SUMMARY: With the increased reliance on multi-omics data for bulk and single-cell analyses, the availability of robust approaches to perform unsupervised learning for clustering, visualization, and feature selection is imperative. We introduce nipalsMCIA, an implementation of multiple co-inertia analysis (MCIA) for joint dimensionality reduction that solves the objective function using an extension to Nonlinear Iterative Partial Least Squares. We applied nipalsMCIA to both bulk and single-cell datasets and observed significant speed-up over other implementations for data with a large sample size and/or feature dimension. AVAILABILITY AND IMPLEMENTATION: nipalsMCIA is available as a Bioconductor package at https://bioconductor.org/packages/release/bioc/html/nipalsMCIA.html, and includes detailed documentation and application vignettes.

Many irregular domains such as social networks, financial transactions, neuron connections, and natural language constructs are represented using graph structures. In recent years, a variety of graph neural networks (GNNs) have been successfully applied for representation learning and prediction on such graphs. In many of the real-world applications, the underlying graph changes over time, however, most of the existing GNNs are inadequate for handling such dynamic graphs. In this paper we propose a novel technique for learning embeddings of dynamic graphs using a tensor algebra framework. Our method extends the popular graph convolutional network (GCN) for learning representations of dynamic graphs using the recently proposed tensor M-product technique. Theoretical results presented establish a connection between the proposed tensor approach and spectral convolution of tensors. The proposed method TM-GCN is consistent with the Message Passing Neural Network (MPNN) framework, accounting for both spatial and temporal message passing. Numerical experiments on real-world datasets demonstrate the performance of the proposed method for edge classification and link prediction tasks on dynamic graphs. We also consider an application related to the COVID-19 pandemic, and show how our method can be used for early detection of infected individuals from contact tracing data.

In recent work [S. Soltani, M. Kilmer, and P. C. Hansen, BIT, 56 (2016)], an algorithm for nonnegative tensor patch dictionary learning in the context of X-ray CT imaging and based on a tensor-tensor product called the t-product [M. E. Kilmer and C. D. Martin, Linear Algebra Appl., 435 (2011), pp. 641--658] was presented. Building on that work, in this paper, we use nonnegative tensor patch--based dictionaries trained on other data, such as facial image data, for the purpose of either compression or image deblurring. We begin with an analysis in which we address issues such as suitability of the tensor-based approach relative to a matrix-based approach, dictionary size, and patch size to balance computational efficiency and qualitative representations. Next, we develop an algorithm that is capable of recovering nonnegative tensor coefficients given a nonnegative tensor dictionary. The algorithm is based on a variant of the modified residual norm steepest descent method. We show how to augment the algorithm to enforce sparsity in the tensor coefficients and note that the approach has broader applicability since it can be applied to the matrix case as well. We illustrate the surprising result that dictionaries trained on image data from one class can be successfully used to represent and compress image data from different classes and across different resolutions. Finally, we address the use of nonnegative tensor dictionaries in image deblurring. We show that tensor treatment of the deblurring problem coupled with nonnegative tensor patch dictionaries can give superior restorations as compared to standard treatment of the nonnegativity constrained deblurring problem.

In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].

The development of energy selective, photon counting X-ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation. Under the assumption of perfect energy resolution, here we propose a tensor-based iterative algorithm that simultaneously reconstructs the X-ray attenuation distribution for each energy. We use a multilinear image model rather than a more standard stacked vector representation in order to develop novel tensor-based regularizers. In particular, we model the multispectral unknown as a three-way tensor where the first two dimensions are space and the third dimension is energy. This approach allows for the design of tensor nuclear norm regularizers, which like its 2D counterpart, is a convex function of the multispectral unknown. The solution to the resulting convex optimization problem is obtained using an alternating direction method of multipliers approach. Simulation results show that the generalized tensor nuclear norm can be used as a standalone regularization technique for the energy selective (spectral) computed tomography problem and when combined with total variation regularization it enhances the regularization capabilities especially at low energy images where the effects of noise are most prominent.

A tensor is a multidimensional array. First-order tensors and second-order tensors can be viewed as vectors and matrices, respectively. Tensors of higher order, with the ability to include more information, appear more frequently nowadays in image and signal processing, data mining, biomedical engineering, and so on. With the recent work of Kilmer and Martin, familiar matrix-based factorizations in linear algebra can be extended in a straightforward way to third-order tensors based on their new tensor multiplication and concepts. Our method has an advantage over a popular tensor-based face recognition algorithm called TensorFaces, which is based on the higher-order SVD of an arrangement of the images in a database, in that it does not require a least squares solve for the coefficients. In this paper, we give a brief introduction to the new tensor framework and apply the induced tensor decompositions to the application of facial recognition. In the numerical results, we compare our new approaches with the traditional principal component analysis method based on matrix decomposition (also known as eigenfaces) and with TensorFaces.

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results in a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, reinitialization, and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the way for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which, used in the proposed manner, provide flexibility in presenting a larger class of shapes with fewer terms. Also they provide a “narrow-banding” advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography, and diffuse optical tomography.

We derive Kronecker product approximations, with the help of tensor decompositions, to construct approximations of severely ill-conditioned matrices that arise in three-dimensional (3-D) image processing applications. We use the Kronecker product approximations to derive preconditioners for iterative regularization techniques; the resulting preconditioned algorithms allow us to restore 3-D images in a computationally efficient manner. Through examples in microscopy and medical imaging, we show that the Kronecker approximation preconditioners provide a powerful tool that can be used to improve efficiency of iterative image restoration algorithms.

The problem of characterizing the geometric structure of an object buried in an inhomogeneous halfspace of unknown composition is considered. The authors develop a nonlinear inverse scattering algorithm based on a low-dimensional parameterization of the unknown object and the background. In particular, they use a low-order polynomial expansion to represent the spatial variations in the real and imaginary parts of the object and background complex permittivities. The boundary separating the target from the unknown background is described using a periodic, quadratic B-spline curve whose control points can be individually manipulated. They determine the unknown control point locations and contrast expansion coefficients using a greedy-type approach to minimize a regularized least-squares cost function. The regularizer used here is designed to constrain the geometric structure of the boundary of the object and is closely related to snake methods employed in the image processing community. They demonstrate the performance of their approach via extensive numerical simulation involving two-dimensional (2D), TM/sub z/ scattering geometries. Their results indicate a strong ability to localize and estimate the shape of the object even in the presence of an unknown and inhomogeneous background.

In this paper, we consider a wavelet based edge-preserving regularization scheme for use in linear image restoration problems. Our efforts build on a collection of mathematical results indicating that wavelets are especially useful for representing functions that contain discontinuities (i.e., edges in two dimensions or jumps in one dimension). We interpret the resulting theory in a statistical signal processing framework and obtain a highly flexible framework for adapting the degree of regularization to the local structure of the underlying image. In particular, we are able to adapt quite easily to scale-varying and orientation-varying features in the image while simultaneously retaining the edge preservation properties of the regularizer. We demonstrate a half-quadratic algorithm for obtaining the restorations from observed data.