arXiv subjects

Misha E. Kilmer

Publications and source records attributed to Misha E. Kilmer.

At least 19 recordsLinked to original sources

arXiv2026

A Symmetry-Preserving Tensor $\star_{\mathbf{M}}$-SVD

Multiway data such as image collections and video is ubiquitous, but the usual approach of flattening them into matrices discards the cross-mode structure that often carries the signal. The t-product and its generalization, the $\star_{\mathbf{M}}$-product, give a matrix-mimetic tensor algebra with a tensor SVD whose truncation is optimal in the Frobenius norm, just as in the matrix case. Much real data also has internal reflective symmetry: frontal faces, manufactured parts, and leaves are all bilaterally symmetric. We define a symmetry-preserving $\star_{\mathbf{M}}$-SVD that extends the matrix symmetry-preserving SVD of Shah and Sorensen to the $\star_{\mathbf{M}}$-algebra. When a tensor's transform-domain frontal slices are reflectively symmetric, the left basis of its $\star_{\mathbf{M}}$-SVD is symmetric too, so only its top half must be stored. Bilaterally symmetric images, turned on their side and stored as lateral slices, give such a tensor. This construction keeps the symmetric part of each image and stores only half the basis. We then recognize new images by projecting them onto this basis and matching to the nearest training image in coefficient space. Across ten datasets of faces, leaves, butterfly wings, and other objects, the symmetric basis matches the recognition rate of ordinary tensor SVD at 2-11x less basis storage, with the exception of MUCT. On faces under varying illumination it exceeds the best rate the ordinary tensor SVD attains at any storage level.

math.NA↗

arXiv2026

Wavelet-based multilevel framework for $\ell_1$-regularized image deblurring

Solving large-scale $\ell_1$-regularized image deblurring problems efficiently while preserving sharp edges remains a significant computational challenge. We propose a wavelet-based multilevel framework that embeds three iterative solvers, Iteratively Reweighted Least Squares (IRLS), Split Bregman (SB), and Majorization-Minimization (MM), within a multilevel V-cycle. Discrete wavelet transforms define the interlevel transfer operators, and regularization parameters are selected automatically by Generalized Cross Validation. Two information transfer strategies are introduced and compared: one transfers only the coarse solution to the fine level, while the other transfers solver-specific auxiliary quantities. Numerical experiments demonstrate substantial computational savings for IRLS, with speedups exceeding an order of magnitude, while MM and SB exhibit more modest computational differences. The experiments generally show that transferring auxiliary iterates performs best with Haar wavelets, whereas transferring only the solution performs best with Daubechies wavelets.

math.NA↗

arXiv2026

Nonlinear RMM-GKS for Large-Scale Dynamic and Streaming Inverse Problems with Uncertain Forward Operators

Many practical imaging systems suffer from uncertainty in acquisition geometry -- such as projection angles in computed tomography or sensor positions in photoacoustic tomography -- leading to nonlinear inverse problems that require joint estimation of both the image and the forward model parameters. Standard approaches that assume a known linear forward operator fail to account for these uncertainties, resulting in significant reconstruction artifacts. We propose a nonlinear recycled majorization-minimization generalized Krylov subspace (NL-RMM-GKS) framework for large-scale inverse problems with uncertain forward operators. The method extends MM-GKS to nonlinear settings by combining majorization-minimization for nonsmooth regularization with Krylov subspace projection and subspace recycling, ensuring bounded memory usage. Two complementary formulations are developed: an alternating minimization approach that alternates between image updates and Gauss-Newton parameter estimation, and a variable projection approach that eliminates the image variable and optimizes directly over the parameters using inexact inner solves. We further introduce streaming variants that process data sequentially, enabling reconstruction from large or dynamically acquired datasets without storing the full operator. For dynamic problems, we incorporate two temporal regularization strategies -- optical flow and anisotropic total variation -- as plug-in choices within the framework. We carry out rigorous numerical experiments in fan-beam computed tomography and photoacoustic tomography to demonstrate that our proposed framework achieves high-quality reconstructions with bounded memory requirements, making it suitable for large-scale dynamic imaging problems.

math.NA↗

arXiv2026

An Efficient Cumulative Edge-Detection Method for Image Reconstruction

When reconstructing images from noisy measurements, such as in medical scans or scientific imaging, we face an inverse problem: recovering an unknown image from indirect, corrupted observations. These problems are typically ill-posed, meaning small amounts of noise can lead to inaccurate reconstructions. Regularization techniques address this by incorporating prior assumptions about the solution, such as smoothness or sparsity. However, standard methods often blur sharp edges--the boundaries between tissues or structures--losing critical detail. A powerful strategy for edge preservation is iterative reweighting, which solves a sequence of weighted subproblems with adaptively updated weights. Non-cumulative schemes derive weights from the current iterate alone and can be solved efficiently using the Recycled Majorization-Minimization Generalized Krylov Subspace method (RMM-GKS). The cumulative approach of Gazzola et al. progressively accumulates edge information across iterations, achieving superior edge preservation but at high computational cost. This work introduces CR-$\ell_q$-RMM-GKS, which combines cumulative edge detection with computational efficiency. We integrate Gazzola's cumulative weighting with RMM-GKS, which handles general $\ell_q$ penalties ($0 < q \le 2$), automatically selects regularization parameters, and recycles Krylov subspaces between iterations, reducing the nested structure to two levels. Numerical experiments in signal deblurring and tomography demonstrate that CR-$\ell_q$-RMM-GKS produces significantly sharper edge reconstructions than standard non-cumulative methods. In particular, CR-$\ell_1$-RMM-GKS outperforms both standard $\ell_1$ methods and CR-$\ell_2$-RMM-GKS, indicating that cumulative weighting and $\ell_1$ penalties are highly complementary.

math.NA↗

arXiv2025

A tensor-based dynamic mode decomposition based on the $\star_{\boldsymbol{M}}$-product

Dynamic mode decomposition (DMD) is a data-driven method for estimating the dynamics of a discrete dynamical system. This paper proposes a tensor-based approach to DMD for applications in which the states can be viewed as tensors. Specifically, we use the $\star_{\boldsymbol{M}}$-product framework for tensor decompositions which we demonstrate offers excellent compression compared to matrix-based methods and can be implemented in a computationally efficient manner. We show how the proposed approach is connected to the traditional DMD and physics-informed DMD frameworks. We give a computational framework for computing the tensor-based DMD and detail the computational costs. We also give a randomized algorithm that enables efficient $\star_{\boldsymbol{M}}$-DMD computations in the streaming setting. The numerical results show that the proposed method achieves equal or better accuracy for the same storage compared to the standard DMD on these examples and is more efficient to compute.

math.NA↗

arXiv2024

Tensor BM-Decomposition for Compression and Analysis of Video Data

Given tensors $\boldsymbol{\mathscr{A}}, \boldsymbol{\mathscr{B}}, \boldsymbol{\mathscr{C}}$ of size $m \times 1 \times n$, $m \times p \times 1$, and $1\times p \times n$, respectively, their Bhattacharya-Mesner (BM) product will result in a third-order tensor of dimension $m \times p \times n$ and BM-rank of 1 (Mesner and Bhattacharya, 1990). Thus, if an arbitrary $m \times p \times n$ third-order tensor can be written as a sum of a small number, relative to $m,p,n$, of such BM-rank 1 terms, this BM-decomposition (BMD) offers an implicitly compressed representation of the tensor. In this paper, we first show that grayscale surveillance video can be accurately captured by a low BM-rank decomposition and give methods for efficiently computing this decomposition. To this end, we first give results that connect rank-revealing matrix factorizations to the BMD. Next, we present a generative model that illustrates that spatio-temporal video data can be expected to have low BM-rank. We combine these observations to derive a regularized alternating least squares (ALS) algorithm to compute an approximate BMD of the video tensor. The algorithm itself is highly parallelizable since the bulk of the computations break down into relatively small regularized least squares problems that can be solved independently. Extensive numerical results compared against the state-of-the-art matrix-based DMD for surveillance video separation show our algorithms can consistently produce results with superior compression properties while simultaneously providing better separation of stationary and non-stationary features in the data. We then introduce a new type of BM-product suitable for color video and provide an algorithm that shows an impressive ability to extract important temporal information from color video while simultaneously compressing the data.

math.NA↗

arXiv2024

Tensor Completion with BMD Factor Nuclear Norm Minimization

This paper is concerned with the problem of recovering third-order tensor data from limited samples. A recently proposed tensor decomposition (BMD) method has been shown to efficiently compress third-order spatiotemporal data. Using the BMD, we formulate a slicewise nuclear norm penalized algorithm to recover a third-order tensor from limited observed samples. We develop an efficient alternating direction method of multipliers (ADMM) scheme to solve the resulting minimization problem. Experimental results on real data show our method to give reconstruction comparable to those of HaLRTC (Liu et al., IEEE Trans Ptrn Anal Mchn Int, 2012), a well-known tensor completion method, in about the same number of iterations. However, our method has the advantage of smaller subproblems and higher parallelizability per iteration.

math.NA↗

arXiv2023

Recycling MMGKS for large-scale dynamic and streaming data

Reconstructing high-quality images with sharp edges requires the use of edge-preserving constraints in the regularized form of the inverse problem. The use of the $\ell_q$-norm on the gradient of the image is a common such constraint. For implementation purposes, the $\ell_q$-norm term is typically replaced with a sequence of $\ell_2$-norm weighted gradient terms with the weights determined from the current solution estimate. While (hybrid) Krylov subspace methods can be employed on this sequence, it would require generating a new Krylov subspace for every new two-norm regularized problem. The majorization-minimization Krylov subspace method (MM-GKS) addresses this disadvantage by combining norm reweighting with generalized Krylov subspaces (GKS). After projecting the problem using a small dimensional subspace - one that expands each iteration - the regularization parameter is selected. Basis expansion repeats until a sufficiently accurate solution is found. Unfortunately, for large-scale problems that require many expansion steps to converge, storage and the cost of repeated orthogonalizations presents overwhelming memory and computational requirements. In this paper we present a new method, recycled MM-GKS (RMM-GKS), that keeps the memory requirements bounded through recycling the solution subspace. Specifically, our method alternates between enlarging and compressing the GKS subspace, recycling directions that are deemed most important via one of our tailored compression routines. We further generalize the RMM-GKS approach to handle experiments where the data is either not all available simultaneously, or needs to be treated as such because of the extreme memory requirements. Numerical examples from dynamic photoacoustic tomography and streaming X-ray computerized tomography (CT) imaging are used to illustrate the effectiveness of the described methods.

math.NA↗

arXiv2023

Multigrid preconditioning for regularized least-squares problems

In this paper, we are concerned with efficiently solving the sequences of regularized linear least squares problems associated with employing Tikhonov-type regularization with regularization operators designed to enforce edge recovery. An optimal regularization parameter, which balances the fidelity to the data with the edge-enforcing constraint term, is typically not known a priori. This adds to the total number of regularized linear least squares problems that must be solved before the final image can be recovered. Therefore, in this paper, we determine effective multigrid preconditioners for these sequences of systems. We focus our approach on the sequences that arise as a result of the edge-preserving method introduced in [6], where we can exploit an interpretation of the regularization term as a diffusion operator; however, our methods are also applicable in other edge-preserving settings, such as iteratively reweighted least squares problems. Particular attention is paid to the selection of components of the multigrid preconditioner in order to achieve robustness for different ranges of the regularization parameter value. In addition, we present a parameter culling approach that, when used with the L-curve heuristic, reduces the total number of solves required. We demonstrate our preconditioning and parameter culling routines on examples in computed tomography and image deblurring.

math.NA↗

arXiv2023

Subspace Recycling for Sequences of Shifted Systems with Applications in Image Recovery

For many applications involving a sequence of linear systems with slowly changing system matrices, subspace recycling, which exploits relationships among systems and reuses search space information, can achieve huge gains in iterations across the total number of linear system solves in the sequence. However, for general (i.e., non-identity) shifted systems with the shift value varying over a wide range, the properties of the linear systems vary widely as well, which makes recycling less effective. If such a sequence of systems is embedded in a nonlinear iteration, the problem is compounded, and special approaches are needed to use recycling effectively. In this paper, we develop new, more efficient, Krylov subspace recycling approaches for large-scale image reconstruction and restoration techniques that employ a nonlinear iteration to compute a suitable regularization matrix. For each new regularization matrix, we need to solve regularized linear systems, ${\bf A} + γ_\ell {\bf E}_k$, for a sequence of regularization parameters, $γ_\ell$, to find the optimally regularized solution that, in turn, will be used to update the regularization matrix. In this paper, we analyze system and solution characteristics to choose appropriate techniques to solve each system rapidly. Specifically, we use an inner-outer recycling approach with a larger, principal recycle space for each nonlinear step and smaller recycle spaces for each shift. We propose an efficient way to obtain good initial guesses from the principle recycle space and smaller shift-specific recycle spaces that lead to fast convergence. Our method is substantially reduces the total number of matrix-vector products that would arise in a naive approach. Our approach is more generally applicable to sequences of shifted systems where the matrices in the sum are positive semi-definite.

math.NA↗

arXiv2021

Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions

In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We have detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes.

math.NA↗

arXiv2021

Structured Matrix Approximations via Tensor Decompositions

We provide a computational framework for approximating a class of structured matrices; here, the term structure is very general, and may refer to a regular sparsity pattern (e.g., block-banded), or be more highly structured (e.g., symmetric block Toeplitz). The goal is to uncover {\it additional latent structure} that will in turn lead to computationally efficient algorithms when the new structured matrix approximations are employed in the place of the original operator. Our approach has three steps: map the structured matrix to tensors, use tensor compression algorithms, and map the compressed tensors back to obtain two different matrix representations -- sum of Kronecker products and block low-rank format. The use of tensor decompositions enables us to uncover latent structure in the problem and leads to compressed representations of the original matrix that can be used efficiently in applications. The resulting matrix approximations are memory efficient, easy to compute with, and preserve the error that is due to the tensor compression in the Frobenius norm. Our framework is quite general. We illustrate the ability of our method to uncover block-low-rank format on structured matrices from two applications: system identification, space-time covariance matrices. In addition, we demonstrate that our approach can uncover sum of structured Kronecker products structure on several matrices from the SuiteSparse collection. Finally, we show that our framework is broad enough to encompass and improve on other related results from the literature, as we illustrate with the approximation of a three-dimensional blurring operator.

math.NA↗

arXiv2021

Dynamic Graph Convolutional Networks Using the Tensor M-Product

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.

cs.LG↗

arXiv2019

An Inner-Outer Iterative Method for Edge Preservation in Image Restoration and Reconstruction

We present a new inner-outer iterative algorithm for edge enhancement in imaging problems. At each outer iteration, we formulate a Tikhonov-regularized problem where the penalization is expressed in the 2-norm and involves a regularization operator designed to improve edge resolution as the outer iterations progress, through an adaptive process. An efficient hybrid regularization method is used to project the Tikhonov-regularized problem onto approximation subspaces of increasing dimensions (inner iterations), while conveniently choosing the regularization parameter (by applying well-known techniques, such as the discrepancy principle or the ${\mathcal L}$-curve criterion, to the projected problem). This procedure results in an automated algorithm for edge recovery that does not involve regularization parameter tuning by the user, nor repeated calls to sophisticated optimization algorithms, and is therefore particularly attractive from a computational point of view. A key to the success of the new algorithm is the design of the regularization operator through the use of an adaptive diagonal weighting matrix that effectively enforces smoothness only where needed. We demonstrate the value of our approach on applications in X-ray CT image reconstruction and in image deblurring, and show that it can be computationally much more attractive than other well-known strategies for edge preservation, while providing solutions of greater or equal quality.

math.NA↗

arXiv2019

Non-negative Tensor Patch Dictionary Approaches for Image Compression and Deblurring Applications

In recent work (Soltani, Kilmer, Hansen, BIT 2016), an algorithm for non-negative tensor patch dictionary learning in the context of X-ray CT imaging and based on a tensor-tensor product called the $t$-product (Kilmer and Martin, 2011) was presented. Building on that work, in this paper, we use of non-negative tensor patch-based dictionaries trained on other data, such as facial image data, for the purposes 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 non-negative tensor coefficients given a non-negative 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 non-negative tensor dictionaries in image deblurring. We show that tensor treatment of the deblurring problem coupled with non-negative tensor patch dictionaries can give superior restorations as compared to standard treatment of the non-negativity constrained deblurring problem.

cs.CV↗

arXiv2019

Randomized algorithms for low-rank tensor decompositions in the Tucker format

Many applications in data science and scientific computing involve large-scale datasets that are expensive to store and compute with, but can be efficiently compressed and stored in an appropriate tensor format. In recent years, randomized matrix methods have been used to efficiently and accurately compute low-rank matrix decompositions. Motivated by this success, we focus on developing randomized algorithms for tensor decompositions in the Tucker representation. Specifically, we present randomized versions of two well-known compression algorithms, namely, HOSVD and STHOSVD. We present a detailed probabilistic analysis of the error of the randomized tensor algorithms. We also develop variants of these algorithms that tackle specific challenges posed by large-scale datasets. The first variant adaptively finds a low-rank representation satisfying a given tolerance and it is beneficial when the target-rank is not known in advance. The second variant preserves the structure of the original tensor, and is beneficial for large sparse tensors that are difficult to load in memory. We consider several different datasets for our numerical experiments: synthetic test tensors and realistic applications such as the compression of facial image samples in the Olivetti database and word counts in the Enron email dataset.

math.NA↗

arXiv2018

Randomized Approach to Nonlinear Inversion Combining Simultaneous Random and Optimized Sources and Detectors

In partial differential equations-based (PDE-based) inverse problems with many measurements, many large-scale discretized PDEs must be solved for each evaluation of the misfit or objective function. In the nonlinear case, evaluating the Jacobian requires solving an additional set of systems. This leads to a tremendous computational cost, and this is by far the dominant cost for these problems. Several authors have proposed randomization and stochastic programming techniques to drastically reduce the number of system solves by estimating the objective function using only a few appropriately chosen random linear combinations of the sources. While some have reported good solution quality at a greatly reduced cost, for our problem of interest, diffuse optical tomography, the approach often does not lead to sufficiently accurate solutions. We propose two improvements. First, to efficiently exploit Newton-type methods, we modify the stochastic estimates to include random linear combinations of detectors, drastically reducing the number of adjoint solves. Second, after solving to a modest tolerance, we compute a few simultaneous sources and detectors that maximize the Frobenius norm of the sampled Jacobian to improve the rate of convergence and obtain more accurate solutions. We complement these optimized simultaneous sources and detectors by random simultaneous sources and detectors constrained to a complementary subspace. Our approach leads to solutions of the same quality as obtained using all sources and detectors but at a greatly reduced computational cost, as the number of large-scale linear systems to be solved is significantly reduced.

math.NA↗

arXiv2016

Computing Reduced Order Models via Inner-Outer Krylov Recycling in Diffuse Optical Tomography

In nonlinear imaging problems whose forward model is described by a partial differential equation (PDE), the main computational bottleneck in solving the inverse problem is the need to solve many large-scale discretized PDEs at each step of the optimization process. In the context of absorption imaging in diffuse optical tomography, one approach to addressing this bottleneck proposed recently (de Sturler, et al, 2015) reformulates the viewing of the forward problem as a differential algebraic system, and then employs model order reduction (MOR). However, the construction of the reduced model requires the solution of several full order problems (i.e. the full discretized PDE for multiple right-hand sides) to generate a candidate global basis. This step is then followed by a rank-revealing factorization of the matrix containing the candidate basis in order to compress the basis to a size suitable for constructing the reduced transfer function. The present paper addresses the costs associated with the global basis approximation in two ways. First, we use the structure of the matrix to rewrite the full order transfer function, and corresponding derivatives, such that the full order systems to be solved are symmetric (positive definite in the zero frequency case). Then we apply MOR to the new formulation of the problem. Second, we give an approach to computing the global basis approximation dynamically as the full order systems are solved. In this phase, only the incrementally new, relevant information is added to the existing global basis, and redundant information is not computed. This new approach is achieved by an inner-outer Krylov recycling approach which has potential use in other applications as well. We show the value of the new approach to approximate global basis computation on two DOT absorption image reconstruction problems.

math.NA↗