arXiv subjects

Irina-Beatrice Haas

Publications and source records attributed to Irina-Beatrice Haas.

4 recordsLinked to original sources

arXiv2026

SuperPCA: subspace analysis and an efficient algorithm for high-dimensional PCA

Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.

stat.ML↗

arXiv2026

A multilevel sketch-and-solve method for overdetermined least squares problems

Sketch-and-solve (SAS) is a very successful method to efficiently estimate the solution of heavily overdetermined large linear least squares problems. It uses random sketching to reduce the size of the problem, hence reducing the computational cost. Several authors have shown that averaging several solutions from SAS further improves the accuracy, which is measured by the residual associated to the approximate solution. Going further, we combine solutions from sketch-and-solve in a multilevel manner, such that the approximate solution is a combination of SAS samples obtained from small sketches and more accurate correction terms obtained from larger sketches. We first consider the variance of the estimator, which depends on the variance of the coarse samples and the correction terms. We show that the variance of the correction terms on each level follows a trend and decreases faster than the variance of the simple SAS estimator. However, we then show that the overall computational cost of our multilevel framework is slightly higher than that of the simple average estimator, so a naive application of multilevel methods appears unattractive for least squares problems.

math.NA↗

arXiv2025

Sharp error bounds for approximate eigenvalues and singular values from subspace methods

Subspace methods are commonly used for finding approximate eigenvalues and singular values of large-scale matrices. Once a subspace is found, the Rayleigh-Ritz method (for symmetric eigenvalue problems) and Petrov-Galerkin projection (for singular values) are the de facto method for extraction of eigenvalues and singular values. In this work we derive quadratic error bounds for approximate eigenvalues of symmetric matrices obtained via the Rayleigh-Ritz process. Our bounds take advantage of the fact that extremal eigenpairs tend to converge faster than the rest, hence having smaller residuals $\|A\widehat x_i-θ_i\widehat x_i\|_2$, where $(θ_i,\widehat x_i)$ is a Ritz pair (approximate eigenpair). The proof uses the structure of the perturbation matrix underlying the Rayleigh-Ritz method to bound the components of its eigenvectors. In this way, we obtain a bound of the form $c\frac{\|A\widehat x_i-θ_i\widehat x_i\|_2^2}{\mbox{Gap}_i}$, where $\mbox{Gap}_i$ is roughly the gap between the $i$th Ritz value and the eigenvalues that are not approximated by the Ritz process, and $c> 1$ is a modest scalar. Our bound is adapted to each Ritz value and is robust to clustered Ritz values, which is a key improvement over existing results. We further show that the bound is asymptotically sharp, and generalize it to singular values of arbitrary real matrices. Finally, we apply these bounds to several methods for computing eigenvalues and singular values, and illustrate the sharpness of our bounds in a number of computational settings, including Krylov methods and randomized algorithms.

math.NA↗

arXiv2025

A nested MLMC framework for efficient simulations on FPGAs

Multilevel Monte Carlo (MLMC) reduces the total computational cost of financial option pricing by combining SDE approximations with multiple resolutions. This paper explores a further avenue for reducing cost and improving power efficiency through the use of low precision calculations on configurable hardware devices such as Field-Programmable Gate Arrays (FPGAs). We propose a new framework that exploits approximate random variables and fixed-point operations with optimised precision to generate most SDE paths with a lower cost and reduce the overall cost of the MLMC framework. We first discuss several methods for the cheap generation of approximate random Normal increments. To set the bit-width of variables in the path generation we then propose a rounding error model and optimise the precision of all variables on each MLMC level. With these key improvements, our proposed framework offers higher computational savings than the existing mixed-precision MLMC frameworks.

q-fin.CP↗