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
The paper establishes the first polynomial lower bounds on the number of parallel rounds required for diffusion sampling with approximate scores. It proves a “~Ω(d^{1/3})” lower bound for sampling smooth, near‑isotropic Gaussian mixtures in ℝ^d, and an Ω(d) lower bound for uniform sampling from anisotropic axis‑aligned boxes inside the unit ball. These bounds apply to any randomized algorithm that can make polynomially many queries per round at arbitrary locations and noise levels, while maintaining inverse‑polynomial score error and constant total variation accuracy.
By Yiwen Kou, Yimeng Wang
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
The paper introduces MASEM, a method for uniform sampling on manifolds that may have multiple disconnected components defined by smooth constraints. It employs a resampling scheme that maximizes the entropy of the empirical distribution using k‑nearest neighbor density estimation, and shows that this approach reduces KL‑divergence exponentially with the number of resampling steps. Experiments on synthetic and robotics benchmarks demonstrate that MASEM achieves faster mixing and improves Sinkhorn distance by an order of magnitude compared to existing alternatives while maintaining competitive runtime.
By Cornelius V. Braun, Tilman Burghoff, Marc Toussaint
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:2610. 22592v1 Announce Type: cross Abstract: Empirical averages are fundamental statistical primitives.
By Ian McPherson, Mauro Maggioni
arXiv:2610.19652v1 Announce Type: new
Abstract: Recent work has sought to transfer first-order nonconvex optimization guarantees to zeroth-order methods through gradient estimators, typically incurri...
By Yuxuan Xia, Xiao-Tong Yuan
arXiv:2607.06644v2 Announce Type: replace
Abstract: Determinantal point processes (DPPs) have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficien...
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
arXiv:2004. 05813v3 Announce Type: replace-cross Abstract: Suppose that we are given independent, identically distributed random samples $x_1,\cdots,x_n$ from a mixture at most $k$ many $d$-dimensional spherical Gaussian distributions $\mu_1,\cdots,\mu_{k_0}$ of identical and known variance $\sigma^2$ in each coordinate, such that the minimum $\ell^2$ distance between two distinct centers $y_l$ and $y_j$ is greater than $2\Delta\sigma \min\{\sqrt{d},\sqrt k\}$, where $\Delta>C_0$, and $C_0$ is a sufficiently large universal constant.
By Somnath Chakraborty, Hariharan Narayanan