arXiv:2606. 17000v1 Announce Type: cross Abstract: We prove that computing approximate stationary points of min-max optimization over the hypercube is PPAD-hard for quadratic polynomials.
arXiv:2606. 17319v1 Announce Type: cross Abstract: Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube.
arXiv:2609.08736v1 Announce Type: new Abstract: We study certificates of positivity and optimality for learning problems whose objectives and constraints need not be polynomial. We isolate an axiomat...
arXiv:2606. 06136v1 Announce Type: cross Abstract: We prove the positive-real $n=9$ case of the Vasc cyclic inequality.
For an arbitrary isotropic log-concave distribution $P$ on $\mathbb{R}^d$, we prove that the polynomial $(Cm)^m\|v\|_2^m - \mathbb{E}_{X\sim P}\langle X,v\rangle^m$ is a sum of squares for every even $m\ge2$, where $C>0$ is a universal constant. This removes the dependence on the Poincaré constant in the theorem of Kothari and Steinhardt (arXiv:1711.
arXiv:2606. 03419v1 Announce Type: cross Abstract: The 2026 disproof of Erd\H{o}s's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$.
The paper investigates how to obtain certificates of positivity and optimality for learning problems whose objectives and constraints are not necessarily polynomial. It isolates an axiomatic core of Fischer's constructive strict and weak Positivstellensätze and extends the resulting theorems to abstract function algebras over ordered fields. The framework distinguishes between objective/constraint functions built from broad classes of continuous or definable operations and auxiliary primitives that satisfy explicit scalar and closure axioms, providing instances over continuous and definable function algebras, including fields not closed under square roots, and analyzing lower-bound and global-optimality certificates as well as computational complexity.
arXiv:2609. 30105v1 Announce Type: new Abstract: For an arbitrary isotropic log-concave distribution $P$ on $\mathbb{R}^d$, we prove that the polynomial $(Cm)^m\|v\|_2^m - \mathbb{E}_{X\sim P}\langle X,v\rangle^m$ is a sum of squares for every even $m\ge2$, where $C>0$ is a universal constant.
arXiv:2608. 08154v1 Announce Type: cross Abstract: The Zarankiewicz number Z(m,n,s,t) is the maximum number of edges in a bipartite graph with parts of orders m and n containing no copy of Ks,t.
arXiv:2610.03321v1 Announce Type: cross Abstract: Low-rank approximation can require additional matrix--vector products to verify that its error meets a prescribed tolerance. We characterize this cer...
arXiv:2608. 02588v1 Announce Type: cross Abstract: In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm.
arXiv:2609. 30877v1 Announce Type: cross Abstract: We study whether the linear condition-number dependence in the stochastic complexity of SAPD+ is necessary for nonconvex-strongly-concave minimax optimization.