arXiv subjects

Johannes Maly

Publications and source records attributed to Johannes Maly.

At least 19 recordsLinked to original sources

arXiv2026

Compressive Sensing - Introduction and Relations to Deep Learning

Compressive sensing predicts that sparse vectors (signals) can be recovered from a small number of linear measurements via efficient algorithms. This finding, which dates back two decades, has triggered a paradigm shift in signal processing and initiated many developments both in practical signal-processing applications, such as medical imaging, radar, and astronomy, and on the theoretical side. More recently, seminal connections to the field of deep learning have led to further advances in the field, such as the use of unrolled neural networks for sparse recovery and the discovery that common training algorithms (variants of gradient descent) favor sparsity in overparameterized scenarios -- the so-called implicit bias phenomenon. This article gives an introduction to compressive sensing and outlines connections to deep learning. In particular, we will discuss generalization for neural networks generated by unrolling sparse recovery algorithms and implicit regularization for gradient descent applied to learning simplified linear neural networks.

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arXiv2026

Efficient computation of the singular value decomposition with linear photonic circuits

In light of today's massive data processing, digital computers are reaching fundamental performance limits due to physical limitations and energy consumption. For specific applications, tailored analog systems offer promising alternatives to digital processors. In this work, we investigate the potential of linear photonic chips for accelerating the computation of the singular value decomposition (SVD) of a matrix. The SVD is a key primitive in linear algebra and forms a crucial component of various modern data processing algorithms. Our main insights are twofold: first, hybrid systems of digital controller and photonic chip asymptotically perform on par with large-scale CPU/GPU systems in terms of runtime. Second, such hybrid systems clearly outperform digital systems in terms of energy consumption.

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arXiv2026

How to induce regularization in linear models: A guide to reparametrizing gradient flow

In this work, we analyze the relation between reparametrizations of gradient flow and the induced implicit bias in linear models, which encompass various basic regression tasks. In particular, we study how reparametrization, loss function, and link function influence the convergence behavior and implicit bias of gradient flow. Our results provide conditions under which the implicit bias can be well-described and convergence of the flow is guaranteed. We furthermore show how to use these insights for designing reparametrization functions that lead to specific implicit biases which are closely connected to $\ell_p$- or trigonometric regularizers.

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arXiv2026

Conflicting Biases at the Edge of Stability: Norm versus Sharpness Regularization

The remarkable generalization properties of overparameterized networks are often attributed to implicit biases, such as norm minimization at small learning rates and low sharpness in the Edge-of-Stability regime. In this work, we argue that a comprehensive understanding of the generalization performance of gradient descent requires analyzing the interaction between these various forms of implicit regularization. We empirically demonstrate that the learning rate interpolates between low parameter norm and low sharpness of the trained model. We furthermore prove that neither implicit bias alone minimizes the generalization error for diagonal linear networks trained on a simple regression task. These findings demonstrate that focusing on a single implicit bias is insufficient to explain good generalization, and they motivate a broader view of implicit regularization that captures the dynamic trade-off between norm and sharpness induced by non-negligible learning rates.

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arXiv2026

Reliable one-bit quantization of bandlimited graph data via single-shot noise shaping

Graph data are ubiquitous in natural sciences and machine learning. In this paper, we consider the problem of quantizing graph structured, bandlimited data to few bits per entry while preserving its information under low-pass filtering. We propose an efficient single-shot noise shaping method that achieves state-of-the-art performance and comes with rigorous error bounds. In contrast to existing methods it allows reliable quantization to arbitrary bit-levels including the extreme case of using a single bit per data coefficient.

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arXiv2025

Subspace and DOA estimation under coarse quantization

We study direction-of-arrival (DOA) estimation from coarsely quantized data. We focus on a two-step approach which first estimates the signal subspace via covariance estimation and then extracts DOA angles by the ESPRIT algorithm. In particular, we analyze two stochastic quantization schemes which use dithering: a one-bit quantizer combined with rectangular dither and a multi-bit quantizer with triangular dither. For each quantizer, we derive rigorous high probability bounds for the distances between the true and estimated signal subspaces and DOA angles. Using our analysis, we identify scenarios in which subspace and DOA estimation via triangular dithering qualitatively outperforms rectangular dithering. We verify in numerical simulations that our estimates are optimal in their dependence on the smallest non-zero eigenvalue of the target matrix. The resulting subspace estimation guarantees are equally applicable in the analysis of other spectral estimation algorithms and related problems.

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arXiv2025

Get rid of your constraints and reparametrize: A study in NNLS and implicit bias

Over the past years, there has been significant interest in understanding the implicit bias of gradient descent optimization and its connection to the generalization properties of overparametrized neural networks. Several works observed that when training linear diagonal networks on the square loss for regression tasks (which corresponds to overparametrized linear regression) gradient descent converges to special solutions, e.g., non-negative ones. We connect this observation to Riemannian optimization and view overparametrized GD with identical initialization as a Riemannian GD. We use this fact for solving non-negative least squares (NNLS), an important problem behind many techniques, e.g., non-negative matrix factorization. We show that gradient flow on the reparametrized objective converges globally to NNLS solutions, providing convergence rates also for its discretized counterpart. Unlike previous methods, we do not rely on the calculation of exponential maps or geodesics. We further show accelerated convergence using a second-order ODE, lending itself to accelerated descent methods. Finally, we establish the stability against negative perturbations and discuss generalization to other constrained optimization problems.

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arXiv2025

More is Less: Inducing Sparsity via Overparameterization

In deep learning it is common to overparameterize neural networks, that is, to use more parameters than training samples. Quite surprisingly training the neural network via (stochastic) gradient descent leads to models that generalize very well, while classical statistics would suggest overfitting. In order to gain understanding of this implicit bias phenomenon we study the special case of sparse recovery (compressed sensing) which is of interest on its own. More precisely, in order to reconstruct a vector from underdetermined linear measurements, we introduce a corresponding overparameterized square loss functional, where the vector to be reconstructed is deeply factorized into several vectors. We show that, if there exists an exact solution, vanilla gradient flow for the overparameterized loss functional converges to a good approximation of the solution of minimal $\ell_1$-norm. The latter is well-known to promote sparse solutions. As a by-product, our results significantly improve the sample complexity for compressed sensing via gradient flow/descent on overparameterized models derived in previous works. The theory accurately predicts the recovery rate in numerical experiments. Our proof relies on analyzing a certain Bregman divergence of the flow. This bypasses the obstacles caused by non-convexity and should be of independent interest.

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arXiv2024

Plug-in Channel Estimation with Dithered Quantized Signals in Spatially Non-Stationary Massive MIMO Systems

As the array dimension of massive MIMO systems increases to unprecedented levels, two problems occur. First, the spatial stationarity assumption along the antenna elements is no longer valid. Second, the large array size results in an unacceptably high power consumption if high-resolution analog-to-digital converters are used. To address these two challenges, we consider a Bussgang linear minimum mean square error (BLMMSE)-based channel estimator for large scale massive MIMO systems with one-bit quantizers and a spatially non-stationary channel. Whereas other works usually assume that the channel covariance is known at the base station, we consider a plug-in BLMMSE estimator that uses an estimate of the channel covariance and rigorously analyze the distortion produced by using an estimated, rather than the true, covariance. To cope with the spatial non-stationarity, we introduce dithering into the quantized signals and provide a theoretical error analysis. In addition, we propose an angular domain fitting procedure which is based on solving an instance of non-negative least squares. For the multi-user data transmission phase, we further propose a BLMMSE-based receiver to handle one-bit quantized data signals. Our numerical results show that the performance of the proposed BLMMSE channel estimator is very close to the oracle-aided scheme with ideal knowledge of the channel covariance matrix. The BLMMSE receiver outperforms the conventional maximum-ratio-combining and zero-forcing receivers in terms of the resulting ergodic sum rate.

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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.

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arXiv2024

Tuning-free one-bit covariance estimation using data-driven dithering

We consider covariance estimation of any subgaussian distribution from finitely many i.i.d. samples that are quantized to one bit of information per entry. Recent work has shown that a reliable estimator can be constructed if uniformly distributed dithers on $[-λ,λ]$ are used in the one-bit quantizer. This estimator enjoys near-minimax optimal, non-asymptotic error estimates in the operator and Frobenius norms if $λ$ is chosen proportional to the largest variance of the distribution. However, this quantity is not known a-priori, and in practice $λ$ needs to be carefully tuned to achieve good performance. In this work we resolve this problem by introducing a tuning-free variant of this estimator, which replaces $λ$ by a data-driven quantity. We prove that this estimator satisfies the same non-asymptotic error estimates - up to small (logarithmic) losses and a slightly worse probability estimate. We also show that by using refined data-driven dithers that vary per entry of each sample, one can construct an estimator satisfying the same estimation error bound as the sample covariance of the samples before quantization -- again up logarithmic losses. Our proofs rely on a new version of the Burkholder-Rosenthal inequalities for matrix martingales, which is expected to be of independent interest.

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arXiv2023

Gradient Descent for Deep Matrix Factorization: Dynamics and Implicit Bias towards Low Rank

In deep learning, it is common to use more network parameters than training points. In such scenarioof over-parameterization, there are usually multiple networks that achieve zero training error so that thetraining algorithm induces an implicit bias on the computed solution. In practice, (stochastic) gradientdescent tends to prefer solutions which generalize well, which provides a possible explanation of thesuccess of deep learning. In this paper we analyze the dynamics of gradient descent in the simplifiedsetting of linear networks and of an estimation problem. Although we are not in an overparameterizedscenario, our analysis nevertheless provides insights into the phenomenon of implicit bias. In fact, wederive a rigorous analysis of the dynamics of vanilla gradient descent, and characterize the dynamicalconvergence of the spectrum. We are able to accurately locate time intervals where the effective rankof the iterates is close to the effective rank of a low-rank projection of the ground-truth matrix. Inpractice, those intervals can be used as criteria for early stopping if a certain regularity is desired. Wealso provide empirical evidence for implicit bias in more general scenarios, such as matrix sensing andrandom initialization. This suggests that deep learning prefers trajectories whose complexity (measuredin terms of effective rank) is monotonically increasing, which we believe is a fundamental concept for thetheoretical understanding of deep learning.

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arXiv2023

Robust Sensing of Low-Rank Matrices with Non-Orthogonal Sparse Decomposition

We consider the problem of recovering an unknown low-rank matrix X with (possibly) non-orthogonal, effectively sparse rank-1 decomposition from measurements y gathered in a linear measurement process A. We propose a variational formulation that lends itself to alternating minimization and whose global minimizers provably approximate X up to noise level. Working with a variant of robust injectivity, we derive reconstruction guarantees for various choices of A including sub-gaussian, Gaussian rank-1, and heavy-tailed measurements. Numerical experiments support the validity of our theoretical considerations.

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arXiv2023

A simple approach for quantizing neural networks

In this short note, we propose a new method for quantizing the weights of a fully trained neural network. A simple deterministic pre-processing step allows us to quantize network layers via memoryless scalar quantization while preserving the network performance on given training data. On one hand, the computational complexity of this pre-processing slightly exceeds that of state-of-the-art algorithms in the literature. On the other hand, our approach does not require any hyper-parameter tuning and, in contrast to previous methods, allows a plain analysis. We provide rigorous theoretical guarantees in the case of quantizing single network layers and show that the relative error decays with the number of parameters in the network if the training data behaves well, e.g., if it is sampled from suitable random distributions. The developed method also readily allows the quantization of deep networks by consecutive application to single layers.

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arXiv2022

Covariance estimation under one-bit quantization

We consider the classical problem of estimating the covariance matrix of a subgaussian distribution from i.i.d. samples in the novel context of coarse quantization, i.e., instead of having full knowledge of the samples, they are quantized to one or two bits per entry. This problem occurs naturally in signal processing applications. We introduce new estimators in two different quantization scenarios and derive non-asymptotic estimation error bounds in terms of the operator norm. In the first scenario we consider a simple, scale-invariant one-bit quantizer and derive an estimation result for the correlation matrix of a centered Gaussian distribution. In the second scenario, we add random dithering to the quantizer. In this case we can accurately estimate the full covariance matrix of a general subgaussian distribution by collecting two bits per entry of each sample. In both scenarios, our bounds apply to masked covariance estimation. We demonstrate the near-optimality of our error bounds by deriving corresponding (minimax) lower bounds and using numerical simulations.

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arXiv2021

Structural Sparsity in Multiple Measurements

We propose a novel sparsity model for distributed compressed sensing in the multiple measurement vectors (MMV) setting. Our model extends the concept of row-sparsity to allow more general types of structured sparsity arising in a variety of applications like, e.g., seismic exploration and non-destructive testing. To reconstruct structured data from observed measurements, we derive a non-convex but well-conditioned LASSO-type functional. By exploiting the convex-concave geometry of the functional, we design a projected gradient descent algorithm and show its effectiveness in extensive numerical simulations, both on toy and real data.

math.NA↗

arXiv2021

New challenges in covariance estimation: multiple structures and coarse quantization

In this self-contained chapter, we revisit a fundamental problem of multivariate statistics: estimating covariance matrices from finitely many independent samples. Based on massive Multiple-Input Multiple-Output (MIMO) systems we illustrate the necessity of leveraging structure and considering quantization of samples when estimating covariance matrices in practice. We then provide a selective survey of theoretical advances of the last decade focusing on the estimation of structured covariance matrices. This review is spiced up by some yet unpublished insights on how to benefit from combined structural constraints. Finally, we summarize the findings of our recently published preprint "Covariance estimation under one-bit quantization" to show how guaranteed covariance estimation is possible even under coarse quantization of the samples.

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arXiv2021

Quantized Compressed Sensing by Rectified Linear Units

This work is concerned with the problem of recovering high-dimensional signals $\mathbf{x} \in \mathbb{R}^n$ which belong to a convex set of low-complexity from a small number of quantized measurements. We propose to estimate the signals via a convex program based on rectified linear units (ReLUs) for two different quantization schemes, namely one-bit and uniform multi-bit quantization. Assuming that the linear measurement process can be modelled by a sensing matrix with i.i.d. subgaussian rows, we obtain for both schemes near-optimal uniform reconstruction guarantees by adding well-designed noise to the linear measurements prior to the quantization step. In the one-bit case, we show that the program is robust against adversarial bit corruptions as well as additive noise on the linear measurements. Further, our analysis quantifies precisely how the rate-distortion relationship of the program changes depending on whether we seek reconstruction accuracies above or below the noise floor. The proofs rely on recent results by Dirksen and Mendelson on non-Gaussian hyperplane tessellations. Finally, we complement our theoretical analysis with numerical experiments which compare our method to other state-of-the-art methodologies.

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