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Steven Heilman

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math.PR Computational Complexity math.FA math.CO Discrete Mathematics math.MG Artificial Intelligence Information Theory Machine Learning math-ph math.DG math.IT math.MP

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preprint2026arXiv

Trees and Graphs with Non Log-concave Dominating Set Sequence via AI Tools

We give new examples of graphs and trees with dominating set sequences that are not log-concave. These examples were generated by PatternBoost, a transformer-based reinforcement learning software developed by Charton-Ellenberg-Wagner-Williamson. We also show: for any positive integer $m$ , there exists a tree whose dominating set sequence is not log-concave for at least $m$ indices by modifying a similar construction of Bautista-Ramos for the independent set sequence. We show that a large class of caterpillar graphs has log-concave dominating set sequences. A continuous analogue of the sequence is also log-concave for all graphs.

preprint2022arXiv

Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

Let $Ω$ be a measurable Euclidean set in $\mathbb{R}^{n}$ that is symmetric, i.e. $Ω=-Ω$ , such that $Ω\times\mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume. We conclude that either $Ω$ or $Ω^{c}$ is convex. Moreover, except for the case $H(x)=\langle x,N(x)\rangle+λ$ with $H\geq0$ and $λ<0$ , we show there exist a radius $r>0$ and an integer $0\leq k\leq n-1$ such that after applying a rotation, the boundary of $Ω$ must satisfy $\partialΩ= rS^{k}\times\mathbb{R}^{n-k-1}$ , with $\sqrt{n-1}\leq r\leq\sqrt{n+1}$ when $k\geq1$ . Here $S^{k}$ denotes the unit sphere of $\mathbb{R}^{k+1}$ centered at the origin, and $n\geq1$ is an integer. One might say this result nearly resolves the symmetric Gaussian conjecture of Barthe from 2001.

preprint2022arXiv

Dimension-Free Noninteractive Simulation from Gaussian Sources

Let $X$ and $Y$ be two real-valued random variables. Let $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots$ be independent identically distributed copies of $(X,Y)$ . Suppose there are two players A and B. Player A has access to $X_{1},X_{2},\ldots$ and player B has access to $Y_{1},Y_{2},\ldots$ . Without communication, what joint probability distributions can players A and B jointly simulate? That is, if $k,m$ are fixed positive integers, what probability distributions on $\{1,\ldots,m\}^{2}$ are equal to the distribution of $(f(X_{1},\ldots,X_{k}),\,g(Y_{1},\ldots,Y_{k}))$ for some $f,g\colon\mathbb{R}^{k}\to\{1,\ldots,m\}$ ? When $X$ and $Y$ are standard Gaussians with fixed correlation $ρ\in(-1,1)$ , we show that the set of probability distributions that can be noninteractively simulated from $k$ Gaussian samples is the same for any $k\geq m^{2}$ . Previously, it was not even known if this number of samples $m^{2}$ would be finite or not, except when $m\leq 2$ . Consequently, a straightforward brute-force search deciding whether or not a probability distribution on $\{1,\ldots,m\}^{2}$ is within distance $0<ε<|ρ|$ of being noninteractively simulated from $k$ correlated Gaussian samples has run time bounded by $(5/ε)^{m(\log(ε/2) / \log|ρ|)^{m^{2}}}$ , improving a bound of Ghazi, Kamath and Raghavendra. A nonlinear central limit theorem (i.e. invariance principle) of Mossel then generalizes this result to decide whether or not a probability distribution on $\{1,\ldots,m\}^{2}$ is within distance $0<ε<|ρ|$ of being noninteractively simulated from $k$ samples of a given finite discrete distribution $(X,Y)$ in run time that does not depend on $k$ , with constants that again improve a bound of Ghazi, Kamath and Raghavendra.

preprint2020arXiv

Independent Sets of Random Trees and of Sparse Random Graphs

An independent set of size $k$ in a finite undirected graph $G$ is a set of $k$ vertices of the graph, no two of which are connected by an edge. Let $x_{k}(G)$ be the number of independent sets of size $k$ in the graph $G$ and let $α(G)=\max\{k\geq0\colon x_{k}(G)\neq0\}$ . In 1987, Alavi, Malde, Schwenk and Erdös asked if the independent set sequence $x_{0}(G),x_{1}(G),\ldots,x_{α(G)}(G)$ of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8\% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk and Erdös is ``four-fifths true'', with high probability. We also show unimodality of the independent set sequence of Erdös-Renyi random graphs, when the expected degree of a single vertex is large (with (exponentially) high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size $k$ as $k$ varies is of interest in probability, statistical physics, combinatorics, and computer science.

preprint2020arXiv

Tree/Endofunction Bijections and Concentration Inequalities

We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is a fixed positive integer. The method uses a bijection between mappings $f\colon\{1,\ldots,n\}\to\{1,\ldots,n\}$ and doubly rooted trees on $n$ vertices. The main application is a concentration inequality for the number of vertices connected to an independent set in a uniformly random tree, which is then used to prove partial unimodality of its independent set sequence. So, we give probabilistic arguments for inequalities that often use combinatorial arguments.

preprint2016arXiv

Low Correlation Noise Stability of Symmetric Sets

We study the Gaussian noise stability of subsets A of Euclidean space satisfying A=-A. It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure. In summary, we provide the first known positive and negative results for the Symmetric Gaussian Problem.

preprint2015arXiv

Strong Contraction and Influences in Tail Spaces

We study contraction under a Markov semi-group and influence bounds for functions in $L^2$ tail spaces, i.e. functions all of whose low level Fourier coefficients vanish. It is natural to expect that certain analytic inequalities are stronger for such functions than for general functions in $L^2$ . In the positive direction we prove an $L^{p}$ Poincaré inequality and moment decay estimates for mean $0$ functions and for all $1<p<\infty$ , proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. That is, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$ , with all influences bounded by $C \log n/n$ for some constants $0<c,C< \infty$ . We also construct a function $f\colon\{-1,1\}^{n}\to\{0,1\}$ with nonzero mean whose remaining Fourier coefficients vanish up to level $c' \log n$ , with the sum of the influences bounded by $C'(\mathbb{E}f)\log(1/\mathbb{E}f)$ for some constants $0<c',C'<\infty$ .

preprint2014arXiv

Euclidean Partitions Optimizing Noise Stability

The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition $\{A_{i}\}_{i=1}^{k}$ of $\mathbb{R}^{n}$ into $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability. That is, for $ρ>0$ , we maximize $\sum_{i=1}^{k}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}1_{A_{i}}(x)1_{A_{i}}(xρ+y\sqrt{1-ρ^{2}}) e^{-(x_{1}^{2}+\cdots+x_{n}^{2})/2}e^{-(y_{1}^{2}+\cdots+y_{n}^{2})/2}dxdy.$ Isaksson and Mossel guessed the best partition for this problem and proved some applications of their conjecture. For example, the Standard Simplex Conjecture implies the Plurality is Stablest Conjecture. For $k=3,n\geq2$ and $0<ρ<ρ_{0}(k,n)$ , we prove the Standard Simplex Conjecture. The full conjecture has applications to theoretical computer science, and to geometric multi-bubble problems (after Isaksson and Mossel).

preprint2014arXiv

Solution of the propeller conjecture in $\mathbb{R}^3$

It is shown that every measurable partition ${A_1,..., A_k}$ of $\mathbb{R}^3$ satisfies $\sum_{i=1}^k||\int_{A_i} xe^{-\frac12||x||_2^2}dx||_2^2\le 9π^2.\qquad(*)$ Let ${P_1,P_2,P_3}$ be the partition of $\mathbb{R}^2$ into $120^\circ$ sectors centered at the origin. The bound is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for $i\in {1,2,3}$ and $A_i=\emptyset$ for $i\in \{4,...,k\}$ (up to measure zero corrections, orthogonal transformations and renumbering of the sets $\{A_1,...,A_k\}$ ). This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of $(*)$ is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with $4 \times 4$ centered and spherical hypothesis matrix equals $\frac{2π}{3}$ .

preprint2014arXiv

Standard Simplices and Pluralities are Not the Most Noise Stable

The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts $k \geq 3$ , for every value $ρ\neq 0$ of the noise and for every prescribed measures for the different parts as long as they are not all equal to $1/k$ . Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell's result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.

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Steven Heilman

Research overview

Is this your work?

10 publications

Trees and Graphs with Non Log-concave Dominating Set Sequence via AI Tools

Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

Dimension-Free Noninteractive Simulation from Gaussian Sources

Independent Sets of Random Trees and of Sparse Random Graphs

Tree/Endofunction Bijections and Concentration Inequalities

Low Correlation Noise Stability of Symmetric Sets

Strong Contraction and Influences in Tail Spaces

Euclidean Partitions Optimizing Noise Stability

Solution of the propeller conjecture in $\mathbb{R}^3$

Standard Simplices and Pluralities are Not the Most Noise Stable

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Steven Heilman

Research overview

Is this your work?

10 publications

Trees and Graphs with Non Log-concave Dominating Set Sequence via AI Tools

Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

Dimension-Free Noninteractive Simulation from Gaussian Sources

Independent Sets of Random Trees and of Sparse Random Graphs

Tree/Endofunction Bijections and Concentration Inequalities

Low Correlation Noise Stability of Symmetric Sets

Strong Contraction and Influences in Tail Spaces

Euclidean Partitions Optimizing Noise Stability

Solution of the propeller conjecture in R3\mathbb{R}^3R3

Standard Simplices and Pluralities are Not the Most Noise Stable

See the researcher in context

Building this map preview

Solution of the propeller conjecture in $\mathbb{R}^3$