Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map. As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles $\theta_1,\dots,\theta_m$), which models computed tomography. In the case when the unknown signal is $s$-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove exact recovery under the condition \[ m\gtrsim s, \] up to logarithmic factors.
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We introduce and study the online pause and resume problem. In this problem, a player attempts to find the $k$ lowest (alternatively, highest) prices in a sequence of fixed length $T$, which is revealed sequentially. At each time step, the player is presented with a price and decides whether to accept or reject it. The player incurs a switching cost whenever their decision changes in consecutive time steps, i.e., whenever they pause or resume purchasing. This online problem is motivated by the goal of carbon-aware load shifting, where a workload may be paused during periods of high carbon intensity and resumed during periods of low carbon intensity and incurs a cost when saving or restoring its state. It has strong connections to existing problems studied in the literature on online optimization, though it introduces unique technical challenges that prevent the direct application of existing algorithms. Extending prior work on threshold-based algorithms, we introduce double-threshold algorithms for both the minimization and maximization variants of this problem. We further show that the competitive ratios achieved by these algorithms are the best achievable by any deterministic online algorithm. Finally, we empirically validate our proposed algorithm through case studies on the application of carbon-aware load shifting using real carbon trace data and existing baseline algorithms.
Nonlinearity parameter tomography leads to the problem of identifying a coefficient in a nonlinear wave equation (such as the Westervelt equation) modeling ultrasound propagation. In this paper we transfer this into frequency domain, where the Westervelt equation gets replaced by a coupled system of Helmholtz equations with quadratic nonlinearities. For the case of the to-be-determined nonlinearity coefficient being a characteristic function of an unknown, not necessarily connected domain $D$, we devise and test a reconstruction algorithm based on weighted point source approximations combined with Newton's method. In a more abstract setting, convergence of a regularised Newton type method for this inverse problem is proven by verifying a range invariance condition of the forward operator and establishing injectivity of its linearisation.
Graph embedding, especially as a subgraph of a grid, is an old topic in VLSI design and graph drawing. In this paper, we investigate related questions concerning the complexity of embedding a graph $G$ in a host graph that is the strong product of a path $P$ with a graph $H$ that satisfies some properties, such as having small treewidth, pathwidth or tree depth. We show that this is NP-hard, even under numerous restrictions on both $G$ and $H$. In particular, computing the row pathwidth and the row treedepth is NP-hard even for a tree of small pathwidth, while computing the row treewidth is NP-hard even for series-parallel graphs.
We propose a data-assisted two-stage method for solving an inverse random source problem of the Helmholtz equation. In the first stage, the regularized Kaczmarz method is employed to generate initial approximations of the mean and variance based on the mild solution of the stochastic Helmholtz equation. A dataset is then obtained by sampling the approximate and corresponding true profiles from a certain a-priori criterion. The second stage is formulated as an image-to-image translation problem, and several data-assisted approaches are utilized to handle the dataset and obtain enhanced reconstructions. Numerical experiments demonstrate that the data-assisted two-stage method provides satisfactory reconstruction for both homogeneous and inhomogeneous media with fewer realizations.
Supersaturated designs, in which the number of factors exceeds the number of runs, are often constructed under a heuristic criterion that measures a design's proximity to an unattainable orthogonal design. Such a criterion does not directly measure a design's quality in terms of screening. To address this disconnect, we develop optimality criteria to maximize the lasso's sign recovery probability. The criteria have varying amounts of prior knowledge about the model's parameters. We show that an orthogonal design is an ideal structure when the signs of the active factors are unknown. When the signs are assumed known, we show that a design whose columns exhibit small, positive correlations are ideal. Such designs are sought after by the Var(s+)-criterion. These conclusions are based on a continuous optimization framework, which rigorously justifies the use of established heuristic criteria. From this justification, we propose a computationally-efficient design search algorithm that filters through optimal designs under different heuristic criteria to select the one that maximizes the sign recovery probability under the lasso.
This paper is concerned with a class of DC composite optimization problems which, as an extension of the convex composite optimization problem and the DC program with nonsmooth components, often arises from robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) which in each step computes an inexact minimizer of a strongly convex majorization constructed by the partial linearization of their objective functions. The generated iterate sequence is shown to be convergent under the Kurdyka-{\L}ojasiewicz (KL) property of a potential function, and the convergence admits a local R-linear rate if the potential function has the KL property of exponent $1/2$ at the limit point. For the latter assumption, we provide a verifiable condition by leveraging the composite structure, and clarify its relation with the regularity used for the convex composite optimization. Finally, the proposed iLPA is applied to a robust factorization model for matrix completions with outliers, DC programs with nonsmooth components, and $\ell_1$-norm exact penalty of DC constrained programs, and numerical comparison with the existing algorithms confirms the superiority of our iLPA in computing time and quality of solutions.
Minimum distance estimation methodology based on an empirical distribution function has been popular due to its desirable properties including robustness. Even though the statistical literature is awash with the research on the minimum distance estimation, the most of it is confined to the theoretical findings: only few statisticians conducted research on the application of the method to real world problems. Through this paper, we extend the domain of application of this methodology to various applied fields by providing a solution to a rather challenging and complicated computational problem. The problem this paper tackles is an image segmentation which has been used in various fields. We propose a novel method based on the classical minimum distance estimation theory to solve the image segmentation problem. The performance of the proposed method is then further elevated by integrating it with the ``segmenting-together" strategy. We demonstrate that the proposed method combined with the segmenting-together strategy successfully completes the segmentation problem when it is applied to the complex, real images such as magnetic resonance images.
In this paper, an efficient ensemble domain decomposition algorithm is proposed for fast solving the fully-mixed random Stokes-Darcy model with the physically realistic Beavers-Joseph (BJ) interface conditions. We utilize the Monte Carlo method for the coupled model with random inputs to derive some deterministic Stokes-Darcy numerical models and use the idea of the ensemble to realize the fast computation of multiple problems. One remarkable feature of the algorithm is that multiple linear systems share a common coefficient matrix in each deterministic numerical model, which significantly reduces the computational cost and achieves comparable accuracy with the traditional methods. Moreover, by domain decomposition, we can decouple the Stokes-Darcy system into two smaller sub-physics problems naturally. Both mesh-dependent and mesh-independent convergence rates of the algorithm are rigorously derived by choosing suitable Robin parameters. Optimized Robin parameters are derived and analyzed to accelerate the convergence of the proposed algorithm. Especially, for small hydraulic conductivity in practice, the almost optimal geometric convergence can be obtained by finite element discretization. Finally, two groups of numerical experiments are conducted to validate and illustrate the exclusive features of the proposed algorithm.
Many real-world dynamical systems are associated with first integrals (a.k.a. invariant quantities), which are quantities that remain unchanged over time. The discovery and understanding of first integrals are fundamental and important topics both in the natural sciences and in industrial applications. First integrals arise from the conservation laws of system energy, momentum, and mass, and from constraints on states; these are typically related to specific geometric structures of the governing equations. Existing neural networks designed to ensure such first integrals have shown excellent accuracy in modeling from data. However, these models incorporate the underlying structures, and in most situations where neural networks learn unknown systems, these structures are also unknown. This limitation needs to be overcome for scientific discovery and modeling of unknown systems. To this end, we propose first integral-preserving neural differential equation (FINDE). By leveraging the projection method and the discrete gradient method, FINDE finds and preserves first integrals from data, even in the absence of prior knowledge about underlying structures. Experimental results demonstrate that FINDE can predict future states of target systems much longer and find various quantities consistent with well-known first integrals in a unified manner.
This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.
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