Randomized numerical linear algebra: why you should care and how you can use it
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Murray, Riley John [Sandia National Laboratories (↗
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Yet Another NLA Library: T-LAPACK
In recent years, Randomized numerical linear algebra (RandNLA) proved to be more than a theoretical novelty: projects like RandLAPACK demonstrate its practical value across architectures, and projects like RandBLAS build trust in randomization as a tool for high-performance NLA. This BoF considers two main questions. First, what are the pressing issues in software standards and implementation that need to be resolved for RandNLA to become a core component of HPC? Second, how can we mobilize a community effort to make progress on these issues? The BoF will engage the audience to discuss the idea of growing the role of RandNLA in high-performance computing and what it would take to scale from niche prototypes to robust, production-quality software libraries.
97 MATHEMATICS AND COMPUTING↗
Randomized algorithms for accelerating linear algebraic computations
The project supported the development of new methodologies for performing matrix computations that form key building blocks in modern scientific computing, such as low rank approximation of matrices, and efficient representations of global operators that arise in simulations of physical phenomena.
97 MATHEMATICS AND COMPUTING↗
Surrogate-Based Autotuning for Randomized Sketching Algorithms in Regression Problems
Algorithms from Randomized Numerical Linear Algebra (RandNLA) are known to be effective in handling high-dimensional computational problems, providing high-quality empirical performance as well as strong probabilistic guarantees. However, their practical application is complicated by the fact that the user needs to set various algorithm-specific tuning parameters which are different from those used in traditional NLA. This paper demonstrates how a surrogate-based autotuning approach can be used to address fundamental problems of parameter selection in RandNLA algorithms. In particular, we provide a detailed investigation of surrogate-based autotuning for sketch-and-precondition (SAP)-based randomized least squares methods, which have been one of the great success stories in modern RandNLA. Empirical results show that our surrogate-based autotuning approach can achieve near-optimal performance with much less tuning cost than a random search (up to about 7.6x fewer trials of different parameter configurations). Moreover, while our experiments focus on least squares, our results demonstrate a general-purpose autotuning pipeline applicable to any kind of RandNLA algorithm.
Cho, Younghyun↗
Randomized Algorithms for Symmetric Nonnegative Matrix Factorization
Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a matrix with a product of a nonnegative, low-rank matrix and it transpose. To design faster and more scalable algorithms for SymNMF we develop two randomized algorithms for its computation. The first method uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to uses this as a low-rank input to rapidly compute a SymNMF. The second methods uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. Here, we prove theoretically that leverage score sampling can approximately solve constrained least squares problems to e-accuracy. Finally we demonstrate both methods work in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.
97 MATHEMATICS AND COMPUTING↗
Randomized Algorithms for Linear Solvers
Recently, randomized algorithms in numerical linear algebra, specifically those centered around random sketching, have gained traction in primarily theoretical research due to their potential to significantly reduce problem dimensionality at the cost of an O(1) multiplicative distortion factor. It has been assumed that this sketching can be done efficiently, but thorough investigation into how precisely to do it has been neglected. Moreover, the theory-based community has argued for sketching’s ability to reduce computational cost via complexity analysis, but has not researched how it affects the stability of the algorithms. At Sandia, efficient linear solvers that scale well on modern HPC architectures while maintaining stability are imperative for practical applications. In this LDRD, we developed a random sketching strategy that is substantially faster than existing ones, and demonstrate its superior performance in practice on a NVIDIA H100 GPU. Moreover, we show how this can be used to significantly outperform existing linear least squares solvers while improving the solver’s stability as well. Additionally, we demonstrate how this sketching strategy can be used to make a fast, stable QR factorization that can subsequently be used in s-step and block Krylov solvers. Finally, we incorporate a sketching-based block orthogonalization scheme into s-step GMRES, which is stable and faster than existing approaches on the Perlmutter supercomputer.
97 MATHEMATICS AND COMPUTING↗
Ensemble Simulation Techniques and Fast Randomized Algorithms
The major goals of the project were to develop and analyze new ensemble simulation techniques, including trajectory stratification and preconditioned MCMC techniques, as well as develop fast numerical linear algebra techniques closely related to ensemble simulation ideas. The trajectory stratification techniques involve simulating in parallel short trajectory fragments of a Markov process confined to a specific region of space‐time and then patching together the statistics gathered to assemble estimates of very general dynamical properties. We have also developed this approach for rare event simulation and extended the techniques to applications requiring a more general framework (such as electronic structure calculations). The preconditioned MCMC techniques involve simulating multiple Markov chains in parallel and then using information from the ensemble to speed the mixing of each individual chain. The fast randomized linear algebra methods are motivated by the diffusion Monte Carlo technique, but are applicable to finding the dominant eigenvalue of (almost) general matrices. For most non‐negative matrices, the schemes result in an error (compared to the power method) that is constant in the dimension of the problem. For more general matrices, we see a very clear sublinear cost trend in computational tests.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Randomized Algorithms for Low-Rank Matrix and Tensor Decompositions
This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality reduction, such as the singular value decomposition (SVD) or interpolative (ID) and CUR decompositions. Recent advances in randomized dimensionality reduction are discussed, including new methods of fast matrix sketching and sampling techniques, which are incorporated into classical matrix algorithms for fast low-rank matrix approximations. The extension of randomized matrix algorithms to tensors is then explored for several low-rank tensor decompositions in the CP and Tucker formats, including the higher-order SVD, ID, and CUR decomposition.
Pearce, Katherine J. [The University of Texas at A↗
Coupling flux balance analysis with reactive transport modeling through machine learning for rapid and stable simulation of microbial metabolic switching
Integrating genome-scale metabolic networks with reactive transport models (RTMs) provides a detailed description of the dynamic changes in microbial growth and metabolism. Despite promising demonstrations in the past, computational inefficiency has been pointed out as a critical issue to overcome because it requires repeated application of linear programming (LP) to obtain flux balance analysis (FBA) solutions in every time step and spatial grid. To address this challenge, we propose a new simulation method where we train and validate artificial neural networks (ANNs) using randomly sampled FBA solutions and incorporate the resulting surrogate FBA model (represented as algebraic equations) into RTMs as source/sink terms. We demonstrate the efficiency of our method via a case study of Shewanella oneidensis MR-1. During aerobic growth on lactate, S. oneidensis produces metabolic byproducts (such as pyruvate and acetate), which are subsequently consumed as alternative carbon sources when the preferred nutrients are depleted. To effectively simulate these complex dynamics, we used a cybernetic approach that models metabolic switches as the outcome of dynamic competition among multiple growth options. In both zero-dimensional batch and one-dimensional column configurations, the ANN-based surrogate models achieved substantial reduction of computational time by several orders of magnitude compared to the original LP-based FBA models. Moreover, the ANN models produced robust solutions without any special measures to prevent numerical instability. These developments significantly promote our ability to utilize genome-scale networks in complex, multi-physics, and multi-dimensional ecosystem modeling.
59 BASIC BIOLOGICAL SCIENCES↗
Window convolution of the galaxy clustering bispectrum
In galaxy survey analysis, the observed clustering statistics do not directly match theoretical predictions but rather have been processed by a window function that arises from the survey geometry including the sky footprint, redshift-dependent background number density and systematic weights. While window convolution of the power spectrum is well studied, for the bispectrum with a larger number of degrees of freedom, it poses a significant numerical and computational challenge. In this work, we consider the effect of the survey window in the tripolar spherical harmonic decomposition of the bispectrum and lay down a formal procedure for their convolution via a series expansion of configuration-space three-point correlation functions, which was first proposed by Sugiyama et al. (2019). We then provide a linear algebra formulation of the full window convolution, where an unwindowed bispectrum model vector can be directly premultiplied by a window matrix specific to each survey geometry. To validate the pipeline, we focus on the Dark Energy Spectroscopic Instrument (DESI) Data Release 1 (DR1) luminous red galaxy (LRG) sample in the South Galactic Cap (SGC) in the redshift bin 0.4 ≤ z ≤ 0.6. We first perform convergence checks on the measurement of the window function from discrete random catalogues, and then investigate the convergence of the window convolution series expansion truncated at a finite of number of terms as well as the performance of the window matrix. This work highlights the differences in window convolution between the power spectrum and bispectrum, and provides a streamlined pipeline for the latter for current surveys such as DESI and the Euclid mission.
79 ASTRONOMY AND ASTROPHYSICS↗
Developing ML/AI Methods for High-Throughput Characterization of Multiple-Sensor Streams of Tokamak Dynamics for High-Speed Control (Final Report)
This project evaluated and developed new mathematical and algorithmic techniques capable of handling (in real-time) the growing amounts of data generated by modern fusion research. While existing numerical linear algebra (NLA) methods provide the backbone to classical data analysis and algorithms, these methods fundamentally do not port to distributed architectures nor do they allow low-latency data reduction for control. Motivated by the needs for modern fusion reactors, this project explored and implemented new numerical methods to characterize plasma dynamics, respond in real-time to discharge evolution, and to process massive-scale data accurately and rapidly more fully. This project links expertise in multiple-sensor diagnostics of tokamak plasma dynamics from Columbia University’s Plasma Physics Laboratory with expertise in massive-scale data reduction and extreme data control algorithms at Columbia University’s Data Science Institute. This interdisciplinary project (i) applied machine learning methods, (ii) implemented a properly-trained neural-network for very fast processing of high-speed plasma videography, and (ii) developed the applied mathematical methods, based on randomized-NLA (rNLA) routines, for data analysis, reduction, and real-time control. The Columbia University High Beta Tokamak-Extended Pulse (HBT-EP) facility provided data to test new algorithms and partnership with Columbia University's Data Sciences Institute evaluated the broader use of new algorithms for many challenging control applications.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗