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Simon Barthelme

Publications and source records attributed to Simon Barthelme.

9 recordsLinked to original sources
arXiv2025

Computing asymptotic eigenvectors and eigenvalues of perturbed symmetric matrices

Computing the eigenvectors and eigenvalues of a perturbed matrix can be remarkably difficult when the unperturbed matrix has repeated eigenvalues. In this work we show how the limiting eigenvectors and eigenvalues of a symmetric matrix $K(\varepsilon)$ as $\varepsilon \to 0$ can be obtained relatively easily from successive Schur complements, provided that the entries scale in different orders of $\varepsilon$. If the matrix does not directly exhibit this structure, we show that putting the matrix into a ``generalised kernel form'' can be very informative. The resulting formulas are much simpler than classical expressions obtained from complex integrals involving the resolvent. We apply our results to the problem of computing the eigenvalues and eigenvectors of kernel matrices in the ``flat limit'', a problem that appears in many applications in statistics and approximation theory. In particular, we prove a conjecture from [SIAM J. Matrix Anal. Appl., 2021, 42(1):17--57] which connects the eigenvectors of kernel matrices to multivariate orthogonal polynomials.

math.NA↗
arXiv2022

Variance Reduction for Inverse Trace Estimation via Random Spanning Forests

The trace $\tr(q(\ma{L} + q\ma{I})^{-1})$, where $\ma{L}$ is a symmetric diagonally dominant matrix, is the quantity of interest in some machine learning problems. However, its direct computation is impractical if the matrix size is large. State-of-the-art methods include Hutchinson's estimator combined with iterative solvers, as well as the estimator based on random spanning forests (a random process on graphs). In this work, we show two ways of improving the forest-based estimator via well-known variance reduction techniques, namely control variates and stratified sampling. Implementing these techniques is easy, and provides substantial variance reduction, yielding comparable or better performance relative to state-of-the-art algorithms.

eess.SP↗
arXiv2022

Graph Tikhonov Regularization and Interpolation via Random Spanning Forests

Novel Monte Carlo estimators are proposed to solve both the Tikhonov regularization (TR) and the interpolation problems on graphs. These estimators are based on random spanning forests (RSF), the theoretical properties of which enable to analyze the estimators' theoretical mean and variance. We also show how to perform hyperparameter tuning for these RSF-based estimators. TR is a component in many well-known algorithms, and we show how the proposed estimators can be easily adapted to avoid expensive intermediate steps in generalized semi-supervised learning, label propagation, Newton's method and iteratively reweighted least squares. In the experiments, we illustrate the proposed methods on several problems and provide observations on their run time.

cs.DM↗
arXiv2018

Optimized Algorithms to Sample Determinantal Point Processes

In this technical report, we discuss several sampling algorithms for Determinantal Point Processes (DPP). DPPs have recently gained a broad interest in the machine learning and statistics literature as random point processes with negative correlation, i.e., ones that can generate a "diverse" sample from a set of items. They are parametrized by a matrix $\mathbf{L}$, called $L$-ensemble, that encodes the correlations between items. The standard sampling algorithm is separated in three phases: 1/~eigendecomposition of $\mathbf{L}$, 2/~an eigenvector sampling phase where $\mathbf{L}$'s eigenvectors are sampled independently via a Bernoulli variable parametrized by their associated eigenvalue, 3/~a Gram-Schmidt-type orthogonalisation procedure of the sampled eigenvectors. In a naive implementation, the computational cost of the third step is on average $\mathcal{O}(Nμ^3)$ where $μ$ is the average number of samples of the DPP. We give an algorithm which runs in $\mathcal{O}(Nμ^2)$ and is extremely simple to implement. If memory is a constraint, we also describe a dual variant with reduced memory costs. In addition, we discuss implementation details often missing in the literature.

stat.CO↗
arXiv2017

Échantillonnage de signaux sur graphes via des processus déterminantaux

We consider the problem of sampling k-bandlimited graph signals, ie, linear combinations of the first k graph Fourier modes. We know that a set of k nodes embedding all k-bandlimited signals always exists, thereby enabling their perfect reconstruction after sampling. Unfortunately, to exhibit such a set, one needs to partially diagonalize the graph Laplacian, which becomes prohibitive at large scale. We propose a novel strategy based on determinantal point processes that side-steps partial diagonalisation and enables reconstruction with only O(k) samples. While doing so, we exhibit a new general algorithm to sample determinantal process, faster than the state-of-the-art algorithm by an order k.

cs.DS↗
arXiv2016

Visualizing the Effects of a Changing Distance on Data Using Continuous Embeddings

Most Machine Learning (ML) methods, from clustering to classification, rely on a distance function to describe relationships between datapoints. For complex datasets it is hard to avoid making some arbitrary choices when defining a distance function. To compare images, one must choose a spatial scale, for signals, a temporal scale. The right scale is hard to pin down and it is preferable when results do not depend too tightly on the exact value one picked. Topological data analysis seeks to address this issue by focusing on the notion of neighbourhood instead of distance. It is shown that in some cases a simpler solution is available. It can be checked how strongly distance relationships depend on a hyperparameter using dimensionality reduction. A variant of dynamical multi-dimensional scaling (MDS) is formulated, which embeds datapoints as curves. The resulting algorithm is based on the Concave-Convex Procedure (CCCP) and provides a simple and efficient way of visualizing changes and invariances in distance patterns as a hyperparameter is varied. A variant to analyze the dependence on multiple hyperparameters is also presented. A cMDS algorithm that is straightforward to implement, use and extend is provided. To illustrate the possibilities of cMDS, cMDS is applied to several real-world data sets.

stat.ML↗
arXiv2014

Fast matrix computations for functional additive models

It is common in functional data analysis to look at a set of related functions: a set of learning curves, a set of brain signals, a set of spatial maps, etc. One way to express relatedness is through an additive model, whereby each individual function $g_{i}\left(x\right)$ is assumed to be a variation around some shared mean $f(x)$. Gaussian processes provide an elegant way of constructing such additive models, but suffer from computational difficulties arising from the matrix operations that need to be performed. Recently Heersink & Furrer have shown that functional additive model give rise to covariance matrices that have a specific form they called quasi-Kronecker (QK), whose inverses are relatively tractable. We show that under additional assumptions the two-level additive model leads to a class of matrices we call restricted quasi-Kronecker, which enjoy many interesting properties. In particular, we formulate matrix factorisations whose complexity scales only linearly in the number of functions in latent field, an enormous improvement over the cubic scaling of naïve approaches. We describe how to leverage the properties of rQK matrices for inference in Latent Gaussian Models.

stat.CO↗