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.

As emerging computing paradigms push beyond the limitations of traditional CMOS-based computing using Von Neumann architectures, there is a growing need to rethink and extend Electronic Design Automation (EDA) methodologies to support their unique characteristics. These paradigms—including Approximate Computing, In-Memory Computing, Reconfigurable Field-Effect Transistors (RFETs), and Photonic Computing—represent diverse and promising directions beyond conventional digital design. Collectively, they offer transformative potential for achieving significant improvements in energy efficiency, computational speed, and architectural scalability. For example, application-specific approximate computing enables the design of custom arithmetic circuits that exploit application-level error resilience, allowing for optimized accuracy–power–performance–area (PPA) trade-offs in error-tolerant applications. Similarly, processing-in-non-volatile memories, such as those based on Ferroelectric Field-effect Transistors (FeFETs), enhances energy efficiency by enabling analog computation—particularly for operations like matrix multiplication—directly within the memory arrays. The intrinsic polymorphism of RFETs supports compact, multifunctional logic gates and introduces new opportunities for circuit-level obfuscation and security-aware design. Likewise, photonic analog wavefront computing offers substantial gains in latency and energy efficiency by encoding and processing information in the analog optical domain, leveraging phenomena such as diffraction and interference to perform computation at the speed of light. However, they also introduce a host of new challenges in circuit and architecture design, such as vast and irregular design spaces, analog and non-Boolean behavior, and new device-level constraints that existing EDA tools are not capable of handling. To this end, the current article focuses on the development of efficient and robust EDA frameworks that can enable the practical realization of circuits and architectures in these emerging domains.

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.

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.

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.

We propose a new algorithm for the problem of recovering data that adheres to multiple, heterogenous low-dimensional structures from linear observations. Focussing 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.

Abstract This paper studies the problem of recovering a signal from one-bit compressed sensing measurements under a manifold model; that is, assuming that the signal lies on or near a manifold of low intrinsic dimension. We provide a convex recovery method based on the Geometric Multi-Resolution Analysis and prove recovery guarantees with a near-optimal scaling in the intrinsic manifold dimension. Our method is the first tractable algorithm with such guarantees for this setting. The results are complemented by numerical experiments confirming the validity of our approach.