arXiv subjects

Simon Buchholz

Publications and source records attributed to Simon Buchholz.

26 records · Page 2Linked to original sources

arXiv2023

AutoML Two-Sample Test

Two-sample tests are important in statistics and machine learning, both as tools for scientific discovery as well as to detect distribution shifts. This led to the development of many sophisticated test procedures going beyond the standard supervised learning frameworks, whose usage can require specialized knowledge about two-sample testing. We use a simple test that takes the mean discrepancy of a witness function as the test statistic and prove that minimizing a squared loss leads to a witness with optimal testing power. This allows us to leverage recent advancements in AutoML. Without any user input about the problems at hand, and using the same method for all our experiments, our AutoML two-sample test achieves competitive performance on a diverse distribution shift benchmark as well as on challenging two-sample testing problems. We provide an implementation of the AutoML two-sample test in the Python package autotst.

cs.LG↗

arXiv2022

Function Classes for Identifiable Nonlinear Independent Component Analysis

Unsupervised learning of latent variable models (LVMs) is widely used to represent data in machine learning. When such models reflect the ground truth factors and the mechanisms mapping them to observations, there is reason to expect that they allow generalization in downstream tasks. It is however well known that such identifiability guaranties are typically not achievable without putting constraints on the model class. This is notably the case for nonlinear Independent Component Analysis, in which the LVM maps statistically independent variables to observations via a deterministic nonlinear function. Several families of spurious solutions fitting perfectly the data, but that do not correspond to the ground truth factors can be constructed in generic settings. However, recent work suggests that constraining the function class of such models may promote identifiability. Specifically, function classes with constraints on their partial derivatives, gathered in the Jacobian matrix, have been proposed, such as orthogonal coordinate transformations (OCT), which impose orthogonality of the Jacobian columns. In the present work, we prove that a subclass of these transformations, conformal maps, is identifiable and provide novel theoretical results suggesting that OCTs have properties that prevent families of spurious solutions to spoil identifiability in a generic setting.

stat.ML↗

arXiv2021

The Inductive Bias of Quantum Kernels

It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classically tractable kernels can correspond to high- or infinite-dimensional reproducing kernel Hilbert spaces (RKHS). We analyze the spectral properties of quantum kernels and find that we can expect an advantage if their RKHS is low dimensional and contains functions that are hard to compute classically. If the target function is known to lie in this class, this implies a quantum advantage, as the quantum computer can encode this inductive bias, whereas there is no classically efficient way to constrain the function class in the same way. However, we show that finding suitable quantum kernels is not easy because the kernel evaluation might require exponentially many measurements. In conclusion, our message is a somewhat sobering one: we conjecture that quantum machine learning models can offer speed-ups only if we manage to encode knowledge about the problem at hand into quantum circuits, while encoding the same bias into a classical model would be hard. These situations may plausibly occur when learning on data generated by a quantum process, however, they appear to be harder to come by for classical datasets.

quant-ph↗

arXiv2021

Assaying Large-scale Testing Models to Interpret COVID-19 Case Numbers

Large-scale testing is considered key to assess the state of the current COVID-19 pandemic. Yet, the link between the reported case numbers and the true state of the pandemic remains elusive. We develop mathematical models based on competing hypotheses regarding this link, thereby providing different prevalence estimates based on case numbers, and validate them by predicting SARS-CoV-2-attributed death rate trajectories. Assuming that individuals were tested based solely on a predefined risk of being infectious implies the absolute case numbers reflect the prevalence, but turned out to be a poor predictor, consistently overestimating growth rates at the beginning of two COVID-19 epidemic waves. In contrast, assuming that testing capacity is fully exploited performs better. This leads to using the percent-positive rate as a more robust indicator of epidemic dynamics, however we find it is subject to a saturation phenomenon that needs to be accounted for as the number of tests becomes larger.

stat.AP↗

arXiv2019

Phase transitions for a class of gradient fields

We consider gradient fields on $\mathbb{Z}^d$ for potentials $V$ that can be expressed as $$e^{-V(x)}=pe^{-\frac{qx^2}{2}}+(1-p)e^{-\frac{x^2}{2}}.$$ This representation allows us to associate a random conductance type model to the gradient fields with zero tilt. We investigate this random conductance model and prove correlation inequalities, duality properties, and uniqueness of the Gibbs measure in certain regimes. Moreover, we show that there is a close relation between Gibbs measures of the random conductance model and gradient Gibbs measures with zero tilt for the potential $V$. Based on these results we can give a new proof for the non-uniqueness of gradient Gibbs measures without using reflection positivity. We also show uniqueness of ergodic zero tilt gradient Gibbs measures for almost all values of $p$ and $q$ and, in dimension $d\geq 4$, for $q$ close to one or for $p(1-p)$ sufficiently small.

math.PR↗

arXiv2018

Probability to be positive for the membrane model in dimensions 2 and 3

We consider the membrane model on a box $V_N\subset \mathbb{Z}^n$ of size $(2N+1)^n$ with zero boundary condition in the subcritical dimensions $n=2$ and $n=3$. We show optimal estimates for the probability that the field is positive in a subset $D_N$ of $V_N$. In particular we obtain for $D_N=V_N$ that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area $N^{n-1}$.

math.PR↗

arXiv2017

Finite Range Decomposition for Gaussian Measures with Improved Regularity

We consider a family of gradient Gaussian vector fields on the torus $(\mathbb{Z}/L^N\mathbb{Z})^d$. Adams, Kotecký, Müller and independently Bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators, i.e., the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter $L^k$. We improve this result and show that the decay behaviour of the kernels in Fourier space can be controlled. Then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition. In particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics.

math-ph↗

arXiv2013

Multivariate Central Limit Theorem in Quantum Dynamics

We consider the time evolution of $N$ bosons in the mean field regime for factorized initial data. In the limit of large $N$, the many body evolution can be approximated by the non-linear Hartree equation. In this paper we are interested in the fluctuations around the Hartree dynamics. We choose $k$ self-adjoint one-particle operators $O_1, \dots, O_k$ on $L^2 (\R^3)$, and we average their action over the $N$-particles. We show that, for every fixed $t \in \R$, expectations of products of functions of the averaged observables approach, as $N \to \infty$, expectations with respect to a complex Gaussian measure, whose covariance matrix can be expressed in terms of a Bogoliubov transformation describing the dynamics of quantum fluctuations around the mean field Hartree evolution. If the operators $O_1, \dots, O_k$ commute, the Gaussian measure is real and positive, and we recover a "classical" multivariate central limit theorem. All our results give explicit bounds on the rate of the convergence (we obtain therefore Berry-Ess{é}en type central limit theorems).

math-ph↗