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This paper studies the robust Hankel recovery problem, which simultaneously removes the sparse outliers and fulfills missing entries from the partial observation. We propose a novel non-convex algorithm, coined Hankel structured Newton-like descent (HSNLD), to tackle the robust Hankel recovery problem. HSNLD is highly efficient with linear convergence, and its convergence rate is independent of the condition number of the underlying Hankel matrix. The recovery guarantee has been established under some mild conditions. Numerical experiments on both synthetic and real datasets show the superior performance of HSNLD against state-of-the-art algorithms. 

In this paper, we focus on a matrix factorization-based approach for robust recovery of low-rank asymmetric matrices from corrupted measurements. We propose an Overparameterized Preconditioned Subgradient Algorithm (OPSA) and provide, for the first time in the literature, linear convergence rates independent of the rank of the sought asymmetric matrix in the presence of gross corruptions. Our work goes beyond existing results in preconditioned-type approaches addressing their current limitation, i.e., the lack of convergence guarantees in the case of asymmetric matrices of unknown rank. By applying our approach to (robust) matrix sensing, we highlight its merits when the measurement operator satisfies a mixed-norm restricted isometry property. Lastly, we present extensive numerical experiments that validate our theoretical results and demonstrate the effectiveness of our approach for different levels of overparameterization and corruption from outliers. 

Tensor dimensionality reduction is one of the fundamental tools for modern data science. To address the high computational overhead, fiber-wise sampled subtensors that preserve the original tensor rank are often used in designing efficient and scalable tensor dimensionality reduction. However, the theory of property inheritance for subtensors is still underdevelopment, that is, how the essential properties of the original tensor will be passed to its subtensors. This paper theoretically studies the property inheritance of the two key tensor properties, namely incoherence and condition number, under the tensor train setting. We also show how tensor train rank is preserved through fiber-wise sampling. The key parameters introduced in theorems are numerically evaluated under various settings. The results show that the properties of interest can be well preserved to the subtensors formed via fiber-wise sampling. Overall, this paper provides several handy analytic tools for developing efficient tensor analysis methods. 

Spatiotemporal systems are ubiquitous in a large number of scientific areas, representing underlying knowledge and patterns in the data. Here, a fundamental question usually arises as how to understand and characterize these spatiotemporal systems with a certain data-driven machine learning framework. In this work, we introduce an unsupervised pattern discovery framework, namely, dynamic autoregressive tensor factorization. Our framework is essentially built on the fact that the spatiotemporal systems can be well described by the time-varying autoregression on multivariate or even multidimensional data. In the modeling process, tensor factorization is seamlessly integrated into the time-varying autoregression for discovering spatial and temporal modes/patterns from the spatiotemporal systems in which the spatial factor matrix is assumed to be orthogonal. To evaluate the framework, we apply it to several real-world spatiotemporal datasets, including fluid flow dynamics, international import/export merchandise trade, and urban human mobility. On the international trade dataset with dimensions {country/region, product type, year}, our framework can produce interpretable import/export patterns of countries/regions, while the low-dimensional product patterns are also important for classifying import/export merchandise and understanding systematical differences between import and export. On the ridesharing mobility dataset with dimensions {origin, destination, time}, our framework is helpful for identifying the shift of spatial patterns of urban human mobility that changed between 2019 and 2022. Empirical experiments demonstrate that our framework can discover interpretable and meaningful patterns from the spatiotemporal systems that are both time-varying and multidimensional. 

This study addresses the problem of convolutional kernel learning in univariate, multivariate, and multidimensional time series data, which is crucial for interpreting temporal patterns in time series and supporting downstream machine learning tasks. First, we propose formulating convolutional kernel learning for univariate time series as a sparse regression problem with a non-negative constraint, leveraging the properties of circular convolution and circulant matrices. Second, to generalize this approach to multivariate and multidimensional time series data, we use tensor computations, reformulating the convolutional kernel learning problem in the form of tensors. This is further converted into a standard sparse regression problem through vectorization and tensor unfolding operations. In the proposed methodology, the optimization problem is addressed using the existing non-negative subspace pursuit method, enabling the convolutional kernel to capture temporal correlations and patterns. To evaluate the proposed model, we apply it to several real-world time series datasets. On the multidimensional ridesharing and taxi trip data from New York City and Chicago, the convolutional kernels reveal interpretable local correlations and cyclical patterns, such as weekly seasonality. For the monthly temperature time series data in North America, the proposed model can quantify the yearly seasonality and make it comparable across different decades. In the context of multidimensional fluid flow data, both local and nonlocal correlations captured by the convolutional kernels can reinforce tensor factorization, leading to performance improvements in fluid flow reconstruction tasks. Thus, this study lays an insightful foundation for automatically learning convolutional kernels from time series data, with an emphasis on interpretability through sparsity and non-negativity constraints. 

The paper discusses derivative-free optimization (DFO), which involves minimizing a function without access to gradients or directional derivatives, only function evaluations. Classical DFO methods such as Nelder-Mead and direct search have limited scalability for high-dimensional problems. Zeroth-order methods, which mimic gradient-based methods, have been gaining popularity due to the demands of large-scale machine learning applications. This paper focuses on the selection of the step size $$\alpha_k$$ in such methods. The proposed approach, called Curvature-Aware Random Search (CARS), uses first- and second-order finite difference approximations to compute a candidate $$\alpha_+$$. A safeguarding step then evaluates $$\alpha_+$$ and chooses an alternate step size in case $$\alpha_+$$ does not decrease the objective function. We prove that for strongly convex objective functions, CARS converges linearly provided that the search direction is drawn from a distribution satisfying very mild conditions. We also present a Cubic Regularized variant of CARS, named CARS-CR, which provably converges at a rate of $O(1/k)$ without the assumption of strong convexity. Numerical experiments show that CARS and CARS-CR match or exceed the state-of-the-art on benchmark problem sets. 

Traditional adversarial attacks typically aim to alter the predicted labels of input images by generating perturbations that are imperceptible to the human eye. However, these approaches often lack explainability. Moreover, most existing work on adversarial attacks focuses on single-stage classifiers, but multi-stage classifiers are largely unexplored. In this paper, we introduce instance-based adversarial attacks for multi-stage classifiers, leveraging Layer-wise Relevance Propagation (LRP), which assigns relevance scores to pixels based on their influence on classification outcomes. Our approach generates explainable adversarial perturbations by utilizing LRP to identify and target key features critical for both coarse and fine-grained classifications. Unlike conventional attacks, our method not only induces misclassification but also enhances the interpretability of the model’s behavior across classification stages, as demonstrated by experimental results. 

This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nystro m method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers. 

Deep unfolding networks have gained increasing attention in the field of compressed sensing (CS) owing to their theoretical interpretability and superior reconstruction performance. However, most existing deep unfolding methods often face the following issues: (1) they learn directly from single-channel images, leading to a simple feature representation that does not fully capture complex features; and (2) they treat various image components uniformly, ignoring the characteristics of different components. To address these issues, we propose a novel wavelet-domain deep unfolding framework named WTDUN, which operates directly on the multi-scale wavelet sub-bands. Our method utilizes the intrinsic sparsity and multi-scale structure of wavelet coefficients to achieve a tree-structured sampling and reconstruction, effectively capturing and highlighting the most important features within images. Specifically, the design of tree-structured reconstruction aims to capture the inter-dependencies among the multi-scale sub-bands, enabling the identification of both fine and coarse features, which can lead to a marked improvement in reconstruction quality. Furthermore, a wavelet domain adaptive sampling method is proposed to greatly improve the sampling capability, which is realized by assigning measurements to each wavelet sub-band based on its importance. Unlike pure deep learning methods that treat all components uniformly, our method introduces a targeted focus on important sub-bands, considering their energy and sparsity. This targeted strategy lets us capture key information more efficiently while discarding less important information, resulting in a more effective and detailed reconstruction. Extensive experimental results on various datasets validate the superior performance of our proposed method. 

The dynamical sampling problem is centered around reconstructing signals that evolve over time according to a dynamical process, from spatial-temporal samples that may be noisy. This topic has been thoroughly explored for one-dimensional signals. Multidimensional signal recovery has also been studied, but primarily in scenarios where the driving operator is a convolution operator. In this work, we shift our focus to the dynamical sampling problem in the context of three-dimensional signal recovery, where the evolution system can be characterized by tensor products. Specifically, we provide a necessary condition for the sampling set that ensures successful recovery of the three-dimensional signal. Furthermore, we reformulate the reconstruction problem as an optimization task, which can be solved efficiently. To demonstrate the effectiveness of our approach, we include some straightforward numerical simulations that showcase the reconstruction performance.