Generative neural networks have been empirically found very promising in providing effective structural priors for compressed sensing, since they can be trained to span low-dimensional data manifolds in high-dimensional signal spaces. Despite the non-convexity of the resulting optimization problem, it has also been shown theoretically that, for neural networks with random Gaussian weights, a signal in the range of the network can be efficiently, approximately recovered from a few noisy measurements. However, a major bottleneck of these theoretical guarantees is a network \emph{expansivity} condition: that each layer of the neural network must be larger than the previous by a logarithmic factor. Our main contribution is to break this strong expansivity assumption, showing that \emph{constant} expansivity suffices to get efficient recovery algorithms, besides it also being information-theoretically necessary. To overcome the theoretical bottleneck in existing approaches we prove a novel uniform concentration theorem for random functions that might not be Lipschitz but satisfy a relaxed notion which we call ``pseudo-Lipschitzness.'' Using this theorem we can show that a matrix concentration inequality known as the \emph{Weight Distribution Condition (WDC)}, which was previously only known to hold for Gaussian matrices with logarithmic aspect ratio, in fact holds for constant aspect ratios too. Since WDC is a fundamental matrix concentration inequality in the heart of all existing theoretical guarantees on this problem, our tighter bound immediately yields improvements in all known results in the literature on compressed sensing with deep generative priors, including one-bit recovery, phase retrieval, and more.

Development of large-area, high-speed electronic skins is a grand challenge for robotics, prosthetics, and human-machine interfaces, but is fundamentally limited by wiring complexity and data bottlenecks. Here, we introduce Single-Pixel Tactile Skin (SPTS), a paradigm that uses compressive sampling to reconstruct rich tactile information from an entire sensor array via a single output channel. This is achieved through a direct circuit-level implementation where each sensing element, equipped with a miniature microcontroller, contributes a dynamically weighted analog signal to a global sum, performing distributed compressed sensing in hardware. Our flexible, daisy-chainable design simplifies wiring to a few input lines and one output, and significantly reduces measurement requirements compared to raster scanning methods. We demonstrate the system's performance by achieving object classification at an effective 3500 FPS and by capturing transient dynamics, resolving an 8 ms projectile impact into 23 frames. A key feature is the support for adaptive reconstruction, where sensing fidelity scales with measurement time. This allows for rapid contact localization using as little as 7% of total data, followed by progressive refinement to a high-fidelity image - a capability critical for responsive robotic systems. This work offers an efficient pathway towards large-scale tactile intelligence for robotics and human-machine interfaces.

In this section we prove Theorem 3.2. The second bound is by concentration of chi-squared with k degrees of freedom. Let x,y, x, y (3 /2) B\ (1/ 2)B . Let d = max( null x y null, null x y null). By Lemma D.4, we can take D = Cnull .

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Compressed sensing is a signal processing technique that allows for the reconstruction of a signal from a small set of measurements. The key idea behind compressed sensing is that many real-world signals are inherently sparse, meaning that they can be efficiently represented in a different space with only a few components compared to their original space representation. In this paper we will explore the mathematical formulation behind compressed sensing, its logic and pathologies, and apply compressed sensing to real world signals.

In this section we prove Theorem 3.2. The second bound is by concentration of chi-squared with k degrees of freedom. Let x,y, x, y (3 /2) B\ (1/ 2)B . Let d = max( null x y null, null x y null). By Lemma D.4, we can take D = Cnull .

In recent years, Compressed Sensing (CS) has gained significant interest as a technique for acquiring high-resolution sensory data using fewer measurements than traditional Nyquist sampling requires. At the same time, autonomous robotic platforms such as drones and rovers have become increasingly popular tools for remote sensing and environmental monitoring tasks, including measurements of temperature, humidity, and air quality. Within this context, this paper presents, to the best of our knowledge, the first investigation into how the structure of CS measurement matrices can be exploited to design optimized sampling trajectories for robotic environmental data collection. We propose a novel Monte Carlo optimization framework that generates measurement matrices designed to minimize both the robot's traversal path length and the signal reconstruction error within the CS framework. Central to our approach is the application of Dictionary Learning (DL) to obtain a data-driven sparsifying transform, which enhances reconstruction accuracy while further reducing the number of samples that the robot needs to collect. We demonstrate the effectiveness of our method through experiments reconstructing $NO_2$ pollution maps over the Gulf region. The results indicate that our approach can reduce robot travel distance to less than $10\%$ of a full-coverage path, while improving reconstruction accuracy by over a factor of five compared to traditional CS methods based on DCT and polynomial dictionaries, as well as by a factor of two compared to previously-proposed Informative Path Planning (IPP) methods.

The trend in modern science and technology is to take vector measurements rather than scalars, ruthlessly scaling to ever higher dimensional vectors. For about two decades now, traditional scalar Compressed Sensing has been synonymous with a Convex Optimization based procedure called Basis Pursuit. In the vector recovery case, the natural tendency is to return to a straightforward vector extension of Basis Pursuit, also based on Convex Optimization. However, Convex Optimization is provably suboptimal, particularly when $B$ is large. In this paper, we propose SteinSense, a lightweight iterative algorithm, which is provably optimal when $B$ is large. It does not have any tuning parameter, does not need any training data, requires zero knowledge of sparsity, is embarrassingly simple to implement, and all of this makes it easily scalable to high vector dimensions. We conduct a massive volume of both real and synthetic experiments that confirm the efficacy of SteinSense, and also provide theoretical justification based on ideas from Approximate Message Passing. Fascinatingly, we discover that SteinSense is quite robust, delivering the same quality of performance on real data, and even under substantial departures from conditions under which existing theory holds.

Summary and Contributions: This paper is about compressed sensing (CS) under generative priors. In such a problem, undersampled linear measurements of a signal of interest are provided, and the signal is sought. The mathematical ambiguity is resolved by finding the feasible point that is in the range of a trained generative model (such as a GAN), which is itself computed by solving an empirical risk minimization. Existing theory establishes a convergence guarantee of an efficient algorithm under an appropriate random model for the weights of the generative prior. The convergence guarantee assumes that the generative model is a multilayer perceptron where the width of each layer grows log-linearly.

In compressed sensing with a random multilayer ReLU neural network as prior, this paper shows that constant expansivity of the weight matrices of the neural network, as opposed to the "strong" expansivity (i.e., with a logarithmic factor) in existing studies, suffices for the existence of a gradient-descent based algorithm with a theoretical recovery guarantee (Theorem 1.1). To prove it, this paper introduced and utilized the novel notion of pseudo-Lipschitzness (Definition 4.2). This paper furthermore succeeded in obtaining several generalizations of Theorem 1.1, as stated informally in Theorem 1.2. The three reviewers rated this paper well above the acceptance threshold. They also agreed that the proof technique developed in this paper will have wider applicability, as well as that this paper is very clearly written.