arXiv subjects

Philip Schär

Publications and source records attributed to Philip Schär.

5 recordsLinked to original sources

arXiv2024

A Dimension-Independent Bound on the Wasserstein Contraction Rate of a Geodesic Random Walk on the Sphere

We theoretically analyze the properties of a geodesic random walk on the Euclidean $d$-sphere. Specifically, we prove that the random walk's transition kernel is Wasserstein contractive with a contraction rate which can be bounded from above independently of the dimension $d$. Our result is of particular interest due to its implications regarding the potential for dimension-independent performance of both geodesic slice sampling on the sphere and Gibbsian polar slice sampling, which are Markov chain Monte Carlo methods for approximate sampling from essentially arbitrary distributions on their respective state spaces.

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arXiv2024

Parallel Affine Transformation Tuning of Markov Chain Monte Carlo

The performance of Markov chain Monte Carlo samplers strongly depends on the properties of the target distribution such as its covariance structure, the location of its probability mass and its tail behavior. We explore the use of bijective affine transformations of the sample space to improve the properties of the target distribution and thereby the performance of samplers running in the transformed space. In particular, we propose a flexible and user-friendly scheme for adaptively learning the affine transformation during sampling. Moreover, the combination of our scheme with Gibbsian polar slice sampling is shown to produce samples of high quality at comparatively low computational cost in several settings based on real-world data.

stat.ME↗

arXiv2023

Dimension-independent spectral gap of polar slice sampling

Polar slice sampling, a Markov chain construction for approximate sampling, performs, under suitable assumptions on the target and initial distribution, provably independent of the state space dimension. We extend the aforementioned result of Roberts & Rosenthal (2002) by developing a theory which identifies conditions, in terms of a generalized level set function, that imply an explicit lower bound on the spectral gap even in a general slice sampling context. Verifying the identified conditions for polar slice sampling yields a lower bound of 1/2 on the spectral gap for arbitrary dimension if the target density is rotationally invariant, log-concave along rays emanating from the origin and sufficiently smooth. The general theoretical result is potentially applicable beyond the polar slice sampling framework.

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arXiv2023

Wasserstein contraction and spectral gap of slice sampling revisited

We propose a new class of Markov chain Monte Carlo methods, called $k$-polar slice sampling ($k$-PSS), as a technical tool that interpolates between and extrapolates beyond uniform and polar slice sampling. By examining Wasserstein contraction rates and spectral gaps of $k$-PSS, we obtain strong quantitative results regarding its performance for different kinds of target distributions. Because $k$-PSS contains uniform and polar slice sampling as special cases, our results significantly advance the theoretical understanding of both of these methods. In particular, we prove realistic estimates of the convergence rates of uniform slice sampling for arbitrary multivariate Gaussian distributions on the one hand, and near-arbitrary multivariate t-distributions on the other. Furthermore, our results suggest that for heavy-tailed distributions, polar slice sampling performs dimension-independently well, whereas uniform slice sampling suffers a rather strong curse of dimensionality.

math.ST↗

arXiv2023

Gibbsian polar slice sampling

Polar slice sampling (Roberts & Rosenthal, 2002) is a Markov chain approach for approximate sampling of distributions that is difficult, if not impossible, to implement efficiently, but behaves provably well with respect to the dimension. By updating the directional and radial components of chain iterates separately, we obtain a family of samplers that mimic polar slice sampling, and yet can be implemented efficiently. Numerical experiments in a variety of settings indicate that our proposed algorithm outperforms the two most closely related approaches, elliptical slice sampling (Murray et al., 2010) and hit-and-run uniform slice sampling (MacKay, 2003). We prove the well-definedness and convergence of our methods under suitable assumptions on the target distribution.

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