arXiv subjects

Hariharan Narayanan

Publications and source records attributed to Hariharan Narayanan.

At least 19 recordsLinked to original sources

arXiv2026

Hives from deformed GUE minor processes

We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices \[ X=\sqrt{n}(wG+uD),\qquad Y=\sqrt{n}(w'G'+u'D'), \] where $D,D'$ are diagonal and have GUE spectra, we use their minor processes to form a double hive and then apply the octahedron recurrence. Under the matching condition \[ \frac{u}{w^2}=\frac{u'}{(w')^2}, \] we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if \[ a^2=w^2+u^2,\qquad b^2=(w')^2+(u')^2, \] then the produced hive density $q_n$ satisfies \[ D_{\mathrm{KL}}\!\left( q_n\, \middle\|\, \operatorname{Density}\bigl(H_n(a\sqrt n,b\sqrt n,c_{**}\sqrt n)\bigr) \right) = O(n\log n). \] The third scale $c_{**}$ is determined by a limiting tetrahedral optimization problem; equivalently, writing $δ=u+u'$, \[ δ^2 = \frac{ 2c_{**}^4(c_{**}^2-a^2-b^2) }{ (c_{**}^2-a^2+b^2)(c_{**}^2+a^2-b^2) }. \] Thus the construction realizes GUE hive laws, up to subleading relative entropy, throughout the right-angled and obtuse regime. The appendix records two explicit surface-tension approximations and numerical comparisons which motivated the construction.

math.PR↗

arXiv2026

On the randomized Horn problem and the surface tension of hives

Given two nonincreasing $n$-tuples of real numbers $λ_n$, $μ_n$, the Horn problem asks for a description of all nonincreasing $n$-tuples of real numbers $ν_n$ such that there exist Hermitian matrices $X_n$, $Y_n$ and $Z_n$ respectively with these spectra such that $X_n + Y_n = Z_n$. There is also a randomized version of this problem where $X_n$ and $Y_n$ are sampled uniformly at random from orbits of Hermitian matrices arising from the conjugacy action by elements of the unitary group. One then asks for a description of the probability measure of the spectrum of the sum $Z_n$. Both the original Horn problem and its randomized version have solutions using the hives introduced by Knutson and Tao. In an asymptotic sense, as $n \rightarrow \infty$, large deviations for the randomized Horn problem were given by Narayanan and Sheffield in terms of the surface tension of hives. In this paper, we provide upper and lower bounds on this surface tension function. We also obtain a closed-form expression for the total entropy of a surface tension minimizing continuum hive with boundary conditions arising from GUE eigenspectra. Finally, we give several empirical results for random hives and lozenge tilings arising from an application of the octahedron recurrence for large $n$ and a numerical approximation of the surface tension function.

math.PR↗

arXiv2026

A Fourier analytique approach to Gaussian mixture learning

Suppose that we are given independent, identically distributed random samples $x_1,\cdots,x_n$ from a mixture at most $k$ many $d$-dimensional spherical Gaussian distributions $μ_1,\cdots,μ_{k_0}$ of identical and known variance $σ^2$ in each coordinate, such that the minimum $\ell^2$ distance between two distinct centers $y_l$ and $y_j$ is greater than $2Δσ\min\{\sqrt{d},\sqrt k\}$, where $Δ>C_0$, and $C_0$ is a sufficiently large universal constant. We develop a randomized algorithm that learns the centers $y_l$'s of the Gaussian components to within an $\ell^2$ distance of $k^{-\tilde C_0}$ -- in presence of arbitrarily large number of components and in arbitrary dimension, when the weights are known to be uniform. Furthermore, if the number of components is $k= Ω(2^d)$, then for arbitrary universal constant $c>0$, even for unknown weights, the algorithm learns the centers to within an $\ell^2$ distance of $d^{-\tilde C_0}$ and the weights up to an accuracy of $cw_{min}$, with probability greater than $1 - \exp(-k/c)$, provided that the weights lie in $[c/k,1/ck]$, and the minimum separation is just $2c\sqrt d$. The number of samples and the computational time is bounded above by $\mathrm{poly}(k, d)$ in either case. Such a bound on the sample and computational complexity was previously unknown in the regime of non-constant dimension, and in particular, when $d$ is not $O(1)$. When $d = O(1)$, this complexity bound follows from work of Regev and Vijayaraghavan, where it has also been shown that the sample complexity of learning a random mixture of Gaussians in a ball of radius $o(\sqrt{d})$ in $d$ dimensions, when $d$ is $Θ( \log k)$, is at least super-polynomial in $k, d$, showing that our result is tight in this case.

cs.DS↗

arXiv2026

Denoising data using convex relaxations

We study the problem of denoising observations $Y_i=X_i+Z_i$, where the latent variables $X_i$ are sampled from a low-dimensional manifold in $\mathbb{R}^n$ and the noise variables $Z_i$ are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

stat.ME↗

arXiv2026

Large deviations for random hives and the spectrum of the sum of two random matrices

Suppose $α, β$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $γ$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $α(0) = γ(0) = 0$, and $α(1) = β(0) = 0$ and $β(1) = γ(1) = 0.$ For an $n \times n$ Hermitian matrix $W$, let $spec(W)$ denote the vector in $\mathbb{R}^n$ whose coordinates are the eigenvalues of $W$ listed in non-increasing order. Let $λ= \partial^- α$, $μ= \partial^- β$ on $(0, 1]$ and $ν= \partial^- γ,$ at all points of $(0, 1]$, where $\partial^-$ is the left derivative, which is monotonically decreasing. Let $λ_n(i) := n^2(α(\frac{i}{n})-α(\frac{i-1}{n}))$, for $i \in [n]$, and similarly, $μ_n(i) := n^2(β(\frac{i}{n})-β(\frac{i-1}{n}))$, and $ν_n(i) := n^2(γ(\frac{i}{n})-γ(\frac{i-1}{n}))$. Let $X_n, Y_n$ be independent random Hermitian matrices from unitarily invariant distributions with spectra $λ_n$, $μ_n$ respectively. We define norm $\|\cdot\|_\mathcal{I}$ to correspond in a certain way to the sup norm of an antiderivative. For suitable $λ$ and $μ$, we prove that the following limit exists. \begin{equation} \lim\limits_{n \rightarrow \infty}\frac{\ln \mathbb{P}\left[\|spec(X_n + Y_n) - ν_n\|_{\mathcal{I}} < n^2 ε\right]}{n^2}.\end{equation} We interpret this limit in terms of the surface tension $σ$ of continuum limits of the discrete hives defined by Knutson and Tao.

math.PR↗

arXiv2026

Deterministically approximating the volume of a Kostka polytope

Polynomial-time deterministic approximation of volumes of polytopes, up to an approximation factor that grows at most sub-exponentially with the dimension, remains an open problem. Recent work on this question has focused on identifying interesting classes of polytopes for which such approximation algorithms can be obtained. In this paper, we focus on one such class of polytopes: the Kostka polytopes. The volumes of Kostka polytopes appear naturally in questions of random matrix theory, in the context of evaluating the probability density that a random Hermitian matrix with fixed spectrum $λ$ has a given diagonal $μ$ (the so-called randomized Schur-Horn problem): the corresponding Kostka polytope is denoted $\mathrm{GT}(λ, μ)$. We give a polynomial-time deterministic algorithm for approximating the volume of a ($Ω(n^2)$ dimensional) Kostka polytope $\mathrm{GT}(λ, μ)$ to within a multiplicative factor of $\exp(O(n\log n))$, when $λ$ is an integral partition with $n$ parts, with entries bounded above by a polynomial in $n$, and $μ$ is an integer vector lying in the interior of the permutohedron (i.e., convex hull of all permutations) of $λ$. The algorithm thus gives asymptotically correct estimates of the log-volume of Kostka polytopes corresponding to such $(λ, μ)$. Our approach is based on a partition function interpretation of a continuous analogue of Schur polynomials.

math.CO↗

arXiv2025

Approximating mixed volumes to arbitrary accuracy

We study the problem of approximating the mixed volume $V(P_1^{(α_1)}, \dots, P_k^{(α_k)})$ of an $k$-tuple of convex polytopes $(P_1, \dots, P_k)$, each of which is defined as the convex hull of at most $m_0$ points in $\mathbb{Z}^n$. We design an algorithm that produces an estimate that is within a multiplicative $1 \pm ε$ factor of the true mixed volume with a probability greater than $1 - δ.$ Let the constant $ \prod_{i=2}^{k} \frac{(α_{i}+1)^{α_{i}+1}}{α_{i}^{\,α_{i}}}$ be denoted by $\tilde{A}$. When each $P_i \subseteq B_\infty(2^L)$, we show in this paper that the time complexity of the algorithm is bounded above by a polynomial in $n, m_0, L, \tilde{A}, ε^{-1}$ and $\log δ^{-1}$. In fact, a stronger result is proved in this paper, with slightly more involved terminology. In particular, we provide the first randomized polynomial time algorithm for computing mixed volumes of such polytopes when $k$ is an absolute constant, but $α_1, \dots, α_k$ are arbitrary. Our approach synthesizes tools from convex optimization, the theory of Lorentzian polynomials, and polytope subdivision.

cs.CG↗

arXiv2025

PRREACH: Probabilistic Risk Assessment Using Reachability for UAV Control

We present a new approach for designing risk-bounded controllers for Uncrewed Aerial Vehicles (UAVs). Existing frameworks for assessing risk of UAV operations rely on knowing the conditional probability of an incident occurring given different causes. Limited data for computing these probabilities makes real-world implementation of these frameworks difficult. Furthermore, existing frameworks do not include control methods for risk mitigation. Our approach relies on UAV dynamics, and employs reachability analysis for a probabilistic risk assessment over all feasible UAV trajectories. We use this holistic risk assessment to formulate a control optimization problem that minimally changes a UAV's existing control law to be bounded by an accepted risk threshold. We call our approach PRReach. Public and readily available UAV dynamics models and open source spatial data for mapping hazard outcomes enables practical implementation of PRReach for both offline pre-flight and online in-flight risk assessment and mitigation. We evaluate PRReach through simulation experiments on real-world data. Results show that PRReach controllers reduce risk by up to 24% offline, and up to 53% online from classical controllers.

eess.SY↗

arXiv2025

Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel

We consider how a closed Riemannian manifold $M$ and its metric tensor $g$ can be approximately reconstructed from local distance measurements. Moreover, we consider an inverse problem of determining $(M,g)$ from limited knowledge on the heat kernel. In the part 1 of the paper, we considered the approximate construction of a smooth manifold in the case when one is given the noisy distances $\tilde d(x,y)=d(x,y)+\varepsilon_{x,y}$ for all points $x,y\in X$, where $X$ is a $δ$-dense subset of $M$ and $|\varepsilon_{x,y}|<δ$. In this part 2 of the paper, we consider a similar problem with partial data, that is, the approximate construction of the manifold $(M,g)$ when we are given $\tilde d(x,y)$ for $x\in X$ and $y \in U\cap X$, where $U$ is an open subset of $M$. In addition, we consider the inverse problem of determining the manifold $(M,g)$ with non-negative Ricci curvature from noisy observations of the heat kernel $G(y,z,t)$. We show that a manifold approximating $(M,g)$ can be determined in a stable way, when for some unknown source points $z_j$ in $X\setminus U$, we are given the values of the heat kernel $G(y,z_k,t)$ for $y\in X\cap U$ and $t\in (0,1)$ with a multiplicative noise. We also give a uniqueness result for the inverse problem in the case when the data does not contain noise and consider applications in manifold learning. A novel feature of the inverse problem for the heat kernel is that the set $M\setminus U$ containing the sources and the observation set $U$ are disjoint.

math.DG↗

arXiv2025

On the limit of random hives with GUE boundary conditions

We show that hives chosen at random with independent GUE boundary conditions on two sides, weighted by a Vandermonde factor depending on the third side (which is necessary in the context of the randomized Horn problem), when normalized so that the eigenvalues at the edge are asymptotically constant, converge in probability to a continuum hive as $n \rightarrow \infty.$ It had previously been shown in joint work with Sheffield and Tao \cite{NST} that the variance of these scaled random hives tends to $0$ and consequently, from compactness, that they converge in probability subsequentially. In the present paper, building on \cite{NST}, we prove convergence in probability to a single continuum hive, without having to pass to a subsequence. We moreover show that the value at a given point $v$ of this continuum hive equals the supremum of a certain functional acting on asymptotic height functions of lozenge tilings.

math.PR↗

arXiv2024

Sampling from convex sets with a cold start using multiscale decompositions

Running a random walk in a convex body $K\subseteq\mathbb{R}^n$ is a standard approach to sample approximately uniformly from the body. The requirement is that from a suitable initial distribution, the distribution of the walk comes close to the uniform distribution $π_K$ on $K$ after a number of steps polynomial in $n$ and the aspect ratio $R/r$ (i.e., when $rB_2 \subseteq K \subseteq RB_{2}$). Proofs of rapid mixing of such walks often require the probability density $η_0$ of the initial distribution with respect to $π_K$ to be at most $\mathrm{poly}(n)$: this is called a "warm start". Achieving a warm start often requires non-trivial pre-processing before starting the random walk. This motivates proving rapid mixing from a "cold start", wherein $η_0$ can be as high as $\exp(\mathrm{poly}(n))$. Unlike warm starts, a cold start is usually trivial to achieve. However, a random walk need not mix rapidly from a cold start: an example being the well-known "ball walk". On the other hand, Lovász and Vempala proved that the "hit-and-run" random walk mixes rapidly from a cold start. For the related coordinate hit-and-run (CHR) walk, which has been found to be promising in computational experiments, rapid mixing from a warm start was proved only recently but the question of rapid mixing from a cold start remained open. We construct a family of random walks inspired by classical decompositions of subsets of $\mathbb{R}^n$ into countably many axis-aligned dyadic cubes. We show that even with a cold start, the mixing times of these walks are bounded by a polynomial in $n$ and the aspect ratio. Our main technical ingredient is an isoperimetric inequality for $K$ for a metric that magnifies distances between points close to the boundary of $K$. As a corollary, we show that the CHR walk also mixes rapidly both from a cold start and from a point not too close to the boundary of $K$.

cs.DS↗

arXiv2023

Fitting a manifold to data in the presence of large noise

We assume that $M_0$ is a $d$-dimensional $C^{2,1}$-smooth submanifold of $R^n$. Let $K_0$ be the convex hull of $M_0,$ and $B^n_1(0)$ be the unit ball. We assume that $ M_0 \subseteq \partial K_0 \subseteq B^n_1(0).$ We also suppose that $M_0$ has volume ($d$-dimensional Hausdorff measure) less or equal to $V$, reach (i.e., normal injectivity radius) greater or equal to $τ$. Moreover, we assume that $M_0$ is $R$-exposed, that is, tangent to every point $x \in M$ there is a closed ball of radius $R$ that contains $M$. Let $x_1, \dots, x_N$ be independent random variables sampled from uniform distribution on $M_0$ and $ζ_1, \dots, ζ_N$ be a sequence of i.i.d Gaussian random variables in $R^n$ that are independent of $x_1, \dots, x_N$ and have mean zero and covariance $σ^2 I_n.$ We assume that we are given the noisy sample points $y_i$, given by $$ y_i = x_i + ζ_i,\quad \hbox{ for }i = 1, 2, \dots,N. $$ Let $ε,η>0$ be real numbers and $k\geq 2$. Given points $y_i$, $i=1,2,\dots,N$, we produce a $C^k$-smooth function which zero set is a manifold $M_{rec}\subseteq R^n$ such that the Hausdorff distance between $M_{rec}$ and $M_0$ is at most $ ε$ and $M_{rec}$ has reach that is bounded below by $cτ/d^6$ with probability at least $1 - η.$ Assuming $d < c \sqrt{\log \log n}$ and all the other parameters are positive constants independent of $n$, the number of the needed arithmetic operations is polynomial in $n$. In the present work, we allow the noise magnitude $σ$ to be an arbitrarily large constant, thus overcoming a drawback of previous work.

math.ST↗

arXiv2023

Sums of GUE matrices and concentration of hives from correlation decay of eigengaps

Associated to two given sequences of eigenvalues $λ_1 \geq \dots \geq λ_n$ and $μ_1 \geq \dots \geq μ_n$ is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as $n \rightarrow \infty$. Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand--Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni--Erdős--Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process.

math.PR↗

arXiv2023

Random discrete concave functions on an equilateral lattice with periodic Hessians

Motivated by connections to random matrices, Littlewood-Richardson coefficients and tilings, we study random discrete concave functions on an equilateral lattice. We show that such functions having a periodic Hessian of a fixed average value $- s$ concentrate around a quadratic function. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$ have a periodic discrete Hessian, with period $n \mathbb L$. We add a convex quadratic of Hessian $s$; the sum is then periodic with period $n \mathbb L$, and view this as a mean zero function $g$ on the set of vertices $V(\mathbb{T}_n)$ of a torus $\mathbb{T}_n := \frac{\mathbb{Z}}{n\mathbb{Z}}\times \frac{\mathbb{Z}}{n\mathbb{Z}}$ whose Hessian is dominated by $s$. The resulting set of semiconcave functions forms a convex polytope $P_n(s)$. The $\ell_\infty$ diameter of $P_n(s)$ is bounded below by $c(s) n^2$, where $c(s)$ is a positive constant depending only on $s$. Our main result is that under certain conditions, that are met for example when $s_0 = s_1 \leq s_2$, for any $ε> 0,$ we have $$\lim_{n \rightarrow 0} \mathbb{P}\left[\|g\|_\infty > n^{\frac{7}{4} + ε}\right] = 0$$ if $g$ is sampled from the uniform measure on $P_n(s)$. Each $g \in P_n(s)$ corresponds to a kind of honeycomb. We obtain concentration results for these as well.

math.PR↗

arXiv2022

A Spectral Approach to Polytope Diameter

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure $1-o(1)$ and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.

math.CO↗

arXiv2022

Fitting a manifold of large reach to noisy data

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than $τ$. Moreover, let $μ$ be a probability measure on ${\mathcal M}$ whose density on ${\mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {\mathcal M}$, $j=1,2,\dots,N$ be $N$ independent random samples from distribution $μ$. Also, let $ξ_j$, $j=1,2,\dots, N$ be independent random samples from a Gaussian random variable in ${\mathbb R}^n$ having covariance $σ^2I$, where $σ$ is less than a certain specified function of $d, V$ and $τ$. We assume that we are given the data points $y_j=x_j+ξ_j,$ $j=1,2,\dots,N$, modelling random points of ${\mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${\mathcal M}_o\subset {\mathbb R}^n$ whose Hausdorff distance to ${\mathcal M}$ is less than $Cdσ^2/τ$ and whose reach is greater than $cτ/d^6$ with universal constants $C,c > 0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on $σ^{-1}$ and exponentially on $d$.

math.ST↗

arXiv2022

On the mixing time of coordinate Hit-and-Run

We obtain a polynomial upper bound on the mixing time $T_{CHR}(ε)$ of the coordinate Hit-and-Run random walk on an $n-$dimensional convex body, where $T_{CHR}(ε)$ is the number of steps needed in order to reach within $ε$ of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex body that is bounded above by a constant). Our upper bound is polynomial in $n, R$ and $\frac{1}ε$, where we assume that the convex body contains the unit $\Vert\cdot\Vert_\infty$-unit ball $B_\infty$ and is contained in its $R$-dilation $R\cdot B_\infty$. Whether coordinate Hit-and-Run has a polynomial mixing time has been an open question.

cs.DS↗

arXiv2021

On a convexity property of tensor products of irreducible, rational representations of $SL(n)$

The aim of this note is to point out a convexity property with respect to the root lattice for the support of the highest weights that occur in a tensor product of irreducible rational representations of $SL(n)$ over the complex numbers. The observation is a consequence of the convexity properties of the saturation cone and the validity of the saturation conjecture for $SL(n)$.

math.RT↗