The paper establishes nearly quadratic lower bounds for randomized algorithms that perform linear optimization and uniform sampling over convex bodies using a membership oracle. It shows that the lower bound for linear optimization matches the best known upper bound up to a polylogarithmic factor in the dimension, while the bound for uniform sampling improves upon the previous linear lower bound. Additionally, the authors demonstrate that their construction yields the same lower bound for volume estimation.
By Santosh S. Vempala
arXiv:2609.15884v1 Announce Type: cross
Abstract: We show that logconcave probability measures along the Gaussian cooling path have thin-shell stability, generalizing the thin-shell theorem. This res...
By Yunbum Kook, Santosh S. Vempala
arXiv:2608. 16506v1 Announce Type: cross Abstract: Dataset alignment is a central step in data analysis across science and engineering, where the goal is to match observations between datasets.
By Keyi Li, Yuval Kluger, Boris Landa
arXiv:2609.06905v1 Announce Type: cross
Abstract: We study the problem of sampling from $\mu(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $...
By Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S Zhang
arXiv:2607. 06644v1 Announce Type: cross Abstract: Determinantal point processes have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficient minibatches, coresets, and other compact representations of large-scale datasets.
By Hoang-Son Tran, Pranav Gupta, Subhroshekhar Ghosh
arXiv:2604. 14614v2 Announce Type: replace-cross Abstract: We give an algorithm for PAC learning intersections of $k$ halfspaces with a $\rho$ margin to within error $\varepsilon$ that runs in time $\textsf{poly}(k, \varepsilon^{-1}, \rho^{-1}) \cdot \exp \left(O(\sqrt{n \log(1/\rho) \log k})\right)$.
By Shyamal Patel, Santosh Vempala