arXiv subjects

Soo Go

Publications and source records attributed to Soo Go.

6 recordsLinked to original sources

arXiv2025

Superfast 1-Norm Estimation

A matrix algorithm is said to be superfast (that is, runs at sublinear cost) if it involves much fewer scalars and flops than the input matrix has entries. Such algorithms have been extensively studied and widely applied in modern computations for matrices with low displacement rank and more recently for low-rank approximation of matrices, even though they are known to fail on worst-case inputs in the latter application. We devise novel superfast algorithms that consistently produce accurate 1-norm estimates for real-world matrices and discuss some promising extensions of our surprisingly simple techniques. With further testing and refinement, our algorithms can potentially be adopted in practical computations.

math.NA↗

arXiv2023

Robust Parameter Estimation for Rational Ordinary Differential Equations

We present a new approach for estimating parameters in rational ODE models from given (measured) time series data. In typical existing approaches, an initial guess for the parameter values is made from a given search interval. Then, in a loop, the corresponding outputs are computed by solving the ODE numerically, followed by computing the error from the given time series data. If the error is small, the loop terminates and the parameter values are returned. Otherwise, heuristics/theories are used to possibly improve the guess and continue the loop. These approaches tend to be non-robust in the sense that their accuracy depend on the search interval and the true parameter values; furthermore, they cannot handle the case where the parameters are locally identifiable. In this paper, we propose a new approach, which does not suffer from the above non-robustness. In particular, it does not require making good initial guesses for the parameter values or specifying search intervals. Instead, it uses differential algebra, interpolation of the data using rational functions, and multivariate polynomial system solving. We also compare the performance of the resulting software with several other estimation software packages.

cs.MS↗

arXiv2022

Fast Approximation of Polynomial Zeros and Matrix Eigenvalues

We approximate the d complex zeros of a univariate polynomial p(x) of a degree d or those zeros that lie in a fixed region of interest on the complex plane such as a disc or a square. Our divide and conquer algorithm of STOC 1995 supports solution of this problem in optimal Boolean time (up to a poly-logarithmic factor), that is, runs nearly as fast as one can access the coefficients of p with the precision necessary to support required accuracy of the output. That record complexity has not been matched by any other algorithm yet, but our root-finder of 1995 is quite involved and has never been implemented. We present alternative nearly optimal root-finders based on our novel variants of the classical subdivision iterations. Unlike our predecessor of 1995, we require randomization of Las Vegas type, allowing us to detect any output error at a dominated computational cost, but our new root-finders are much simpler to implement than their predecessor of 1995. According to the results of extensive test with standard test polynomials for their preliminary version, which incorporates only a part of our novel techniques, the new root-finders compete and for a large class of inputs significantly supersedes the package of root-finding subroutines MPSolve, which for decades has been user's choice package. Unlike our predecessor of 1995 and all known fast algorithms for the cited tasks of polynomial root-finding, our new algorithms can be also applied to a polynomial given by a black box oracle for its evaluation rather than by its coefficients. This makes our root-finders particularly efficient for polynomials p(x) that can be evaluated fast such as the Mandelbrot polynomials or those given by the sum of a small number of shifted monomials. Our algorithm can be readily extended to fast approximation of the eigenvalues of a matrix or a matrix polynomial.

cs.SC↗

arXiv2019

CUR Low Rank Approximation of a Matrix at Sublinear Cost

Low rank approximation of a matrix (hereafter LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with an LRA at sublinear cost -- by using much fewer memory cells and flops than an input matrix M has entries. For worst case inputs one cannot compute even a reasonably close LRA at sublinear cost, but in computational practice accurate LRAs, even in their memory efficient form of CUR LRAs, are routinely obtained at sublinear cost for large and important classes of matrices, in particular by means of Cross-Approximation iterations, which specialize Alternating Direction techniques to LRA. We identify some classes of matrices for which CUR LRA are computed at sublinear cost as well as some sublinear cost LRA algorithms that are empirically accurate for large classes of inputs. Some of our techniques and concepts can be of independent interests.

math.NA↗

arXiv2019

Iterative Refinement and Oversampling for Low Rank Approximation

Iterative refinement is particularly popular for numerical solution of linear systems of equations. We extend it to Low Rank Approximation of a matrix (LRA) and observe close link of the resulting algorithm to oversampling techniques, commonly used in randomized LRA algorithms. We elaborate upon this link and revisit oversampling and some efficient randomized LRA algorithms. Applied with sparse sketch matrices they run significantly faster and in particular yield Very Low Rank Approximation (VLRA) at sublinear cost, using much fewer scalars and flops than the input matrix has entries. This is achieved at the price of deterioration of output accuracy, but according to our formal and empirical study subsequent oversampling improves accuracy to near-optimal level under the spectral norm for a large sub-class of matrices with fast decaying spectra of singular values.

math.NA↗

arXiv2018

Superfast Low Rank Approximation

Low rank approximation of a matrix (LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with an LRA superfast -- by using much fewer memory cells and flops than an input matrix has entries. Can we, however, compute an LRA of a matrix superfast? YES and NO. For worst case inputs, any LRA algorithm fails miserably unless it involves all input entries, but in computational practice worst case inputs seem to appear rarely, and accurate LRA are routinely computed superfast for large and important classes of matrices, in particular in the memory efficient form of CUR, widely used in data analysis. We advance formal study of this YES and NO coexistence by proving novel universal upper bounds on the spectral output error norms of all CUR LRA algorithms and, under a fixed probabilistic structure in the space of input matrices, on both spectral and Frobenius error norms of nearly all sketching LRA algorithms. These bounds imply that superfast LRA algorithms of the two kinds fail miserably only for a very narrow input class. Furthermore, in our numerical tests such superfast algorithms were consistently much more accurate than our upper estimates ensure and usually were reasonably close to optimal.

math.NA↗