arXiv subjects

Maike Meier

Publications and source records attributed to Maike Meier.

6 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

Reducing acquisition time and radiation damage: data-driven subsampling for spectro-microscopy

Spectro-microscopy is an experimental technique which can be used to observe spatial variations in chemical state and changes in chemical state over time or under experimental conditions. As a result it has broad applications across areas such as energy materials, catalysis, environmental science and biological samples. However, the technique is often limited by factors such as long acquisition times and radiation damage. We present two measurement strategies that allow for significantly shorter experiment times and total doses applied. The strategies are based on taking only a small subset of all the measurements (e.g. sparse acquisition or subsampling), and then computationally reconstructing all unobserved measurements using mathematical techniques. The methods are data-driven, using spectral and spatial importance subsampling distributions to identify important measurements. As a result, taking as little as 4-6\% of the measurements is sufficient to capture the same information as in a conventional scan.

physics.med-ph↗

arXiv2025

Fast randomized least-squares solvers can be just as accurate and stable as classical direct solvers

One of the greatest success stories of randomized algorithms for linear algebra has been the development of fast, randomized algorithms for highly overdetermined linear least-squares problems. However, none of the existing algorithms is backward stable, preventing them from being deployed as drop-in replacements for existing QR-based solvers. This paper introduces sketch-and-precondition with iterative refinement (SPIR) and FOSSILS, two provably backward stable randomized least-squares solvers. SPIR and FOSSILS combine iterative refinement with a preconditioned iterative method applied to the normal equations and converge at the same rate as existing randomized least-squares solvers. This work offers the promise of incorporating randomized least-squares solvers into existing software libraries while maintaining the same level of accuracy and stability as classical solvers.

math.NA↗

arXiv2024

Fast randomized numerical rank estimation for numerically low-rank matrices

Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an ad-hoc fashion in large-scale settings. In this work we develop a randomized algorithm for estimating the numerical rank of a (numerically low-rank) matrix. The algorithm is based on sketching the matrix with random matrices from both left and right; the key fact is that with high probability, the sketches preserve the orders of magnitude of the leading singular values. We prove a result on the accuracy of the sketched singular values and show that gaps in the spectrum are detected. For an $m\times n$ $(m\geq n)$ matrix of numerical rank $r$, the algorithm runs with complexity $O(mn\log n+r^3)$, or less for structured matrices. The steps in the algorithm are required as a part of many low-rank algorithms, so the additional work required to estimate the rank can be even smaller in practice. Numerical experiments illustrate the speed and robustness of our rank estimator.

math.NA↗

arXiv2023

Are sketch-and-precondition least squares solvers numerically stable?

Sketch-and-precondition techniques are efficient and popular for solving large least squares (LS) problems of the form $Ax=b$ with $A\in\mathbb{R}^{m\times n}$ and $m\gg n$. This is where $A$ is ``sketched" to a smaller matrix $SA$ with $S\in\mathbb{R}^{\lceil cn\rceil\times m}$ for some constant $c>1$ before an iterative LS solver computes the solution to $Ax=b$ with a right preconditioner $P$, where $P$ is constructed from $SA$. Prominent sketch-and-precondition LS solvers are Blendenpik and LSRN. We show that the sketch-and-precondition technique in its most commonly used form is not numerically stable for ill-conditioned LS problems. For provable and practical backward stability and optimal residuals, we suggest using an unpreconditioned iterative LS solver on $(AP)z=b$ with $x=Pz$. Provided the condition number of $A$ is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a backward error comparable to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to argue that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems. Additionally, we provide experimental evidence that using the sketch-and-solve solution as a starting vector in sketch-and-precondition algorithms (as suggested by Rokhlin and Tygert in 2008) should be highly preferred over the zero vector. The initialization often results in much more accurate solutions -- albeit not always backward stable ones.

math.NA↗

arXiv2022

Randomized algorithms for Tikhonov regularization in linear least squares

We describe two algorithms to efficiently solve regularized linear least squares systems based on sketching. The algorithms compute preconditioners for $\min \|Ax-b\|^2_2 + λ\|x\|^2_2$, where $A\in\mathbb{R}^{m\times n}$ and $λ>0$ is a regularization parameter, such that LSQR converges in $\mathcal{O}(\log(1/ε))$ iterations for $ε$ accuracy. We focus on the context where the optimal regularization parameter is unknown, and the system must be solved for a number of parameters $λ$. Our algorithms are applicable in both the underdetermined $m\ll n$ and the overdetermined $m\gg n$ setting. Firstly, we propose a Cholesky-based sketch-to-precondition algorithm that uses a `partly exact' sketch, and only requires one sketch for a set of $N$ regularization parameters $λ$. The complexity of solving for $N$ parameters is $\mathcal{O}(mn\log(\max(m,n)) +N(\min(m,n)^3 + mn\log(1/ε)))$. Secondly, we introduce an algorithm that uses a sketch of size $\mathcal{O}(\text{sd}_λ(A))$ for the case where the statistical dimension $\text{sd}_λ(A)\ll\min(m,n)$. The scheme we propose does not require the computation of the Gram matrix, resulting in a more stable scheme than existing algorithms in this context. We can solve for $N$ values of $λ_i$ in $\mathcal{O}(mn\log(\max(m,n)) + \min(m,n)\,\text{sd}_{\minλ_i}(A)^2 + Nmn\log(1/ε))$ operations.

math.NA↗