Large Matrices Research Articles

Some Involutory Pascal Matrices Can Make More Involutory Pascal Matrices

Summary There is an intuitive way to flip signs within a lower triangular matrix n × n matrix whose entries form the first n – 1 rows of Pascal’s triangle to cause the matrix to be its own multiplicative inverse. For any given positive integer n, we find how many unintuitive choices of flipped signs will also cause the n × n matrix to be its own multiplicative inverse by first seeing how such matrices may or may not fit into larger matrices from the same family.

An extension of the Walsh-Hadamard transform to calculate and model epistasis in genetic landscapes of arbitrary shape and complexity.

Accurate models describing the relationship between genotype and phenotype are necessary in order to understand and predict how mutations to biological sequences affect the fitness and evolution of living organisms. The apparent abundance of epistasis (genetic interactions), both between and within genes, complicates this task and how to build mechanistic models that incorporate epistatic coefficients (genetic interaction terms) is an open question. The Walsh-Hadamard transform represents a rigorous computational framework for calculating and modeling epistatic interactions at the level of individual genotypic values (known as genetical, biological or physiological epistasis), and can therefore be used to address fundamental questions related to sequence-to-function encodings. However, one of its main limitations is that it can only accommodate two alleles (amino acid or nucleotide states) per sequence position. In this paper we provide an extension of the Walsh-Hadamard transform that allows the calculation and modeling of background-averaged epistasis (also known as ensemble epistasis) in genetic landscapes with an arbitrary number of states per position (20 for amino acids, 4 for nucleotides, etc.). We also provide a recursive formula for the inverse matrix and then derive formulae to directly extract any element of either matrix without having to rely on the computationally intensive task of constructing or inverting large matrices. Finally, we demonstrate the utility of our theory by using it to model epistasis within both simulated and empirical multiallelic fitness landscapes, revealing that both pairwise and higher-order genetic interactions are enriched between physically interacting positions.

Statistical approaches enabling technology-specific assay interference prediction from large screening data sets

High throughput screening (HTS) technologies allow the biological testing of hundreds of thousands of compounds per day. Typically, a substantial proportion of the initial hits obtained by HTS are artifacts caused by assay interference. Therefore, global and technology-specific in silico models for identifying and predicting compounds interfering with biological assays have been developed. The global models benefit from training on large screening data sets, while the specialized models benefit from training on assay technology-specific experimental data. In this work, we develop and explore strategies for generating better predictors of technology-specific assay interference by utilizing the large bioactivity data matrices global models are trained on and employing partially new compound labeling approaches to maintain the assay technology awareness of specialized models. We demonstrate the utility of the statistically derived interference labels in machine learning using fluorescence-based assay interference as a representative example. Our random forest and multi-layer perceptron classifiers showed improved performance compared to existing models, achieving Matthews correlation coefficients (MCCs) of up to 0.47 on holdout data and up to 0.45 on an external test set. These results demonstrate that accurate assay-specific interference labels can be derived from large bioactivity data matrices, enabling the development of new machine-learning models without the need for further experimental data.

Efficient subspace clustering and feature extraction via [formula omitted]-norm and [formula omitted]-norm minimization

The nuclear norm-based Latent Low-Rank Representation (LatLRR) has gained much attention due to its success in subspace clustering and feature extraction. However, it suffers from high computational costs due to the calculation of singular value decomposition for large matrices. To this end, we develop an efficient subspace clustering and feature extraction method (ESCFE) which substitutes the nuclear norm with the ℓ2,1-norm and ℓ1,2-norm respectively. Theoretically proof shows both the ℓ2,1-norm and ℓ1,2-norm can serve as the convex surrogates of the nuclear norm while can derive closed-form solutions. Furthermore, the ℓ2,1-norm (or ℓ1,2-norm) regularization promotes column (or row) structure sparsity due to the discriminative nature inherited from the ℓ1-norm. Thus the proposed ESCFE is robust to outliers in data and can extract features with joint sparsity. Extensive experiments on multiple benchmark datasets demonstrate the superiority of our method in both efficiency and effectiveness.

Linear mathematical model for the underlying mechanism of extended ptychographic iterative engine

A linear model by writing the recorded diffractive intensities as a set of linear equations was put forwarded to determine the complex amplitude of the illumination and sample analytically for the extended ptychographic iterative engine (ePIE) imaging. An efficient computing method that can process large matrices was proposed to improve computing efficiency. The influences of detector noise and the positioning error of the sample were studied to explain why satisfactory reconstruction can be obtained experimentally when the linear equations of diffractive intensities are incorrect. This study demonstrates for the first time that ePIE technique has a unique and deterministic solution even with noisy experimental data, overcoming the main obstacle hindering the application of the ePIE algorithm in optical measurement and optical metrology, where a mathematically unique solution is crucial.

On the accurate computation of the Newton form of the Lagrange interpolant

AbstractIn recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great precision, even for large ill-conditioned matrices. In this framework, the present work provides the factorization of the collocation matrices of Newton bases—of relevance when considering the Lagrange interpolation problem—together with an algorithm that allows to numerically compute it to high relative accuracy. This further allows to determine the coefficients of the interpolating polynomial and to compute the singular values and the inverse of the collocation matrix. Conditions that guarantee high relative accuracy for these methods and, in the former case, for the classical recursion formula of divided differences, are determined. Numerical errors due to imprecise computer arithmetic or perturbed input data in the computation of the factorization are analyzed. Finally, numerical experiments illustrate the accuracy and effectiveness of the proposed methods with several algebraic problems, in stark contrast with traditional approaches.

Log determinant of large correlation matrices under infinite fourth moment

Log determinant of large correlation matrices under infinite fourth moment

Data-Driven Compression of Electron-Phonon Interactions

First-principles calculations of electron interactions in materials have seen rapid progress in recent years, with electron-phonon (e−ph) interactions being a prime example. However, these techniques use large matrices encoding the interactions on dense momentum grids, which reduces computational efficiency and obscures interpretability. For e−ph interactions, existing interpolation techniques leverage locality in real space, but the high dimensionality of the data remains a bottleneck to balance cost and accuracy. Here we show an efficient way to compress e−ph interactions based on singular value decomposition (SVD), a widely used matrix and image compression technique. Leveraging (un)constrained SVD methods, we accurately predict material properties related to e−ph interactions—including charge mobility, spin relaxation times, band renormalization, and superconducting critical temperature—while using only a small fraction (1%–2%) of the interaction data. These findings unveil the hidden low-dimensional nature of e−ph interactions. Furthermore, they accelerate state-of-the-art first-principles e−ph calculations by about 2 orders of magnitude without sacrificing accuracy. Our Pareto-optimal parametrization of e−ph interactions can be readily generalized to electron-electron and electron-defect interactions, as well as to other couplings, advancing quantitative studies of condensed matter. Published by the American Physical Society 2024

Inverse matrix estimations by iterative methods with weight functions and their stability analysis

In this paper, we construct a parametric family of iterative methods to compute the inverse of a nonsingular matrix. This class is free of inverse operators. We prove the third-order of convergence under some conditions involving the parameter of the family. Moreover, a dynamical analysis is made for the first time to a matrix iterative method, finding intervals of stability, that include but are wider than those found in the convergence analysis. Numerical tests on large random matrices confirm the results found.

SHREG: Mitigating register redundancy in GPUs

Graphics Processing Units (GPUs) have become dominant accelerators for Machine Learning (ML) and High-Performance Computing (HPC) applications due to their massive parallelism capabilities, through the utilization of general matrix-to-matrix multiplication (GEMM) kernels. However, GEMM kernels often suffer from duplicated memory requests, mainly caused by matrix tiling used for handling large matrices. While GPUs have adopted programmable shared memory to mitigate this issue by preserving frequently reused data in shared memory, GEMM still introduces duplication in register files. Our observations show that the matrix tiling issues memory requests to the same shared memory address for neighboring threads, and this results in a substantial increase in the number of duplicated data in the register files. Such duplication degrades GPU performance by limiting warp-level parallelism due to the register shortage and redundant memory requests to shared memory. We find that the data duplication can be categorized into two types that occur with fixed patterns during the matrix tiling. Based on these observations, we introduce SHREG, an architecture design that enables different threads to share registers for overlapped data from shared memory, effectively reducing duplicated data within the register files. By leveraging the duplication patterns, SHREG utilizes register sharing and improves performance with minimal hardware overhead. Our evaluation shows that SHREG improves performance by 31.4% on various ML applications over the baseline GPU.

Temporally-correlated massive access: joint user activity detection and channel estimation via vector approximate message passing

In the paper, we investigate a massive machine-type communication (mMTC), where numerous single-antenna users communicate with a single-antenna base station while being active. However, the status of user can undergoes multiple transitions between active and inactive states across whole consecutive intervals. Then, we formulate the problem of joint user activity detection and channel estimation within the dynamic compressed sensing (DCS) framework, considering the temporally-correlated user activity across the entire consecutive intervals. To be specific, we introduce a new hybrid vector approximate message passing algorithm for DCS (HyVAMP-DCS). The proposed algorithm comprises a VAMP block for estimating channel and a loopy belief propagation (LBP) block for detecting user activity. Moreover, these two blocks can exchange messages, enhancing the performance of both channel estimation and user activity detection. Importantly, compared to the fragile GAMP algorithm, VAMP is robust and applicable to a much broader class of large random matrices. Furthermore, the fixed points of VAMP’s state evolution align with the replica prediction of the minimum mean-squared error. The simulation results illustrate the superiority of HyVAMP-DCS, demonstrating its significant outperformance over HyGAMP-DCS.

The limiting spectral distribution of large random permutation matrices

Abstract We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let $\textbf X = (\textbf x_1,\ldots,\textbf x_n)$ $\in \mathbb{C} ^{m \times n}$ be an $m \times n$ data matrix after self-normalization (n samples and m features), where $\textbf x_j = (x_{1j}^{*},\ldots, x_{mj}^{*} )^{*}$ . Specifically, we generate a permutation matrix $\textbf X_\pi$ by permuting the entries of $\textbf x_j$ $(j=1,\ldots,n)$ and demonstrate that the empirical spectral distribution of $\textbf {B}_n = ({m}/{n})\textbf{U} _{n} \textbf{X} _\pi \textbf{X} _\pi^{*} \textbf{U} _{n}^{*}$ weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where $\textbf{U} _n$ is a sequence of $p \times m$ non-random complex matrices. The conditions we require are $p/n \to c >0$ and $m/n \to \gamma > 0$ .

Bayesian reconstruction of Cartesian product graph signals with general patterns of missing data

In this paper, we address the challenge of signal reconstruction on Cartesian product graphs, which frequently arises in applications where node-level data may be corrupted by equipment failure or difficult to reliably collect. A recent approach to Graph Signal Reconstruction (GSR) is to formulate it as a Bayesian inference problem, where the underlying signal is presumed to be smooth with respect to the graph topology. However, the resulting linear system derived from this model specification is often intractable to solve directly for general Cartesian product graphs. To overcome this limitation, we introduce two robust and efficient iterative procedures for computing the posterior mean. The first method is a stationary iterative technique based on matrix splitting, while the second employs a conjugate gradient approach with a symmetric graph-spectral preconditioner. We provide a theoretical and numerical comparison of the convergence behaviour of these algorithms, demonstrating that the optimal choice depends on the model hyperparameters. Finally, we propose a novel supervised learning-based technique, enhanced by an active learning strategy, for estimating the posterior marginal variance. This circumvents the need to instantiate, invert or decompose large matrices directly in memory and results in an R2 statistic of over 0.95 on a synthetic test data set. This paper contributes to the development of advanced signal reconstruction techniques applicable in various domains, including large-scale sensor networks and social network analysis.

Feed Forward Neural Network With Column-Wise Matrix–Vector Multiplication on FPGAs

This article presents a reconfigurable accelerator for recurrent neural networks with fine grained column wise matrix vector multiplication (RENOWN).We propose a novel latency hiding architecture for recurrent neural network accelerator using column wise matrix vector multiplication instead of the state of the art row wise operation. This hardware (HW) architecture can eliminate data dependencies to improve the throughput of RNN inference systems. Besides, we introduce a configurable checkerboard tiling strategy which allows large weight matrices, while incorporating various configurations of element-based parallelism (EP) and vector-based parallelism (VP). These optimizations improve the exploitation of parallelism to increase HW utilization and enhance system throughput. Evaluation results show that our design can achieve over 29.6 tera operations per second (TOPS) which would be among the highest for field-programmable gate array (FPGA)-based RNN designs. Compared to state-of-the-art accelerators on FPGAs, our design achieves 3.7–14.8 times better performance and has the highest HW.

Engineering fibronectin-templated multi-component fibrillar extracellular matrices to modulate tissue-specific cell response

Cells assemble fibronectin, the major extracellular matrix (ECM) protein, into fibrillar matrices, which serve as 3D architectural scaffolds to provide, together with other ECM proteins tissue-specific environments. Although recent approaches enable to bioengineer 3D fibrillar fibronectin matrices in vitro, it remains elusive how fibronectin can be co-assembled with other ECM proteins into complex 3D fibrillar matrices that recapitulate tissue-specific compositions and cellular responses. Here, we introduce the engineering of fibrillar fibronectin-templated 3D matrices that can be complemented with other ECM proteins, including vitronectin, collagen, and laminin to resemble ECM architectures observed in vivo. For the co-assembly of different ECM proteins, we employed their innate fibrillogenic mechanisms including shear forces, pH-dependent electrostatic interactions, or specific binding domains. Through recapitulating various tissue-specific ECM compositions and morphologies, the large scale multi-composite 3D fibrillar ECM matrices can guide fibroblast adhesion, 3D fibroblast tissue formation, or tissue morphogenesis of epithelial cells. In other examples, we customize multi-composite 3D fibrillar matrices to support the growth of signal propagating neuronal networks and of human brain organoids. We envision that these 3D fibrillar ECM matrices can be tailored in scale and composition to modulate tissue-specific responses across various biological length scales and systems, and thus to advance manyfold studies of cell biological systems.

Fast annual daylighting simulation and high dynamic range image processing using NumPy

Annual daylighting and point-in-time glare calculations depend on manipulating large matrices representing annual light levels and image pixels, respectively. Advances in computation speed have allowed modern green building standards like LEED, WELL, and EN 17037 to require annual daylighting calculations at a fine resolution across a building’s floorplate. As parallel ray tracing speeds become faster and cameras increase in resolution, matrix manipulation itself has emerged as a bottleneck to delivering fast simulation results. This paper describes a how vectorized computation methods can be applied to annual daylighting simulation and high dynamic range (HDR) image processing. Vectorization allows a computer to perform simple operations on large amounts of data simultaneously. We use the NumPy library in Python to vectorize multiple steps in the calculation of spatial daylight autonomy and annual renderings for glare as example climate-based daylighting metrics, achieving a speedup of two orders of magnitude. We also show that NumPy can manipulate large HDR images with speedups for individual operations up to six orders of magnitude.

Coupling randomisation and sparse modelling for the exploratory analysis of large hyperspectral datasets

Sparse-based models are a powerful tools for data compression, variable reduction, and model complexity reduction. Nevertheless, their major issue is the high computational time needed in large matrices. This manuscript proposes, for the first time, to couple randomised decomposition as a first step before sparsity calculations, followed by a projection of the full data onto a reduced-sparse set of loadings that will drastically reduce the time needed for calculations and built models that are equally reliable as their sparse-based homologous. While this new approach might be valid for several scenarios (exploration, regression and classification), we will focus on exploration methods (like Principal Component Analysis – PCA) applied to large datasets of hyperspectral images. Two datasets of different complexity have been tested, and the benefits of the coupled randomisation and sparse PCA (rsPCA) are extensively studied.

On inference for modularity statistics in structured networks

This paper revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional results in the large-network limit for networks exhibiting nodal community structure. Our work facilitates testing for network differences and can be used in conjunction with existing theoretical guarantees for stochastic blockmodel random graphs. Our results are enabled by recent advances in the study of low-rank truncations of large network adjacency matrices. We provide confirmatory simulation studies and real data analysis pertaining to the network neuroscience study of psychosis, specifically schizophrenia. Collectively, this paper contributes to the limited existing literature to date on statistical inference for modularity-based network analysis. Supplemental materials for this article are available online.

Profile likelihoods for parameters in trans-Gaussian geostatistical models

Profile likelihoods are rarely used in geostatistical models due to the computational burden imposed by repeated decompositions of large variance matrices. Accounting for uncertainty in covariance parameters can be highly consequential in geostatistical models as some covariance parameters are poorly identified, the problem is severe enough that the differentiability parameter of the Matern correlation function is typically treated as fixed. The problem is compounded with anisotropic spatial models as there are two additional parameters to consider. In this paper, we make the following contributions: Firstly, a methodology is created for profile likelihoods for Gaussian spatial models with Matérn family of correlation functions, including anisotropic models. This methodology adopts a novel reparameterization for generation of representative points, and uses GPUs for parallel profile likelihoods computation in software implementation. Then, we show the profile likelihood of the Matérn shape parameter is often quite flat but still identifiable, it can usually rule out very small values. Finally, simulation studies and applications on real data examples show that profile-based confidence intervals of covariance parameters and regression parameters have superior coverage to the traditional standard Wald type confidence intervals.

Deblending and interpolation of subsampled blended seismic data based on damped randomized singular spectrum analysis

AbstractWhen compared to traditional seismic data acquisition, irregular blended acquisition significantly promotes the acquisition efficiency. Yet, the blending noise of subsampled blended data introduces new obstacles for the subsequent processing of seismic data. Due to the predictability of linear events in the frequency–space domain, the constructed Hankel matrices exhibit low‐rank characteristics. However, the blending noise of subsampled blended data increases the rank, so deblending and interpolation can be implemented via rank‐reduction algorithms such as the singular spectrum analysis. The significant computing cost of the singular value decomposition, however, makes the traditional singular spectrum analysis inefficient. An alternative algorithm, known as the randomized singular spectrum analysis, employs the randomized singular value decomposition instead of the traditional singular value decomposition for rank‐reduction. The randomized singular spectrum analysis significantly enhances the efficiency of the decomposition process, particularly when dealing with large Hankel matrices. There still remains some random noise when using the singular spectrum analysis or randomized singular spectrum analysis for subsampled blended data, because the noise subspace and signal subspace are coupled together. Thus, we incorporate a damping operator into the randomized singular value decomposition and propose a novel damped randomized singular spectrum analysis method. The damped randomized singular spectrum analysis combines the advantages of the randomized singular value decomposition and the damping operator to enhance the computational efficiency and suppress the remaining noise. Moreover, an iterative projected gradient descent strategy is introduced to achieve deblended and interpolated seismic data for subsequent processing. Examples from synthetic data and field data are used to demonstrate the effectiveness and superiority of the proposed damped randomized singular spectrum analysis method, which enhances the accuracy and efficiency during simultaneous deblending and interpolation.