Latent Archetypes of the Spatial Patterns of Cancer.

The cancer atlas edited by several countries is the main resource for the analysis of the geographic variation of cancer risk. Correlating the observed spatial patterns with known or hypothesized risk factors is time-consuming work for epidemiologists who need to deal with each cancer separately, breaking down the patterns according to sex and race. The recent literature has proposed to study more than one cancer simultaneously looking for common spatial risk factors. However, this previous work has two constraints: they consider only a very small (2-4) number of cancers previously known to share risk factors. In this article, we propose an exploratory method to search for latent spatial risk factors of a large number of supposedly unrelated cancers. The method is based on the singular value decomposition and nonnegative matrix factorization, it is computationally efficient, scaling easily with the number of regions and cancers. We carried out a simulation study to evaluate the method's performance and apply it to cancer atlas from the USA, England, France, Australia, Spain, and Brazil. We conclude that with very few latent maps, which can represent a reduction of up to 90% of atlas maps, most of the spatial variability is conserved. By concentrating on the epidemiological analysis of these few latent maps a substantial amount of work is saved and, at the same time, high-level explanations affecting many cancers simultaneously can be reached.

LoRA-TV: read depth profile-based clustering of tumor cells in single-cell sequencing.

Single-cell sequencing has revolutionized our ability to dissect the heterogeneity within tumor populations. In this study, we present LoRA-TV (Low Rank Approximation with Total Variation), a novel method for clustering tumor cells based on the read depth profiles derived from single-cell sequencing data. Traditional analysis pipelines process read depth profiles of each cell individually. By aggregating shared genomic signatures distributed among individual cells using low-rank optimization and robust smoothing, the proposed method enhances clustering performance. Results from analyses of both simulated and real data demonstrate its effectiveness compared with state-of-the-art alternatives, as supported by improvements in the adjusted Rand index and computational efficiency.

From low-rank retractions to dynamical low-rank approximation and back.

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston-Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

Tracking online low-rank approximations of higher-order incomplete streaming tensors.

In this paper, we propose two new provable algorithms for tracking online low-rank approximations of high-order streaming tensors with missing data. The first algorithm, dubbed adaptive Tucker decomposition (ATD), minimizes a weighted recursive least-squares cost function to obtain the tensor factors and the core tensor in an efficient way, thanks to an alternating minimization framework and a randomized sketching technique. Under the canonical polyadic (CP) model, the second algorithm, called ACP, is developed as a variant of ATD when the core tensor is imposed to be identity. Both algorithms are low-complexity tensor trackers that have fast convergence and low memory storage requirements. A unified convergence analysis is presented for ATD and ACP to justify their performance. Experiments indicate that the two proposed algorithms are capable of streaming tensor decomposition with competitive performance with respect to estimation accuracy and runtime on both synthetic and real data.

Low-rank latent matrix-factor prediction modeling for generalized high-dimensional matrix-variate regression.

Motivated by diagnosing the COVID-19 disease using two-dimensional (2D) image biomarkers from computed tomography (CT) scans, we propose a novel latent matrix-factor regression model to predict responses that may come from an exponential distribution family, where covariates include high-dimensional matrix-variate biomarkers. A latent generalized matrix regression (LaGMaR) is formulated, where the latent predictor is a low-dimensional matrix factor score extracted from the low-rank signal of the matrix variate through a cutting-edge matrix factor model. Unlike the general spirit of penalizing vectorization plus the necessity of tuning parameters in the literature, instead, our prediction modeling in LaGMaR conducts dimension reduction that respects the geometric characteristic of intrinsic 2D structure of the matrix covariate and thus avoids iteration. This greatly relieves the computation burden, and meanwhile maintains structural information so that the latent matrix factor feature can perfectly replace the intractable matrix-variate owing to high-dimensionality. The estimation procedure of LaGMaR is subtly derived by transforming the bilinear form matrix factor model onto a high-dimensional vector factor model, so that the method of principle components can be applied. We establish bilinear-form consistency of the estimated matrix coefficient of the latent predictor and consistency of prediction. The proposed approach can be implemented conveniently. Through simulation experiments, the prediction capability of LaGMaR is shown to outperform some existing penalized methods under diverse scenarios of generalized matrix regressions. Through the application to a real COVID-19 dataset, the proposed approach is shown to predict efficiently the COVID-19.

Bayesian generalized linear low rank regression models for the detection of vaccine-adverse event associations.

We propose a generalized linear low-rank mixed model (GLLRM) for the analysis of both high-dimensional and sparse responses and covariates where the responses may be binary, counts, or continuous. This development is motivated by the problem of identifying vaccine-adverse event associations in post-market drug safety databases, where an adverse event is any untoward medical occurrence or health problem that occurs during or following vaccination. The GLLRM is a generalization of a generalized linear mixed model in that it integrates a factor analysis model to describe the dependence among responses and a low-rank matrix to approximate the high-dimensional regression coefficient matrix. A sampling procedure combining the Gibbs sampler and Metropolis and Gamerman algorithms is employed to obtain posterior estimates of the regression coefficients and other model parameters. Testing of response-covariate pair associations is based on the posterior distribution of the corresponding regression coefficients. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures on binary and count outcomes. We further illustrate the GLLRM via a real data example based on the Vaccine Adverse Event Reporting System.

Low-rank Parareal: a low-rank parallel-in-time integrator.

In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge-Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.

Fast computational E-field dosimetry for transcranial magnetic stimulation using adaptive cross approximation and auxiliary dipole method (ACA-ADM).

Transcranial Magnetic Stimulation (TMS) is a non-invasive brain stimulation technique that uses a coil to induce an electric field (E-field) in the brain and modulate its activity. Many applications of TMS call for the repeated execution of E-field solvers to determine the E-field induced in the brain for different coil placements. However, the usage of solvers for these applications remains impractical because each coil placement requires the solution of a large linear system of equations. We develop a fast E-field solver that enables the rapid evaluation of the E-field distribution for a brain region of interest (ROI) for a large number of coil placements, which is achieved in two stages. First, during the pre-processing stage, the mapping between coil placement and brain ROI E-field distribution is approximated from E-field results for a few coil placements. Specifically, we discretize the mapping into a matrix with each column having the ROI E-field samples for a fixed coil placement. This matrix is approximated from a few of its rows and columns using adaptive cross approximation (ACA). The accuracy, efficiency, and applicability of the new ACA approach are determined by comparing its E-field predictions with analytical and standard solvers in spherical and MRI-derived head models. During the second stage, the E-field distribution in the brain ROI from a specific coil placement is determined by the obtained rows and columns in milliseconds. For many applications, only the E-field distribution for a comparatively small ROI is required. For example, the solver can complete the pre-processing stage in approximately 4 hours and determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error enabling its use for neuro-navigation and other applications. Highlight: We developed a fast solver for TMS computational E-field dosimetry, which can determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error.

Enhancing linguistic research through 2-mm isotropic 3D dynamic speech MRI optimized by sparse temporal sampling and low-rank reconstruction.

PURPOSE: To enable a more comprehensive view of articulations during speech through near-isotropic 3D dynamic MRI with high spatiotemporal resolution and large vocal-tract coverage. METHODS: Using partial separability model-based low-rank reconstruction coupled with a sparse acquisition of both spatial and temporal models, we are able to achieve near-isotropic resolution 3D imaging with a high frame rate. The total acquisition time of the speech acquisition is shortened by introducing a sparse temporal sampling that interleaves one temporal navigator with four randomized phase and slice-encoded imaging samples. Memory and computation time are improved through compressing coils based on the region of interest for low-rank constrained reconstruction with an edge-preserving spatial penalty. RESULTS: The proposed method has been evaluated through experiments on several speech samples, including a standard reading passage. A near-isotropic 1.875 × 1.875 × 2 mm3 spatial resolution, 64-mm through-plane coverage, and a 35.6-fps temporal resolution are achieved. Investigations and analysis on specific speech samples support novel insights into nonsymmetric tongue movement, velum raising, and coarticulation events with adequate visualization of rapid articulatory movements. CONCLUSION: Three-dimensional dynamic images of the vocal tract structures during speech with high spatiotemporal resolution and axial coverage is capable of enhancing linguistic research, enabling visualization of soft tissue motions that are not possible with other modalities.

Joint denoising of diffusion-weighted images via structured low-rank patch matrix approximation.

PURPOSE: To develop a joint denoising method that effectively exploits natural information redundancy in MR DWIs via low-rank patch matrix approximation. METHODS: A denoising method is introduced to jointly reduce noise in DWI dataset by exploiting nonlocal self-similarity as well as local anatomical/structural similarity within multiple 2D DWIs acquired with the same anatomical geometry but different diffusion directions. Specifically, for each small 3D reference patch sliding within 2D DWI, nonlocal but similar patches are searched by matching image contents within entire DWI dataset and then structured into a patch matrix. The resulting patch matrices are denoised by enforcing low-rankness via weighted nuclear norm minimization and finally are back-distributed to DWI space. The proposed procedure was evaluated with simulated and in vivo brain diffusion tensor imaging (DTI) datasets and then compared to existing Marchenko-Pastur principal component analysis denoising method. RESULTS: The proposed method achieved significant noise reduction while preserving structural details in all DWIs for both simulated and in vivo datasets. Quantitative evaluation of error maps demonstrated it consistently outperformed Marchenko-Pastur principal component analysis method. Further, the denoised DWIs led to substantially improved DTI parametric maps, exhibiting significantly less noise and revealing more microstructural details. CONCLUSION: The proposed method denoises DWI dataset by utilizing both nonlocal self-similarity and local structural similarity within DWI dataset. This weighted nuclear norm minimization-based low-rank patch matrix denoising approach is effective and highly applicable to various diffusion MRI applications, including DTI as a postprocessing procedure.