Erroneous correspondences between samples and their respective channel or target commonly arise in several real-world applications. For instance, whole-brain calcium imaging of freely moving organisms, multiple target tracking or multi-person contactless vital sign monitoring may be severely affected by mismatched sample-channel assignments. To systematically address this fundamental problem, we pose it as a signal reconstruction problem where we have lost correspondences between the samples and their respective channels. We show that under the assumption that the signals of interest admit a sparse representation over an overcomplete dictionary, unique signal recovery is possible. Our derivations reveal that the problem is equivalent to a structured unlabeled sensing problem without precise knowledge of the sensing matrix. Unfortunately, existing methods are neither robust to errors in the regressors nor do they exploit the structure of the problem. Therefore, we propose a novel robust two-step approach for the reconstruction of shuffled sparse signals. The performance and robustness of the proposed approach is illustrated in an application of whole-brain calcium imaging in computational neuroscience. The proposed framework can be generalized to sparse signal representations other than the ones considered in this work to be applied in a variety of real-world problems with imprecise measurement or channel assignment.

为了解决逆问题，已经开发了插件（PNP）方法，可以用呼叫特定于应用程序的DeNoiser在凸优化算法中替换近端步骤，该算法通常使用深神经网络（DNN）实现。尽管这种方法已经成功，但可以改进它们。例如，Denoiser通常经过设计/训练以消除白色高斯噪声，但是PNP算法中的DINOISER输入误差通常远非白色或高斯。近似消息传递（AMP）方法提供了白色和高斯DEOISER输入误差，但仅当正向操作员是一个大的随机矩阵时。在这项工作中，对于基于傅立叶的远期运营商，我们提出了一种基于普遍期望一致性（GEC）近似的PNP算法 - AMP的紧密表弟 - 在每次迭代时提供可预测的错误统计信息，以及新的DNN利用这些统计数据的Denoiser。我们将方法应用于磁共振成像（MRI）图像恢复，并证明其优于现有的PNP和AMP方法。

We consider the nonlinear inverse problem of learning a transition operator $\mathbf{A}$ from partial observations at different times, in particular from sparse observations of entries of its powers $\mathbf{A},\mathbf{A}^2,\cdots,\mathbf{A}^{T}$. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if $\mathbf{A}$ is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than $\mathcal{O}(rn \log(nT))$ space-time samples are sufficient to ensure accurate recovery of a rank-$r$ operator $\mathbf{A}$ of size $n \times n$. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order $O(r n T)$ and per-iteration time complexity linear in $n$. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.

在许多工程应用中，例如雷达/声纳/超声成像等许多工程应用中，稀疏多通道盲卷（S-MBD）的问题经常出现。为了降低其计算和实施成本，我们提出了一种压缩方法，该方法可以及时从更少的测量值中进行盲目恢复。提出的压缩通过过滤器随后进行亚采样来测量信号，从而大大降低了实施成本。我们得出理论保证，可从压缩测量中识别和回收稀疏过滤器。我们的结果允许设计广泛的压缩过滤器。然后，我们提出了一个由数据驱动的展开的学习框架，以学习压缩过滤器并解决S-MBD问题。编码器是一个经常性的推理网络，该网络将压缩测量结果映射到稀疏过滤器的估计值中。我们证明，与基于优化的方法相比，我们展开的学习方法对源形状的选择更为强大，并且具有更好的恢复性能。最后，在具有有限数据的应用程序（少数图）的应用中，我们强调了与传统深度学习相比，展开学习的卓越概括能力。

神经影像动物和超越的几个问题需要对多任务稀疏分层回归模型参数的推断。示例包括M / EEG逆问题，用于基于任务的FMRI分析的神经编码模型，以及气候或CPU和GPU的温度监测。在这些域中，要推断的模型参数和测量噪声都可以表现出复杂的时空结构。现有工作要么忽略时间结构，要么导致计算苛刻的推论方案。克服这些限制，我们设计了一种新颖的柔性等级贝叶斯框架，其中模型参数和噪声的时空动态被建模为具有Kronecker产品协方差结构。我们的框架中的推断是基于大大化最小化优化，并有保证的收敛属性。我们高效的算法利用了时间自传矩阵的内在riemannian几何学。对于Toeplitz矩阵描述的静止动力学，采用了循环嵌入的理论。我们证明了Convex边界属性并导出了结果算法的更新规则。在来自M / EEG的合成和真实神经数据上，我们证明了我们的方法导致性能提高。

假设我们在$ \ mathbb {r} ^ d $和predictor x中的响应变量y在$ \ mathbb {r} ^ d $，以便为$ d \ geq 1 $。在置换或未解释的回归中，我们可以访问x和y上的单独无序数据，而不是在通常回归中的（x，y）-pabes上的数据。到目前为止，在文献中，案件$ d = 1 $已收到关注，请参阅例如近期的纸张和杂草[信息和推理，8,619--717]和Balabdaoui等人。 [J.马赫。学习。 res，22（172），1-60]。在本文中，我们考虑使用$ d \ geq 1 $的一般多变量设置。我们表明回归函数的周期性单调性的概念足以用于置换/未解释的回归模型中的识别和估计。我们在允许的回归设置中研究置换恢复，并在基于Kiefer-WolfoItz的基于代索的计算高效且易用算法[ANN。数学。统计部。，27,887--906]非参数最大似然估计和来自最佳运输理论的技术。我们在高斯噪声的相关均方方向误差误差上提供显式上限。与之前的案件的工作$ d = 1 $一样，置换/未解释的设置涉及潜在的解卷积问题的慢速（对数）收敛率。数值研究证实了我们的理论分析，并表明所提出的方法至少根据上述事先工作中的方法进行了比例，同时在计算复杂性方面取得了大量减少。

我们研究了趋势过滤的多元版本，称为Kronecker趋势过滤或KTF，因为设计点以$ D $维度形成格子。 KTF是单变量趋势过滤的自然延伸（Steidl等，2006; Kim等人，2009; Tibshirani，2014），并通过最大限度地减少惩罚最小二乘问题，其罚款术语总和绝对（高阶）沿每个坐标方向估计参数的差异。相应的惩罚运算符可以编写单次趋势过滤惩罚运营商的Kronecker产品，因此名称Kronecker趋势过滤。等效，可以在$ \ ell_1 $ -penalized基础回归问题上查看KTF，其中基本功能是下降阶段函数的张量产品，是一个分段多项式（离散样条）基础，基于单变量趋势过滤。本文是Sadhanala等人的统一和延伸结果。 （2016,2017）。我们开发了一套完整的理论结果，描述了$ k \ grone 0 $和$ d \ geq 1 $的$ k ^ {\ mathrm {th}} $ over kronecker趋势过滤的行为。这揭示了许多有趣的现象，包括KTF在估计异构平滑的功能时KTF的优势，并且在$ d = 2（k + 1）$的相位过渡，一个边界过去（在高维对 - 光滑侧）线性泡沫不能完全保持一致。我们还利用Tibshirani（2020）的离散花键来利用最近的结果，特别是离散的花键插值结果，使我们能够将KTF估计扩展到恒定时间内的任何偏离晶格位置（与晶格数量的大小无关）。

新磁共振（MR）成像方式可以量化血流动力学，但需要长时间的采集时间，妨碍其广泛用于早期诊断心血管疾病。为了减少采集​​时间，常规使用来自未采样测量的重建方法，使得利用旨在提高图像可压缩性的表示。重建的解剖和血液动力学图像可能存在视觉伪影。尽管这些工件中的一些基本上是重建错误，因此欠采样的后果，其他人可能是由于测量噪声或采样频率的随机选择。另有说明，重建的图像变为随机变量，并且其偏差和其协方差都可以导致视觉伪影;后者会导致可能误解的空间相关性以用于视觉信息。虽然前者的性质已经在文献中已经研究过，但后者尚未得到关注。在这项研究中，我们研究了从重建过程产生的随机扰动的理论特性，并对模拟和主动脉瘤进行了许多数值实验。我们的结果表明，当基于$ \ ell_1 $ -norm最小化的高斯欠采样模式与恢复算法组合时，相关长度保持限制为2到三个像素。然而，对于其他欠采样模式，相关长度可以显着增加，较高的欠采样因子（即8倍或16倍压缩）和不同的重建方法。

Outier-bubust估计是一个基本问题，已由统计学家和从业人员进行了广泛的研究。在过去的几年中，整个研究领域的融合都倾向于“算法稳定统计”，该统计数据的重点是开发可拖动的异常体 - 固定技术来解决高维估计问题。尽管存在这种融合，但跨领域的研究工作主要彼此断开。本文桥接了有关可认证的异常抗衡器估计的最新工作，该估计是机器人技术和计算机视觉中的几何感知，并在健壮的统计数据中并行工作。特别是，我们适应并扩展了最新结果对可靠的线性回归（适用于<< 50％异常值的低外壳案例）和列表可解码的回归（适用于>> 50％异常值的高淘汰案例）在机器人和视觉中通常发现的设置，其中（i）变量（例如旋转，姿势）属于非convex域，（ii）测量值是矢量值，并且（iii）未知的异常值是先验的。这里的重点是绩效保证：我们没有提出新算法，而是为投入测量提供条件，在该输入测量值下，保证现代估计算法可以在存在异常值的情况下恢复接近地面真相的估计值。这些条件是我们所谓的“估计合同”。除了现有结果的拟议扩展外，我们认为本文的主要贡献是（i）通过指出共同点和差异来统一平行的研究行，（ii）在介绍先进材料（例如，证明总和证明）中的统一行为。对从业者的可访问和独立的演讲，（iii）指出一些即时的机会和开放问题，以发出异常的几何感知。

Discriminative features extracted from the sparse coding model have been shown to perform well for classification. Recent deep learning architectures have further improved reconstruction in inverse problems by considering new dense priors learned from data. We propose a novel dense and sparse coding model that integrates both representation capability and discriminative features. The model studies the problem of recovering a dense vector $\mathbf{x}$ and a sparse vector $\mathbf{u}$ given measurements of the form $\mathbf{y} = \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$. Our first analysis proposes a geometric condition based on the minimal angle between spanning subspaces corresponding to the matrices $\mathbf{A}$ and $\mathbf{B}$ that guarantees unique solution to the model. The second analysis shows that, under mild assumptions, a convex program recovers the dense and sparse components. We validate the effectiveness of the model on simulated data and propose a dense and sparse autoencoder (DenSaE) tailored to learning the dictionaries from the dense and sparse model. We demonstrate that (i) DenSaE denoises natural images better than architectures derived from the sparse coding model ($\mathbf{B}\mathbf{u}$), (ii) in the presence of noise, training the biases in the latter amounts to implicitly learning the $\mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$ model, (iii) $\mathbf{A}$ and $\mathbf{B}$ capture low- and high-frequency contents, respectively, and (iv) compared to the sparse coding model, DenSaE offers a balance between discriminative power and representation.

最近，刘和张研究了从压缩传感的角度研究了时间序列预测的相当具有挑战性的问题。他们提出了一个没有学习的方法，名为卷积核规范最小化（CNNM），并证明了CNNM可以完全从其观察到的部分恢复一系列系列的部分，只要该系列是卷积的低级。虽然令人印象深刻，但是每当系列远离季节性时可能不满足卷积的低秩条件，并且实际上是脆弱的趋势和动态的存在。本文试图通过将学习，正常的转换集成到CNNM中，以便将一系列渐开线结构转换为卷积低等级的常规信号的目的。我们证明，由于系列的变换是卷积低级的转换，所以，所产生的模型是基于学习的基于学习的CNNM（LBCNM），严格成功地识别了一个系列的未来部分。为了学习可能符合所需成功条件的适当转换，我们设计了一种基于主成分追求（PCP）的可解释方法。配备了这种学习方法和一些精心设计的数据论证技巧，LBCNM不仅可以处理时间序列的主要组成部分（包括趋势，季节性和动态），还可以利用其他一些预测方法提供的预测;这意味着LBCNNM可以用作模型组合的一般工具。从时间序列数据库（TSDL）和M4竞争（M4）的100,452个现实世界时间序列的大量实验证明了LBCNNM的卓越性能。

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets.This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed-either explicitly or implicitly-to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis.The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k)) floating-point operations (flops) in contrast with O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multi-processor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

近年来，在诸如denoing，压缩感应，介入和超分辨率等反问题中使用深度学习方法的使用取得了重大进展。尽管这种作品主要是由实践算法和实验驱动的，但它也引起了各种有趣的理论问题。在本文中，我们调查了这一作品中一些突出的理论发展，尤其是生成先验，未经训练的神经网络先验和展开算法。除了总结这些主题中的现有结果外，我们还强调了一些持续的挑战和开放问题。

许多现代数据集，从神经影像和地统计数据等领域都以张量数据的随机样本的形式来说，这可以被理解为对光滑的多维随机功能的嘈杂观察。来自功能数据分析的大多数传统技术被维度的诅咒困扰，并且随着域的尺寸增加而迅速变得棘手。在本文中，我们提出了一种学习从多维功能数据样本的持续陈述的框架，这些功能是免受诅咒的几种表现形式的。这些表示由一组可分离的基函数构造，该函数被定义为最佳地适应数据。我们表明，通过仔细定义的数据的仔细定义的减少转换的张测仪分解可以有效地解决所得到的估计问题。使用基于差分运算符的惩罚，并入粗糙的正则化。也建立了相关的理论性质。在模拟研究中证明了我们对竞争方法的方法的优点。我们在神经影像动物中得出真正的数据应用。

由于其快速和低功率配置，可重新配置的智能表面（RISS）最近被视为未来无线网络的节能解决方案，这在实现大规模连通性和低延迟通信方面具有增加的潜力。基于RIS的系统中的准确且低空的通道估计是通常的RIS单元元素及其独特的硬件约束，这是最关键的挑战之一。在本文中，我们专注于RIS授权的多用户多用户多输入单输出（MISO）上行链路通信系统的上行链路，并根据并行因子分解提出了一个通道估计框架，以展开所得的级联通道模型。我们为基站和RIS之间的渠道以及RIS与用户之间的渠道提供了两种迭代估计算法。一个基于交替的最小二乘（ALS），而另一个使用向量近似消息传递到迭代的迭代中，从估计的向量重建了两个未知的通道。为了从理论上评估基于ALS的算法的性能，我们得出了其估计值CRAM \'ER-RAO BOND（CRB）。我们还通过估计的通道和基本站的不同预码方案讨论了可实现的总和率计算。我们的广泛仿真结果表明，我们的算法表现优于基准方案，并且ALS技术可实现CRB。还证明，使用估计通道的总和率总是在各种设置下达到完美通道的总和，从而验证了提出的估计算法的有效性和鲁棒性。

This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the 1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.

In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models which includes the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that in general the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level is above some critical threshold and the nonlinearities exhibit sufficiently fast growth. We derive explicit bounds on the critical noise threshold in the case of Gaussian white noise and provide an explicit characterization of these spurious terms in the case of trigonometric and/or polynomial model nonlinearities. However, a silver lining to this negative result is that if the data is suitably denoised (a simple moving average filter is sufficient), then we recover unconditional asymptotic consistency on the class of models with locally-Lipschitz nonlinearities. Altogether, our results reveal several important aspects of weak-form equation learning which may be used to improve future algorithms. We demonstrate our results numerically using the Lorenz system, the cubic oscillator, a viscous Burgers growth model, and a Kuramoto-Sivashinsky-type higher-order PDE.

我们调查与高斯的混合的数据分享共同但未知，潜在虐待协方差矩阵的数据。我们首先考虑具有两个等级大小的组件的高斯混合，并根据最大似然估计导出最大切割整数程序。当样品的数量在维度下线性增长时，我们证明其解决方案实现了最佳的错误分类率，直到对数因子。但是，解决最大切割问题似乎是在计算上棘手的。为了克服这一点，我们开发了一种高效的频谱算法，该算法达到最佳速率，但需要一种二次样本量。虽然这种样本复杂性比最大切割问题更差，但我们猜测没有多项式方法可以更好地执行。此外，我们收集了支持统计计算差距存在的数值和理论证据。最后，我们将MAX-CUT程序概括为$ k $ -means程序，该程序处理多组分混合物的可能性不平等。它享有相似的最优性保证，用于满足运输成本不平等的分布式的混合物，包括高斯和强烈的对数的分布。

Countless signal processing applications include the reconstruction of signals from few indirect linear measurements. The design of effective measurement operators is typically constrained by the underlying hardware and physics, posing a challenging and often even discrete optimization task. While the potential of gradient-based learning via the unrolling of iterative recovery algorithms has been demonstrated, it has remained unclear how to leverage this technique when the set of admissible measurement operators is structured and discrete. We tackle this problem by combining unrolled optimization with Gumbel reparametrizations, which enable the computation of low-variance gradient estimates of categorical random variables. Our approach is formalized by GLODISMO (Gradient-based Learning of DIscrete Structured Measurement Operators). This novel method is easy-to-implement, computationally efficient, and extendable due to its compatibility with automatic differentiation. We empirically demonstrate the performance and flexibility of GLODISMO in several prototypical signal recovery applications, verifying that the learned measurement matrices outperform conventional designs based on randomization as well as discrete optimization baselines.

约束的张量和矩阵分子化模型允许从多道数据中提取可解释模式。因此，对于受约束的低秩近似度的可识别性特性和有效算法是如此重要的研究主题。这项工作涉及低秩近似的因子矩阵的列，以众所周知的和可能的过度顺序稀疏，该模型包括基于字典的低秩近似（DLRA）。虽然早期的贡献集中在候选列字典内的发现因子列，即一稀疏的近似值，这项工作是第一个以大于1的稀疏性解决DLRA。我建议专注于稀疏编码的子问题，在解决DLRA时出现的混合稀疏编码（MSC）以交替的优化策略在解决DLRA时出现。提供了基于稀疏编码启发式的几种算法（贪婪方法，凸起放松）以解决MSC。在模拟数据上评估这些启发式的性能。然后，我展示了如何基于套索来调整一个有效的MSC求解器，以计算高光谱图像处理和化学测量学的背景下的基于词典的基于矩阵分解和规范的多adic分解。这些实验表明，DLRA扩展了低秩近似的建模能力，有助于降低估计方差并提高估计因子的可识别性和可解释性。