142 citations · 214 across the 30 of their papers we have counts for
28 papers · 1 filter
Revisiting column subset selection through the lens of submodularity
Ilse C. F. Ipsen, Arvind K. Saibaba
The problem is to select k columns with maximal volume from a real mxn matrix X. We show that the logarithm of the volume is a set submodular function on columns of X, and for full…
Numerically Stable Cholesky-QR on GPU via Mixed-Precision Randomized Preconditioning
James E. Garrison, Chao Chen, Ilse C. F. Ipsen
Cholesky-QR is among the fastest algorithms for computing the thin QR factorization of tall-and-skinny matrices on GPUs, relying entirely on BLAS-3 operations. However, it is numer…
Perturbation Analysis for Preconditioned Normal Equations in Mixed Precision
James E. Garrison, Ilse C. F. Ipsen
For real matrices of full column-rank, we analyze the conditioning of several types of normal equations that are preconditioned by a randomized preconditioner computed in lower pre…
Many (most?) column subset selection criteria are NP hard for a few columns
Ilse C. F. Ipsen, Arvind K. Saibaba
We consider a variety of criteria for selecting k representative columns from a real mxn matrix A, when sufficiently few columns are required, i.e., 1<= k<= min{rank(A), m/3}. The…
Solution of Least Squares Problems with Randomized Preconditioned Normal Equations
Ilse C. F. Ipsen
We consider the solution of full column-rank least squares problems by means of normal equations that are preconditioned, symmetrically or non-symmetrically, with a randomized prec…
Stochastic Rounding 2.0, with a View towards Complexity Analysis
Petros Drineas, Ilse C. F. Ipsen
Stochastic Rounding is a probabilistic rounding mode that is surprisingly effective in large-scale computations and low-precision arithmetic. Its random nature promotes error cance…
Stable Rank and Intrinsic Dimension of Real and Complex Matrices
Ilse C. F. Ipsen, Arvind K. Saibaba
The notion of `stable rank' of a matrix is central to the analysis of randomized matrix algorithms, covariance estimation, deep neural networks, and recommender systems. We compare…
A randomized preconditioned Cholesky-QR algorithm
James E. Garrison, Ilse C. F. Ipsen
We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR ha…
Small singular values can increase in lower precision
Christos Boutsikas, Petros Drineas, Ilse C. F. Ipsen
We perturb a real matrix of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our…
Statistical Properties of the Probabilistic Numeric Linear Solver BayesCG
Tim W. Reid, Ilse C. F. Ipsen, Jon Cockayne +1
We analyse the calibration of BayesCG under the Krylov prior, a probabilistic numeric extension of the Conjugate Gradient (CG) method for solving systems of linear equations with s…
Robust Parameter Identifiability Analysis via Column Subset Selection
Katherine J. Pearce, Ilse C. F. Ipsen, Mansoor A. Haider +2
We advocate a numerically reliable and accurate approach for practical parameter identifiability analysis: Applying column subset selection (CSS) to the sensitivity matrix, instead…
Precision-aware Deterministic and Probabilistic Error Bounds for Floating Point Summation
Eric Hallman, Ilse C. F. Ipsen
We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the prec…
Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix
Eric Hallman, Ilse C. F. Ipsen, Arvind Saibaba
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds…
Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms
Eric Hallman, Ilse C. F. Ipsen
We analyse the forward error in the floating point summation of real numbers, from algorithms that do not require recourse to higher precision or better hardware. We derive informa…
BayesCG As An Uncertainty Aware Version of CG
Tim W. Reid, Ilse C. F. Ipsen, Jon Cockayne +1
The Bayesian Conjugate Gradient method (BayesCG) is a probabilistic generalization of the Conjugate Gradient method (CG) for solving linear systems with real symmetric positive def…
Multiplicative Perturbation Bounds for Multivariate Multiple Linear Regression in Schatten -Norms
Jocelyn T. Chi, Ilse C. F. Ipsen
Multivariate multiple linear regression (MMLR), which occurs in a number of practical applications, generalizes traditional least squares (multivariate linear regression) to multip…
Probabilistic Error Analysis for Inner Products
Ilse C. F. Ipsen, Hua Zhou
Probabilistic models are proposed for bounding the forward error in the numerically computed inner product (dot product, scalar product) between of two real -vectors. We derive…
A Probabilistic Subspace Bound with Application to Active Subspaces
John T. Holodnak, Ilse C. F. Ipsen, Ralph C. Smith
Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative erro…
Low-Rank Matrix Approximations Do Not Need a Singular Value Gap
Petros Drineas, Ilse C. F. Ipsen
This is a systematic investigation into the sensitivity of low-rank approximations of real matrices. We show that the low-rank approximation errors, in the two-norm, Frobenius norm…
Structural Convergence Results for Approximation of Dominant Subspaces from Block Krylov Spaces
Petros Drineas, Ilse Ipsen, Eugenia-Maria Kontopoulou +1
This paper is concerned with approximating the dominant left singular vector space of a real matrix of arbitrary dimension, from block Krylov spaces generated by the matrix $AA…
Randomized Matrix-free Trace and Log-Determinant Estimators
Arvind K. Saibaba, Alen Alexanderian, Ilse C. F. Ipsen
We present randomized algorithms for estimating the trace and deter- minant of Hermitian positive semi-definite matrices. The algorithms are based on subspace iteration, and access…
kappa_SQ: A Matlab package for randomized sampling of matrices with orthonormal columns
Thomas Wentworth, Ilse Ipsen
The kappa_SQ software package is designed to assist researchers working on randomized row sampling. The package contains a collection of Matlab functions along with a GUI that ties…
Conditioning of Leverage Scores and Computation by QR Decomposition
John T. Holodnak, Ilse C. F. Ipsen, Thomas A. Wentworth
The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range(A). We show that corresponding leverage scores of two matrices A and…
Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds
John T. Holodnak, Ilse C. F. Ipsen
Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for…