In this work, a tensor completion problem is studied, which aims to perfectly recover the tensor from partial observations. Existing theoretical guarantee requires the involved transform to be orthogonal, which hinders its applications. In this paper, jumping out of the constraints of isotropy or self-adjointness, the theoretical guarantee of exact tensor completion with arbitrary linear transforms is established. To that end, we define a new tensor-tensor product, which leads us to a new definition of the tensor nuclear norm. Equipped with these tools, an efficient algorithm based on alternating direction of multipliers is designed to solve the transformed tensor completion program and the theoretical bound is obtained. Our model and proof greatly enhance the flexibility of tensor completion and extensive experiments validate the superiority of the proposed method.

The recently proposed tensor robust principal component analysis (TRPCA) methods based on tensor singular value decomposition (t-SVD) have achieved numerous successes in many fields. However, most of these methods are only applicable to third-order tensors, whereas the data obtained in practice are often of higher order, such as fourth-order color videos, fourth-order hyperspectral videos, and fifth-order light-field images. Additionally, in the t-SVD framework, the multi-rank of a tensor can describe more fine-grained low-rank structure in the tensor compared with the tubal rank. However, determining the multi-rank of a tensor is a much more difficult problem than determining the tubal rank. Moreover, most of the existing TRPCA methods do not explicitly model the noises except the sparse noise, which may compromise the accuracy of estimating the low-rank tensor. In this work, we propose a novel high-order TRPCA method, named as Low-Multi-rank High-order Bayesian Robust Tensor Factorization (LMH-BRTF), within the Bayesian framework. Specifically, we decompose the observed corrupted tensor into three parts, i.e., the low-rank component, the sparse component, and the noise component. By constructing a low-rank model for the low-rank component based on the order-$d$ t-SVD and introducing a proper prior for the model, LMH-BRTF can automatically determine the tensor multi-rank. Meanwhile, benefiting from the explicit modeling of both the sparse and noise components, the proposed method can leverage information from the noises more effectivly, leading to an improved performance of TRPCA. Then, an efficient variational inference algorithm is established for parameters estimation. Empirical studies on synthetic and real-world datasets demonstrate the effectiveness of the proposed method in terms of both qualitative and quantitative results.

Distributed algorithms, particularly Diffusion Least Mean Square, are widely favored for their reliability, robustness, and fast convergence in various industries. However, limited observability of the target can compromise the integrity of the algorithm. To address this issue, this paper proposes a framework for analyzing combination strategies by drawing inspiration from signal flow analysis. A thresholding-based algorithm is also presented to identify and utilize the support vector in scenarios with missing information about the target vector's support. The proposed approach is demonstrated in two combination scenarios, showcasing the effectiveness of the algorithm in situations characterized by sparse observations in the time and transform domains.

We describe a method for signal parameter estimation using the signed cumulative distribution transform (SCDT), a recently introduced signal representation tool based on optimal transport theory. The method builds upon signal estimation using the cumulative distribution transform (CDT) originally introduced for positive distributions. Specifically, we show that Wasserstein-type distance minimization can be performed simply using linear least squares techniques in SCDT space for arbitrary signal classes, thus providing a global minimizer for the estimation problem even when the underlying signal is a nonlinear function of the unknown parameters. Comparisons to current signal estimation methods using $L_p$ minimization shows the advantage of the method.

The recent outbreak of COVID-19 has motivated researchers to contribute in the area of medical imaging using artificial intelligence and deep learning. Super-resolution (SR), in the past few years, has produced remarkable results using deep learning methods. The ability of deep learning methods to learn the non-linear mapping from low-resolution (LR) images to their corresponding high-resolution (HR) images leads to compelling results for SR in diverse areas of research. In this paper, we propose a deep learning based image super-resolution architecture in Tchebichef transform domain. This is achieved by integrating a transform layer into the proposed architecture through a customized Tchebichef convolutional layer (TCL).

A relatively new set of transport-based transforms (CDT, R-CDT, LOT) have shown their strength and great potential in various image and data processing tasks such as parametric signal estimation, classification, cancer detection among many others. It is hence worthwhile to elucidate some of the mathematical properties that explain the successes of these transforms when they are used as tools in data analysis, signal processing or data classification. In particular, we give conditions under which classes of signals that are created by algebraic generative models are transformed into convex sets by the transport transforms. Such convexification of the classes simplify the classification and other data analysis and processing problems when viewed in the transform domain. More specifically, we study the extent and limitation of the convexification ability of these transforms under an algebraic generative modeling framework. We hope that this paper will serve as an introduction to these transforms and will encourage mathematicians and other researchers to further explore the theoretical underpinnings and algorithmic tools that will help understand the successes of these transforms and lay the groundwork for further successful applications.

Signal models based on sparse representation have received considerable attention in recent years. Compared to synthesis dictionary learning, sparsifying transform learning involves highly efficient sparse coding and operator update steps. In this work, we propose a Multi-layer Residual Sparsifying Transform (MRST) learning model wherein the transform domain residuals are jointly sparsified over layers. In particular, the transforms for the deeper layers exploit the more intricate properties of the residual maps. We investigate the application of the learned MRST model for low-dose CT reconstruction using Penalized Weighted Least Squares (PWLS) optimization. Experimental results on Mayo Clinic data show that the MRST model outperforms conventional methods such as FBP and PWLS methods based on edge-preserving (EP) regularizer and single-layer transform (ST) model, especially for maintaining some subtle details.

Signal models based on sparsity, low-rank and other properties have been exploited for image reconstruction from limited and corrupted data in medical imaging and other computational imaging applications. In particular, sparsifying transform models have shown promise in various applications, and offer numerous advantages such as efficiencies in sparse coding and learning. This work investigates pre-learning a multi-layer extension of the transform model for image reconstruction, wherein the transform domain or filtering residuals of the image are further sparsified over the layers. The residuals from multiple layers are jointly minimized during learning, and in the regularizer for reconstruction. The proposed block coordinate descent optimization algorithms involve highly efficient updates. Preliminary numerical experiments demonstrate the usefulness of a two-layer model over the previous related schemes for CT image reconstruction from low-dose measurements.

We introduce a general method of performing Residual Network inference and learning in the JPEG transform domain that allows the network to consume compressed images as input. Our formulation leverages the linearity of the JPEG transform to redefine convolution and batch normalization with a tune-able numerical approximation for ReLu. The result is mathematically equivalent to the spatial domain network up to the ReLu approximation accuracy. A formulation for image classification and a model conversion algorithm for spatial domain networks are given as examples of the method. We show that the sparsity of the JPEG format allows for faster processing of images with little to no penalty in the network accuracy.

We propose a novel multilinear dynamical system (MLDS) in a transform domain, named $\mathcal{L}$-MLDS, to model tensor time series. With transformations applied to a tensor data, the latent multidimensional correlations among the frontal slices are built, and thus resulting in the computational independence in the transform domain. This allows the exact separability of the multi-dimensional problem into multiple smaller LDS problems. To estimate the system parameters, we utilize the expectation-maximization (EM) algorithm to determine the parameters of each LDS. Further, $\mathcal{L}$-MLDSs significantly reduce the model parameters and allows parallel processing. Our general $\mathcal{L}$-MLDS model is implemented based on different transforms: discrete Fourier transform, discrete cosine transform and discrete wavelet transform. Due to the nonlinearity of these transformations, $\mathcal{L}$-MLDS is able to capture the nonlinear correlations within the data unlike the MLDS \cite{rogers2013multilinear} which assumes multi-way linear correlations. Using four real datasets, the proposed $\mathcal{L}$-MLDS is shown to achieve much higher prediction accuracy than the state-of-the-art MLDS and LDS with an equal number of parameters under different noise models. In particular, the relative errors are reduced by $50\% \sim 99\%$. Simultaneously, $\mathcal{L}$-MLDS achieves an exponential improvement in the model's training time than MLDS.