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Konstantin Tikhomirov

Publications and source records attributed to Konstantin Tikhomirov.

At least 19 recordsLinked to original sources
arXiv2026

The threshold for online balancing of i.i.d. binary vectors

Consider the task of online vector balancing for stochastic arrivals $X_1,\ldots,{X_T}$, where the $X_i$ are independent uniformly random $d$--sparse binary vectors in $\{0,1\}^n$. This is a random analogue of the online Beck--Fiala problem. We show that uniformly for $2\le d\le n/2$ and $T = Θ(n)$, the optimal online prefix discrepancy $\max\limits_{t\leq T}\left\|\sum_{i=1}^tσ_i X_i\right\|_\infty$ is of order \[ Θ\big(\max\{\sqrt d,\log\log n\}\big). \] The upper bound is achieved by an efficient online algorithm. Thus, for $d\le(\log\log n)^2$, the optimal discrepancy is $Θ(\log\log n)$ and is independent of the sparsity up to constant factors, whereas above this scale it is $Θ(\sqrt d)$, matching the order of the offline discrepancy. This identifies the threshold at which sparsity begins to govern the online discrepancy of the random Beck--Fiala model.

math.PR↗
arXiv2026

Online Permutation Embedding: Optimal Stopping and Scaling Laws

We study optimal online algorithms for embedding a permutation $π$ of $[k]$ into an iid stream of uniform $[0,1]$ random variables. This problem is a broad generalization of the classical online monotone subsequence selection problem, recovered in the special case $π=\mathrm{Id}_k$. Our first contribution is an efficiently solvable dynamic program for the optimal embedding time of any $k$-permutation $π$. This dynamic program also yields an explicit optimal online embedding algorithm. We then investigate the asymptotic scaling of the optimal embedding time for uniformly random target permutations, as well as the extremal problem of identifying the permutations with largest expected online embedding time. Our second main result shows that, to first order, random permutations are strictly faster to embed than monotone permutations, which in turn are strictly faster to embed than the extremal permutations. This separation stands in sharp contrast to prevailing conjectures and heuristics in the offline theory of permutation embeddings.

math.PR↗
arXiv2026

Discrete Poincaré inequalities and universal approximators for random graphs

Nonlinear Poincaré inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon Kleinberg (2013), asks whether the optimal nonlinear Poincaré constant for maps between two independent $3$-regular random graphs is dimension-free, i.e., independent of vertex-set sizes. We give a complete and affirmative resolution to Kleinberg's problem, also allowing for arbitrary graph degrees. As a corollary, we obtain a stochastic construction of $O(1)\text{-universal}$ approximators for random graphs, answering a question of Mendel and Naor.

math.MG↗
arXiv2026

Level-set entropy and sparse randomized embeddings

Let $Π$ be a $k\times n$ sparse random matrix. For a fixed $r$-dimensional subspace $V\subset{\mathbb R}^n$, let $U_V:{\mathbb R}^r\to{\mathbb R}^n$ denote an isometry from ${\mathbb R}^r$ onto $V$. The product $ΠU_V$ is a central model in randomized dimension reduction and has been studied primarily through trace and Gaussian comparison inequalities. In this work, we develop an approach to the spectral norm of the matrix product $ΠU_V$, based on entropy estimates for level sets of vectors $x\in V$. Combining the method with existing estimates, we show the following. Assume that \[ k\ge C\,r(\log\log r)^2,\qquad p\ge (\log k)/k. \] Let $Π$ be a $k\times n$ matrix with i.i.d. entries equidistributed with the product $b\,ξ$, where $b$ is a Bernoulli($p$) random variable and $ξ$ is mean-zero, independent of $b$, and satisfies $|ξ|\le1$ almost surely. Then with high probability \[ \|ΠU_V\|\le C\sqrt{kp}. \] Matching results hold for other random models with negatively associated entries.

math.PR↗
arXiv2026

Online Beck--Fiala Down to Logarithmic Sparsity

The Beck--Fiala conjecture asserts that every matrix $A\in\{0,1\}^{n\times T}$ with at most $d$ nonzero entries in each column has discrepancy $O(\sqrt d)$. A major breakthrough result of Bansal and Jiang recently established the validity of the conjecture for $d \ge \log(T)^2$. The present article extends the validity of the classical \textit{offline} Beck--Fiala conjecture to $d \ge \log(T)^{1+o(1)}$; moreover, the main thrust of the result is that it is actually obtained by an efficient \textit{online} algorithm that minimizes prefix discrepancy. The result is also essentially optimal, since online prefix discrepancy is known to scale as $ω(\sqrt{d})$ for $d =o(\log T)$. As an immediate corollary, the open question of online vector balancing in the Spencer setting is also resolved. The algorithm is based on a compactly supported Metropolis fixed-point walk, constructed by combining ideas from several recent works on the online Komlós problem. The proof was generated in conversation with ChatGPT 5.6 Pro; the authors provided high-level guidance in several rounds of prompting, followed by manual checking and rewriting of the proof.

math.CO↗
arXiv2026

Well-invertible column subsets of sparse matrices are rare

A random $n\times k$ matrix $S$ is an \emph{$(r,α)$-oblivious subspace injection} (OSI) if $\mathbb{E}\|S^\top x\|_2^2=\|x\|_2^2$ for every $x\in\mathbb{R}^n$, and for every fixed $r$-dimensional subspace $V\subset\mathbb{R}^n$, with probability close to one, one has $α\|x\|_2^2\le\|S^\top x\|_2^2$ for all $x\in V$. In this work, we show that in the regime $r=Ω(k)$ and $α=Ω(1)$, and under a mild additional structural assumption, no constant-row-sparsity matrix $S$ is OSI, thereby answering, in a strong form, a question raised by Camaño, Epperly, Meyer, and Tropp. We show that the failure of the OSI property for sparse random matrices stems from a general deterministic phenomenon, thereby reducing a probabilistic problem to a non-probabilistic one. This phenomenon is related to the restricted invertibility principle introduced in the seminal work of Bourgain--Tzafriri. Let $(n_k)_{k\in\mathbb{N}}$ be a sequence of integers satisfying $\frac{n_k}{k}\to\infty$. For each $k$, let $S^{(k)}$ be a $n_k\times k$ non-random matrix with $O(1)$ nonzero entries per row, whose nonzero entries have average magnitude $O(1)$, and such that the total number of pairs of rows with supports overlapping at two or more indices is $o({n_k}^2/k)$. We prove that for every constant $\varepsilon>0$, as $k\to\infty$, the overwhelming majority of $k\times \lfloor\varepsilon k\rfloor$ submatrices of $(S^{(k)})^\top$ have the smallest singular value $o(1)$. Thus, the well-invertible submatrices whose existence is guaranteed by the Bourgain--Tzafriri theorem are rare. The proof is itself based on probabilistic tools.

math.PR↗
arXiv2026

A universal threshold for geometric embeddings of trees

A graph $G=(V,E)$ is geometrically embeddable into a normed space $X$ when there is a mapping $ζ: V\to X$ such that $\|ζ(v)-ζ(w)\|_X\leqslant 1$ if and only if $\{v,w\}\in E$, for all distinct $v,w\in V$. Our result is the following universal threshold for the embeddability of trees. Let $Δ\geqslant 3$, and let $N$ be sufficiently large in terms of $Δ$. Every $N$--vertex tree of maximal degree at most $Δ$ is embeddable into any normed space of dimension at least $64\,\frac{\log N}{\log\log N}$, and complete trees are non-embeddable into any normed space of dimension less than $\frac{1}{2}\,\frac{\log N}{\log\log N}$. In striking contrast, spectral expanders and random graphs are known to be non-embeddable in sublogarithmic dimension. Our result is based on a randomized embedding whose analysis utilizes the recent breakthroughs on Bourgain's slicing problem.

math.CO↗
arXiv2026

On the optimal objective value of random linear programs

We consider the problem of maximizing $\langle c,x \rangle$ subject to the constraints $Ax \leq \mathbf{1}$, where $x\in R^n$, $A$ is an $m\times n$ matrix with mutually independent centered subgaussian entries of unit variance, and $c$ is a cost vector of unit Euclidean length. In the asymptotic regime $n\to\infty$, $\frac{m}{n}\to\infty$, and under some mild assumptions on $c$, we prove that the optimal objective value $z^*$ of the linear program satisfies $$ \lim\limits_{n\to\infty}\sqrt{2\log(m/n)}\,z^*= 1\quad \mbox{almost surely}. $$ We provide numerical experiments as supporting data for the theoretical predictions. Further, we carry out numerical studies of the limiting distribution and the standard deviation of $z^*$.

math.PR↗
arXiv2026

Cotype of random polytopes

For $N\geq n$, let $P_{N,n}$ be a random polytope in ${\mathbb R}^n$ with vertices $\pm X_i$, $1\leq i\leq N$, where $X_1,\dots,X_N$ are i.i.d standard Gaussian vectors in ${\mathbb R}^n$. Random polytopes $P_{N,n}$, as well as their duals, are classical objects of interest in high-dimensional convex geometry and local Banach space theory. In this paper, we provide a {\it dimension-independent} bound on the cotype of the corresponding normed space $({\mathbb R}^n,\|\cdot\|_{P_{N,n}})$, generated by $P_{N,n}$. Let $K'\geq K>1$, and assume that $K'\geq \frac{N}{n}\geq K$. We show that with probability $1-o(1)$, for any $k\geq 1$, and any collection $y_1,\dots,y_k$ of vectors in ${\mathbb R}^n$, $$ {\mathbb E}_σ\,\Big\|\sum_{i=1}^k σ_i y_i\Big\|_{P_{N,n}}^q \geq \frac{1}{C_q^q}\sum_{i=1}^k \big\|y_i\big\|_{P_{N,n}}^q, $$ where $σ=(σ_1,\dots,σ_k)$ is a vector of random signs, and where $q\in [2,\infty)$ and $C_q\in[1,\infty)$ may only depend on $K,K'$. We discuss the result in context of infinite-dimensional Banach spaces.

math.FA↗
arXiv2026

Metric Poincaré inequalities for graphs

This article obtains purely metric counterparts of cornerstone results in the theory of embedding graphs into normed spaces. Our first main result is a metric analogue of Matoušek's extrapolation relating the Poincaré constants $γ(G,\varrho^p)$ and $γ(G,\varrho^q)$ for any exponents $0 < p,q < \infty$, any bounded-degree expander graph $G$, and any target metric space $\mathcal{M}=(M,\varrho)$. Our second main result provides a sharp estimate of the Poincaré constant $γ(G,\varrho)$ in terms of the cardinalities of the vertex set of $G$ and the metric space $\mathcal{M}=(M,\varrho)$, in the setting of \textit{random} graphs. This yields optimal estimates on the minimum cardinality of (bi-Lipschitz) universal metric spaces for graphs, finally establishing a nonlinear analogue of Matoušek's celebrated "incompressibility" theorem (1996). Further, we obtain estimates on the nonlinear spectral gap of metric snowflakes and sharp lower bounds on the distortion of random regular graphs into arbitrary metric spaces. Our proofs develop new nonlinear techniques, including random compression methods and a novel structural dichotomy for metric embeddings.

math.MG↗
arXiv2025

A threshold for online balancing of sparse i.i.d. vectors

Consider the task of \textit{online} vector balancing for stochastic arrivals $(X_i)_{i \in [T]}$, where the time horizon satisfies $T = Θ(n)$, and the $X_i$ are i.i.d uniform $d$--sparse $n$--dimensional binary vectors, with $2\leq d \le (\log\log n)^2/\log\log\log n$. We show that for this range of parameters, every online algorithm incurs discrepancy at least $Ω(\log \log n)$, and there is an efficient algorithm which achieves a matching discrepancy bound of $O(\log\log n)$ w.h.p. This establishes an asymptotic gap, both existential and algorithmic, between the online and offline versions of the average--case Beck--Fiala problem. Strikingly, the optimal online discrepancy in the considered setting is order $\log \log n$, independent of $d$ and the norms of the vectors $(X_i)_i$. Our assumptions on $d$ are nearly optimal, as this independence ceases when $d=ω((\log\log n)^2)$.

math.PR↗
arXiv2025

On the probability that convex hull of random points contains the origin

The classical theorem of Wendel provides an exact formula for the probability that the convex hull of independent symmetrically distributed vectors in ${\mathbb R}^d$ contains the origin as long as the distributions of the vectors are continuous. In this note, we provide an extension to Wendel's theorem for independent random vectors $X_1,\dots,X_n$ with i.i.d components having a (possibly discrete) symmetric distribution of unit variance. As a related observation, we give sharp estimates on the probability that a random linear program of the form ``$\max\langle x,{\mathfrak c}\rangle\quad\mbox{subject to }\langle X_i,x\rangle\leq 1,\;i\leq n$'', is bounded.

math.MG↗
arXiv2025

Metric dimension reduction modulus for superlogarithmic distortion

The metric dimension reduction modulus $k^α_n(\ell_\infty)$ is the smallest $k$ such that every $n$--point metric space can be embedded into some $k$-dimensional normed space, with bi--Lipschitz distortion at most $α$. Determining sharp asymptotics for $k^α_n(\ell_\infty)$ is a fundamental task in metric geometry, with $α=Θ(\log n)$ bearing particular interest. A line of advances over the past decades has led to an upper bound on $k^α_n(\ell_\infty)$ for $α= Ω(\log n)$, but a matching lower bound has remained open. We close this gap, establishing: for every fixed $β> 0$, $$ k^α_n(\ell_\infty) =Θ\bigg(\frac{\log n}{\log(\fracα{\log n}+1)}\bigg)\quad \mbox{for every $α\geq β\log n$}. $$ This resolves a question from Naor's 2018 ICM plenary lecture. Our result is obtained by characterizing the minimum dimension $d$ for which, with high probability, a random regular graph admits an $α$--embedding into some $d$--dimensional normed space.

math.MG↗
arXiv2025

A combinatorial approach to nonlinear spectral gaps

A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincaré inequality for scalar functions, also satisfies an analogous inequality for functions taking values in \textit{any} normed space with non-trivial cotype. Motivated by applications, it is also greatly important to quantify the dependence of the corresponding optimal Poincaré constant on the cotype $q$. Works of Odell--Schlumprecht (1994), Ozawa (2004), and Naor (2014) make substantial progress on the former question by providing a positive answer for normed spaces which also have an unconditional basis, in addition to finite cotype. However, little is known in the way of quantitative estimates: the mentioned results imply a bound on the Poincaré constant depending super-exponentially on $q$. We introduce a novel combinatorial framework for proving quantitative nonlinear spectral gap estimates. The centerpiece is a property of regular graphs that we call \emph{long range expansion}, which holds with high probability for random regular graphs. Our main result is that any regular graph with the long-range expansion property satisfies a discrete Poincaré inequality for any normed space with an unconditional basis and cotype $q$, with a Poincaré constant that depends \emph{polynomially} on $q$, which is optimal. As an application, any normed space with an unconditional basis which admits a low distortion embedding of an $n$-vertex random regular graph, must have cotype at least polylogarithmic in $n$. This extends a celebrated lower-bound of Matoušek for low distortion embeddings of random graphs into $\ell_q$ spaces.

math.MG↗
arXiv2025

Universal geometric non-embedding of random regular graphs

Let $Δ\ge 3$ be fixed, $n \ge n_Δ$ be a large integer. It is a classical result that $Δ$--regular expanders on $n$ vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than $c \log n$, for some universal constant $c$. We show that for typical $Δ$-regular graphs, this obstruction is universal with respect to the choice of norm. More precisely, for a uniform random $Δ$-regular graph $G$ on $n$ vertices, it holds with high probability: there is no normed space of dimension less than $c\log n$ which admits a geometric graph isomorphic to $G$. The proof is based on a seeded multiscale $\varepsilon$--net argument.

math.MG↗
arXiv2024

Locally seeded embeddings, and Ramsey numbers of bipartite graphs with sublinear bandwidth

A seminal result of Lee asserts that the Ramsey number of any bipartite $d$-degenerate graph $H$ satisfies $\log r(H) = \log n + O(d)$. In particular, this bound applies to every bipartite graph of maximal degree $Δ$. It remains a compelling challenge to identify conditions that guarantee that an $n$-vertex graph $H$ has Ramsey number linear in $n$, independently of $Δ$. Our contribution is a characterization of bipartite graphs with linear-size Ramsey numbers in terms of graph bandwidth, a notion of local connectivity. We prove that for any $n$-vertex bipartite graph $H$ with maximal degree at most $Δ$ and bandwidth $b(H)$ at most $\exp(-CΔ\logΔ)\,n$, we have $\log r(H) = \log n + O(1)$. This characterization is nearly optimal: for every $Δ$ there exists an $n$-vertex bipartite graph $H$ of degree at most $Δ$ and $b(H) \leq \exp(-cΔ)\,n$, such that $\log r(H) = \log n + Ω(Δ)$. We also provide bounds interpolating between these two bandwidth regimes.

math.CO↗
arXiv2024

Average-case analysis of the Gaussian Elimination with Partial Pivoting

The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical evidence strongly suggests that for a {\it typical} square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random $n\times n$ standard Gaussian coefficient matrix $A$, the {\it growth factor} of the Gaussian Elimination with Partial Pivoting is at most polynomially large in $n$ with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve $Ax = b$ to $m$ bits of accuracy using GEPP is $m+O(\log n)$, which improves an earlier estimate $m+O(\log^2 n)$ of Sankar, and which we conjecture to be optimal by the order of magnitude. We further provide tail estimates of the growth factor which can be used to support the empirical observation that GEPP is more stable than the Gaussian Elimination with no pivoting.

math.NA↗