Victor Reis

Victor Reis, Thomas Rothvoss

In a seminal paper, Kannan and Lovász (1988) considered a quantity $μ_{KL}(Λ,K)$ which denotes the best volume-based lower bound on the covering radius $μ(Λ,K)$ of a convex body $K$ with respect to a lattice $Λ$. Kannan and Lovász proved that $μ(Λ,K) \leq n \cdot μ_{KL}(Λ,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\log(2n))$ factor suffices, which would match the lower bound from the work of Kannan and Lovász. We settle this conjecture up to a constant in the exponent by proving that $μ(Λ,K) \leq O(\log^{3}(2n)) \cdot μ_{KL} (Λ,K)$. Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012, 2019), we obtain a $(\log(2n))^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal flatness constant of $O(n \log^{2}(2n))$, improving on the previous bound of $O(n^{4/3} \log^{O(1)} (2n))$.

Victor Reis

A zonotope is a linear image of the cube $[-1,1]^m$ for some $m \in \mathbb{N}$. We show that there is a universal constant $C$ such that, for every zonotope $Z\subset \mathbb{R}^d$ and vectors $v_1,\dots,v_n\in Z$, there are signs $x_1,\dots,x_n\in\{-1,1\}$ with \[ \sum_{i=1}^n x_i v_i \in C\sqrt d\, Z. \] This resolves a 2002 question of Schechtman and generalizes Spencer's six standard deviations theorem, which corresponds to the case $Z=[-1,1]^d$.

Shuaiqi Wang, Vikas Raunak, Arturs Backurs et al.

Differentially private (DP) synthetic data generation is a promising technique for utilizing private datasets that otherwise cannot be exposed for model training or other analytics. While much research literature has focused on generating private unstructured text and image data, in enterprise settings, structured data (e.g., tabular) is more common, often including natural language fields or components. Existing synthetic data evaluation techniques (e.g., FID) struggle to capture the structural properties and correlations of such datasets. In this work, we propose Struct-Bench, a framework and benchmark for evaluating synthetic datasets derived from structured datasets that contain natural language data. The Struct-Bench framework requires users to provide a representation of their dataset structure as a Context-Free Grammar (CFG). Our benchmark comprises 5 real-world and 2 synthetically generated datasets, each annotated with CFGs. We show that these datasets demonstrably present a great challenge even for state-of-the-art DP synthetic data generation methods. Struct-Bench also includes reference implementations of different metrics and a leaderboard, thereby providing researchers a standardized evaluation platform to benchmark and investigate privacy-preserving synthetic data generation methods. Further, we also present a case study showing how to use Struct-Bench to improve the synthetic data quality of Private Evolution (PE) on structured data. The benchmark and the leaderboard have been publicly made available at https://struct-bench.github.io.