems range in complexity from those admitting simple closed-form solutions to those requiring advanced and novel techniques. We compare different Procrustes problems in which we align two sets of points after various perturbations by minimizing the norm of the difference between one matrix and an orthogonal transformation of the other. The minimization problem depends significantly on the choice of matrix norm; we highlight recent developments in nonsmooth Riemannian optimization and characterize which choices of norm work best for each perturbation. We show that in several applications, from low-dimensional alignments to hypothesis testing for random networks, when Procrustes alignment with the spectral or robust norm is the appropriate choice, it is often feasible to replace the computationally more expensive spectral and robust minimizers with their closed-form Frobenius-norm counterpart. Our work reinforces the synergy between optimization, geometry, and statistics.

stat.ME↗

arXiv2025

An Exposition of Pathfinding Strategies Within Lightning Network Clients

The Lightning Network is a peer-to-peer network designed to address Bitcoin's scalability challenges, facilitating rapid, cost-effective, and instantaneous transactions through bidirectional, blockchain-backed payment channels among network peers. Due to a source-based routing of payments, different pathfinding strategies are used in practice, trading off different objectives for each other such as payment reliability and routing fees. This paper explores differences within pathfinding strategies used by prominent Lightning Network node implementations, which include different underlying cost functions and different constraints, as well as different greedy algorithms of shortest path-type. Surprisingly, we observe that the pathfinding problems that most LN node implementations attempt to solve are NP-complete, and cannot be guaranteed to be optimally solved by the variants of Dijkstra's algorithm currently deployed in production. Through comparative analysis and simulations, we evaluate efficacy of different pathfinding strategies across metrics such as success rate, fees, path length, and timelock. Our experiments indicate that the strategies used by Eclair are advantageous in terms of payment reliability and result in paths with low fees. LND exhibits moderate success rates, while LDK results in paths with higher fee levels for smaller payment amounts; furthermore, CLN stands out for its minimal timelock paths. Additionally, we investigate the impact of Lightning node connectivity levels on routing efficiency. The findings of our analysis provide insights towards future improvements of pathfinding strategies and algorithms used within the Lightning Network.

cs.NI↗

arXiv2024

Sample-Efficient Geometry Reconstruction from Euclidean Distances using Non-Convex Optimization

The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this paper, we aim to solve this problem given a minimal number of distance samples. To this end, we leverage continuous and non-convex rank minimization formulations of the problem and establish a local convergence guarantee for a variant of iteratively reweighted least squares (IRLS), which applies if a minimal random set of observed distances is provided. As a technical tool, we establish a restricted isometry property (RIP) restricted to a tangent space of the manifold of symmetric rank-$r$ matrices given random Euclidean distance measurements, which might be of independent interest for the analysis of other non-convex approaches. Furthermore, we assess data efficiency, scalability and generalizability of different reconstruction algorithms through numerical experiments with simulated data as well as real-world data, demonstrating the proposed algorithm's ability to identify the underlying geometry from fewer distance samples compared to the state-of-the-art.

cs.LG↗

arXiv2024

Sparse Recovery for Overcomplete Frames: Sensing Matrices and Recovery Guarantees

Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core question is, by exploiting the intrinsic low dimensional structure of the signal, how to design the sensing process and decoder in a way that the number of measurements is essentially close to the complexity of the signal set. This chapter provides a survey of important results in answering this question, with an emphasis on a basis pursuit like convex optimization decoder that admits a wide range of random sensing matrices. The results are quite established in the case signals are sparse in an orthonormal basis, while the case with frame sparse signals is much less explored. In addition to presenting the latest results on recovery guarantee and how few random heavier-tailed measurements fulfill these recovery guarantees, this chapter also aims to provide some insights in proof techniques. We also take the opportunity of this book chapter to publish an interesting result (Theorem 3.10) about a restricted isometry like property related to a frame.

cs.IT↗

arXiv2024

UnitNorm: Rethinking Normalization for Transformers in Time Series

Normalization techniques are crucial for enhancing Transformer models' performance and stability in time series analysis tasks, yet traditional methods like batch and layer normalization often lead to issues such as token shift, attention shift, and sparse attention. We propose UnitNorm, a novel approach that scales input vectors by their norms and modulates attention patterns, effectively circumventing these challenges. Grounded in existing normalization frameworks, UnitNorm's effectiveness is demonstrated across diverse time series analysis tasks, including forecasting, classification, and anomaly detection, via a rigorous evaluation on 6 state-of-the-art models and 10 datasets. Notably, UnitNorm shows superior performance, especially in scenarios requiring robust attention mechanisms and contextual comprehension, evidenced by significant improvements by up to a 1.46 decrease in MSE for forecasting, and a 4.89% increase in accuracy for classification. This work not only calls for a reevaluation of normalization strategies in time series Transformers but also sets a new direction for enhancing model performance and stability. The source code is available at https://anonymous.4open.science/r/UnitNorm-5B84.

cs.LG↗

arXiv2024

Recovering Simultaneously Structured Data via Non-Convex Iteratively Reweighted Least Squares

We propose a new algorithm for the problem of recovering data that adheres to multiple, heterogeneous low-dimensional structures from linear observations. Focusing on data matrices that are simultaneously row-sparse and low-rank, we propose and analyze an iteratively reweighted least squares (IRLS) algorithm that is able to leverage both structures. In particular, it optimizes a combination of non-convex surrogates for row-sparsity and rank, a balancing of which is built into the algorithm. We prove locally quadratic convergence of the iterates to a simultaneously structured data matrix in a regime of minimal sample complexity (up to constants and a logarithmic factor), which is known to be impossible for a combination of convex surrogates. In experiments, we show that the IRLS method exhibits favorable empirical convergence, identifying simultaneously row-sparse and low-rank matrices from fewer measurements than state-of-the-art methods. Code is available at https://github.com/ckuemmerle/simirls.

cs.LG↗

arXiv2022

Learning Transition Operators From Sparse Space-Time Samples

We consider the nonlinear inverse problem of learning a transition operator $\mathbf{A}$ from partial observations at different times, in particular from sparse observations of entries of its powers $\mathbf{A},\mathbf{A}^2,\cdots,\mathbf{A}^{T}$. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if $\mathbf{A}$ is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than $\mathcal{O}(rn \log(nT))$ space-time samples are sufficient to ensure accurate recovery of a rank-$r$ operator $\mathbf{A}$ of size $n \times n$. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order $O(r n T)$ and per-iteration time complexity linear in $n$. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.

cs.IT↗

arXiv2022

Global Linear and Local Superlinear Convergence of IRLS for Non-Smooth Robust Regression

We advance both the theory and practice of robust $\ell_p$-quasinorm regression for $p \in (0,1]$ by using novel variants of iteratively reweighted least-squares (IRLS) to solve the underlying non-smooth problem. In the convex case, $p=1$, we prove that this IRLS variant converges globally at a linear rate under a mild, deterministic condition on the feature matrix called the \textit{stable range space property}. In the non-convex case, $p\in(0,1)$, we prove that under a similar condition, IRLS converges locally to the global minimizer at a superlinear rate of order $2-p$; the rate becomes quadratic as $p\to 0$. We showcase the proposed methods in three applications: real phase retrieval, regression without correspondences, and robust face restoration. The results show that (1) IRLS can handle a larger number of outliers than other methods, (2) it is faster than competing methods at the same level of accuracy, (3) it restores a sparsely corrupted face image with satisfactory visual quality. https://github.com/liangzu/IRLS-NeurIPS2022

math.OC↗

arXiv2021

Iteratively Reweighted Least Squares for Basis Pursuit with Global Linear Convergence Rate

The recovery of sparse data is at the core of many applications in machine learning and signal processing. While such problems can be tackled using $\ell_1$-regularization as in the LASSO estimator and in the Basis Pursuit approach, specialized algorithms are typically required to solve the corresponding high-dimensional non-smooth optimization for large instances. Iteratively Reweighted Least Squares (IRLS) is a widely used algorithm for this purpose due its excellent numerical performance. However, while existing theory is able to guarantee convergence of this algorithm to the minimizer, it does not provide a global convergence rate. In this paper, we prove that a variant of IRLS converges with a global linear rate to a sparse solution, i.e., with a linear error decrease occurring immediately from any initialization, if the measurements fulfill the usual null space property assumption. We support our theory by numerical experiments showing that our linear rate captures the correct dimension dependence. We anticipate that our theoretical findings will lead to new insights for many other use cases of the IRLS algorithm, such as in low-rank matrix recovery.

math.OC↗

arXiv2021

Dictionary-Sparse Recovery From Heavy-Tailed Measurements

The recovery of signals that are sparse not in a basis, but rather sparse with respect to an over-complete dictionary is one of the most flexible settings in the field of compressed sensing with numerous applications. As in the standard compressed sensing setting, it is possible that the signal can be reconstructed efficiently from few, linear measurements, for example by the so-called $\ell_1$-synthesis method. However, it has been less well-understood which measurement matrices provably work for this setting. Whereas in the standard setting, it has been shown that even certain heavy-tailed measurement matrices can be used in the same sample complexity regime as Gaussian matrices, comparable results are only available for the restrictive class of sub-Gaussian measurement vectors as far as the recovery of dictionary-sparse signals via $\ell_1$-synthesis is concerned. In this work, we fill this gap and establish optimal guarantees for the recovery of vectors that are (approximately) sparse with respect to a dictionary via the $\ell_1$-synthesis method from linear, potentially noisy measurements for a large class of random measurement matrices. In particular, we show that random measurements that fulfill only a small-ball assumption and a weak moment assumption, such as random vectors with i.i.d. Student-$t$ entries with a logarithmic number of degrees of freedom, lead to comparable guarantees as (sub-)Gaussian measurements. As a technical tool, we show a bound on the expectation of the sum of squared order statistics under very general assumptions, which might be of independent interest. As a corollary of our results, we also obtain a slight improvement on the weakest assumption on a measurement matrix with i.i.d. rows sufficient for uniform recovery in standard compressed sensing, improving on results by Lecué and Mendelson and Dirksen, Lecué and Rauhut.

cs.IT↗

arXiv2021

A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples

We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermore, we show that the linear systems to be solved are well-conditioned even for very ill-conditioned ground truth matrices. We provide extensive experiments, indicating that unlike many state-of-the-art approaches, our method is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples, while being competitive in its scalability.

math.OC↗

arXiv2020

On the robustness of noise-blind low-rank recovery from rank-one measurements

We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from underdetermined linear measurements. Our results are applicable for symmetric rank-one measurements as used in a formulation of the phase retrieval problem. We obtain these results by establishing that with high probability rank-one measurement operators defined by i.i.d. Gaussian vectors exhibit the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit ball. We complement our analysis by numerical experiments comparing the solutions of noise-blind and noise-aware formulations. These experiments confirm that noise-blind optimization methods exhibit comparable robustness to noise-aware formulations. Keywords: low-rank matrix recovery, phase retrieval, quotient property, noise-blind, robustness, nuclear norm minimization

cs.IT↗