多观测线性反问题稀疏解的新算法

基于Heavy-ball的最优阈值算法用于稀疏线性逆问题

Heavy-ball-based optimal thresholding algorithms for sparse linear inverse problems

本文针对稀疏线性逆问题中最优$k$-阈值化方法在大规模问题下运行时间较长的问题，提出了基于Heavy-ball的加速算法HBOT及其松弛变体，并在限制等距性质下证明了其收敛性，数值仿真验证了其在含噪信号与图像重建中的鲁棒性。

Continuous-Domain Solutions of Linear Inverse Problems with Tikhonov vs. Generalized TV Regularization

We consider linear inverse problems that are formulated in the continuous domain. The object of recovery is a function that is assumed to minimize a convex objective functional. The solutions are constrained by imposing a continuous-domain regularization. We derive the parametric form of the solution (representer theorems) for Tikhonov (quadratic) and generalized total-variation (gTV) regularizations. We show that, in both cases, the solutions are splines that are intimately related to the regularization operator. In the Tikhonov case, the solution is smooth and constrained to live in a fixed …

用于检测稀疏泄漏排放源反问题的稀疏传感器配置

Sparse Sensor Allocation for Inverse Problems of Detecting Sparse Leaking Emission Sources

针对稀疏泄漏排放源检测中经典贝叶斯反演的局限性，本文基于双层优化框架研究传感器稀疏最优配置，提出两种算法并验证其可降低估计误差，还提供了GPU加速的开源代码。

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Extracting information from nonlinear measurements is a fundamental challenge in data analysis. In this work, we consider separable inverse problems, where the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. These parameters may represent neuronal activity in a human brain, frequencies of electromagnetic waves, fluorescent probes in a cell, or magnetic relaxation times of biological tissues. Separable nonlinear inverse problems can be reformulated as underdetermined sparse-recovery problems, and solved using convex programming. T…

反问题的压缩感知与稀疏Radon变换的样本复杂度

Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform

本文为涉及任意前向算子的抽象反问题发展了无限维压缩感知的一般理论，并首次为建模计算机断层扫描的稀疏Radon变换提供了严格的恢复估计，证明了在平行束和扇形束设置下，当信号关于紧支撑小波基是$s$-稀疏时，仅需$m\gtrsim s$（受对数因子影响）的测量即可实现稳定恢复。

The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension

Non-convex methods for linear inverse problems with low-dimensional models have emerged as an alternative to convex techniques. We propose a theoretical framework where both finite dimensional and infinite dimensional linear inverse problems can be studied. We show how the size of the the basins of attraction of the minimizers of such problems is linked with the number of available measurements. This framework recovers known results about low-rank matrix estimation and off-the-grid sparse spike estimation, and it provides new results for Gaussian mixture estimation from linear measurements. ke…

反问题的压缩感知 II：在反卷积、源恢复和 MRI 中的应用

Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI

本文将面向稀疏 Radon 变换的无限维压缩感知样本复杂度理论推广到反卷积、椭圆 PDE 源恢复和 MRI 傅里叶采样三类反问题，并通过验证准对角化与相干性界建立恢复保证。

NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problems

Solving inverse problems is a fundamental component of science, engineering and mathematics. With the advent of deep learning, deep neural networks have significant potential to outperform existing state-of-the-art, model-based methods for solving inverse problems. However, it is known that current data-driven approaches face several key issues, notably hallucinations, instabilities and unpredictable generalization, with potential impact in critical tasks such as medical imaging. This raises the key question of whether or not one can construct deep neural networks for inverse problems with exp…

分数Laplacian的左逆与稀疏随机过程

Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes

分数Laplacian $(-\triangle)^{γ/2}$与欧氏空间$\RR^d$中的基本配位变换（伸缩、平移和旋转）互换，并与样条、分形和稳定Levy过程有着紧密的联系。对于$0<γ<d$，它的逆函数是经典的Riesz势$I_γ$，它是膨胀不变的，也是膨胀不变的.在这项工作中，我们调查的功能特性（连续性，衰减和可逆性）的扩展类的微分算子，共享这些不变性。特别地，我们将经典Riesz势$I_γ$的定义推广到任何大于$d$的非整数$γ$，并证明了它是膨胀不变和膨胀不变的分数拉普拉斯算子$(-\triangle)^{γ/2}$的唯一左逆.我们观察到，对于任何$1\le p\le \infty$和$γ\ge d(1-1/p)$，存在一个Schwartz函数$f$使得$I_γf$不是$p$可积的.然后，我们引入新的唯一的分数拉普拉斯算子$(-\triangle)^{γ/2}$的左逆$I_{γ, p}$的性质，$I_{γ, p}$是膨胀不变的（但不是膨胀不变的）和$I_{γ, p}f$是$p$ -可积的任何施瓦茨函数$f$。最后，我们应用线性算子$I_{γ, p}$和$p=1$求解了以白泊松噪声为驱动项的随机偏微分方程$(-\triangle)^{γ/2} Φ=w$。

A data-dependent weighted LASSO under Poisson noise

Sparse linear inverse problems appear in a variety of settings, but often the noise contaminating observations cannot accurately be described as bounded by or arising from a Gaussian distribution. Poisson observations in particular are a feature of several real-world applications. Previous work on sparse Poisson inverse problems encountered several limiting technical hurdles. This paper describes a novel alternative analysis approach for sparse Poisson inverse problems that (a) sidesteps the technical challenges in previous work, (b) admits estimators that can readily be computed using off-the…

Sparse image reconstruction on the sphere: a general approach with uncertainty quantification

Inverse problems defined naturally on the sphere are becoming increasingly of interest. In this article we provide a general framework for evaluation of inverse problems on the sphere, with a strong emphasis on flexibility and scalability. We consider flexibility with respect to the prior selection (regularization), the problem definition - specifically the problem formulation (constrained/unconstrained) and problem setting (analysis/synthesis) - and optimization adopted to solve the problem. We discuss and quantify the trade-offs between problem formulation and setting. Crucially, we consider…