@article{RudolfSchaer2023, author = {Rudolf, Daniel and Sch{\"a}r, Philip}, title = {Dimension-independent spectral gap of polar slice sampling}, series = {Statistics and Computing}, volume = {34}, journal = {Statistics and Computing}, number = {1}, publisher = {Springer Nature}, address = {Berlin}, doi = {10.1007/s11222-023-10335-y}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2024011821064231531382}, pages = {14 Seiten}, year = {2023}, abstract = {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 and Rosenthal (Stoch Model 18(2):257-280, 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.}, language = {en} } @article{ProchnoRudolf2024, author = {Prochno, Joscha and Rudolf, Daniel}, title = {The minimal spherical dispersion}, series = {The Journal of Geometric Analysis}, volume = {34}, journal = {The Journal of Geometric Analysis}, number = {3}, publisher = {Springer US}, address = {New York}, issn = {1050-6926}, doi = {10.1007/s12220-023-01527-z}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2405020842543.936733306281}, pages = {14 Seiten}, year = {2024}, abstract = {We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy in (Anz {\"O}sterreich Akad Wiss Math Nat Kl 132:3-10, 1995). In particular, we see that the inverse N(ε,d)of the minimal spherical dispersion is, for fixed ε>0, linear in the dimension d of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse N~(ε,d), our bounds are optimal with respect to the dependence on ε.}, language = {en} }