arXiv subjects
Publications and source records attributed to Stefan Tudose.
We prove that any random vector in $\mathbb{R}^n$ which is dominated in convex order by a standard Gaussian vector can be written as the sum of three standard Gaussian vectors. This implies that any $1$-subgaussian random vector in $\mathbb{R}^n$ is the sum of a universal number of Gaussian vectors. It also solves M. Talagrand's convexity problem, which in turn implies a weak version of a combinatorial analogue to the problem.
We prove that for every Cayley graph $\mathcal{G}$ over any finite abelian group $G$, there is a weighted Cayley graph with $O(\log |G|)$ generators that is a spectral sparsifier for $\mathcal{G}$. This bound is optimal. Applying our bound to the group $G = \mathbb{F}_2^n$, yields, as a corollary, $O(n/\varepsilon^2)$-sized code sparsifiers for $\mathbb{F}_2$-linear codes, improving on the work of Khanna, Putterman and Sudan (SODA'24) who obtained a similar result with an additional $\mathrm{polylog}(n)$ loss. Our proof is strongly inspired by a recent work of Reis and Rothvoss for the construction of $\ell_1$-sparsifiers. Following their work, the abelian Cayley sparsification problem can be reduced to establishing a lower bound for the volume of a certain natural convex body. This volume bound follows from a short, elementary argument that relies on character symmetry.
Brouwer's Laplacian conjecture states that the sum of the largest $k$ eigenvalues of a graph's Laplacian is less than or equal to the number of edges plus $\binom{k+1}{2}$. We give a proof of this conjecture. Our proof relies on the Grone--Merris--Bai theorem for \emph{split} graphs. We also show the converse, thereby establishing an equivalence between Brouwer's conjecture and the Grone--Merris--Bai theorem.