Query lower bounds for log-concave sampling

Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following query lower bounds: (1) sampling from strongly log-concave and log-smooth distributions in dimension $d\ge 2$ requires $\Omega(\log \kappa)$ queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in dimension $d$ (hence also from general log-concave and log-smooth distributions in dimension $d$) requires $\widetilde \Omega(\min(\sqrt\kappa \log d, d))$ queries, which is nearly sharp for the class of Gaussians. Here $\kappa$ denotes the condition number of the target distribution. Our proofs rely upon (1) a multiscale construction inspired by work on the Kakeya conjecture in harmonic analysis, and (2) a novel reduction that demonstrates that block Krylov algorithms are optimal for this problem, as well as connections to lower bound techniques based on Wishart matrices developed in the matrix-vector query literature.Comment: 46 pages, 2 figure

Part I: Uniform estimates for operators involving polynomial curves. Part II: Decoupling estimates for fractal and product sets.

The first part of the thesis focuses on the uniformity of harmonic analysis estimates on curves. We first show a decomposition theorem for polynomial curves on local fields as a bounded number of perturbations of monomial curves. Using this theorem, we extend uniform restriction estimates for real curves to the endpoint case, show uniform decoupling for those curves, and show novel uniform restriction estimates for curves over C, and Qp. We then show uniform estimates for the discrete analog to this problem in a restricted range of exponent.The second part focuses on decoupling estimates for sets with a product or self-similar structure. A recurring phenomenon for those sets is that functions with constant Fourier transform on their support are far from extremizers. As applications we will show a de- coupling estimate for fractal subsets of the parabola, and study subsets of cubes with high additive energy compared to their cardinality

Additive energies on discrete cubes

We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and $\log_{2}(2^k+2)$ is the best possible exponent. We also show that if $d\geq 0$ and $2\leq k\leq 10$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$) is at most $|A|^{\log_2{ \binom{2k}{k}}}$, and $\log_2{ \binom{2k}{k}}$ is the best possible exponent. We discuss the analogous problems for the sets $\{0,1,\dots,n\}^d$ for $n\geq 2$.Comment: 16 pages, 3 figure