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Kavita Ramanan

Publications and source records attributed to Kavita Ramanan.

At least 19 recordsLinked to original sources
arXiv2026

Gibbs conditioning, atypical consensus and splitting Gibbs measures on random regular graphs

Given n independent Bernoulli(p) random variables X_i, i = 1, ..., n, representing the opinions of individuals connected by an underlying random k-regular graph G_n on {1, ..., n}, we show that when conditioned on an atypical empirical consensus, which is the normalized sum of X_i X_j over neighboring vertices i, j, the joint distribution of the random variables converges, as n goes to infinity, to an Ising measure on the infinite k-regular tree T^k with a specific external field that depends only on the bias parameter p, and a temperature that depends on both p and the atypical consensus value. In particular, we show that conditional on the empirical consensus being smaller (respy, larger) than typical, the limit is a translation-invariant splitting (TIS) antiferromagnetic (respy, ferromagnetic) Ising measure on T^k. Moreover, if the bias is zero, then there is a phase transition: when the consensus exceeds k/(k-1), the conditional limits could be either the plus or minus boundary condition Ising measures. Furthermore, when X_i, i = 1, ..., n, are i.i.d. on a finite space, we show that when conditioned on an atypical value of the scaled sum of h(X_i, X_j) over neighboring vertices i and j, for any symmetric edge potential h, the limiting joint distribution of {X_i} lies in the set of (possibly degenerate) TIS Gibbs measures on T^k. The proofs leverage a tractable form of the large deviation rate function for component empirical measures of random regular graphs with i.i.d. marks and Gibbs conditioning principles, and entail careful analyses of associated non-convex constrained optimization problems. As a by-product of our results, we also obtain an (asymptotic) analog of the maximum entropy principle for Gibbs measures on random regular graphs.

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arXiv2025

Marginal dynamics of probabilistic cellular automata on trees

We study locally interacting processes in discrete time, often called probabilistic cellular automata, indexed by locally finite graphs. For infinite regular trees and certain generalized Galton-Watson trees, we show that the marginal evolution at a single vertex and its neighborhood can be characterized by an autonomous stochastic recursion referred to as the local-field equation. This evolution can be viewed as a nonlinear or measure-dependent chain, but the measure dependence arises from the symmetries of the underlying tree rather than from any mean field interactions. We discuss applications to simulation of marginal dynamics and approximations of empirical measures of interacting chains on several generic classes of large-scale finite graphs that are locally tree-like. In addition to the symmetries of the tree, a key role is played by a second-order Markov random field property, which we establish for general graphs along with some other novel Gibbs measure properties.

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arXiv2025

A case study of the long-time behavior of the Gaussian local-field equation

For any integer $κ\geq 2$, the $κ$-local-field equation ($κ$-LFE) characterizes the limit of the neighborhood path empirical measure of interacting diffusions on $κ$-regular random graphs, as the graph size goes to infinity. It has been conjectured that the long-time behavior of the (in general non-Markovian) $κ$-LFE coincides with that of a certain more tractable Markovian analog, the Markov $κ$-local-field equation. In the present article, we prove this conjecture for the case when $κ= 2$ and the diffusions are one-dimensional with affine drifts. As a by-product of our proof, we also show that for interacting diffusions on the $n$-cycle (or 2-regular random graph on $n$ vertices), the limits $n \rightarrow \infty$ and $t\rightarrow \infty$ commute. Along the way, we also establish well-posedness of the Markov $κ$-local field equations with affine drifts for all $κ\geq 2$, which may be of independent interest.

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arXiv2025

Tractable description of hydrodynamic limits of a class of interacting jump processes on sparse graphs

We consider dynamics of the empirical measure of vertex neighborhood states of Markov interacting jump processes on sparse random graphs, in a suitable asymptotic limit as the graph size goes to infinity. Under the assumption of a certain acyclic structure on single-particle transitions, we provide a tractable autonomous description of the evolution of this hydrodynamic limit in terms of a finite coupled system of ordinary differential equations. Key ingredients of the proof include a characterization of the hydrodynamic limit of the neighborhood empirical measure in terms of a certain local-field equation, well-posedness of its Markovian projection, and a Markov random field property of the time-marginals, which may be of independent interest. We also show how our results lead to principled approximations for classes of interacting jump processes and illustrate its efficacy via simulations on several examples, including an idealized model of seizure spread in the brain.

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arXiv2024

An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees

We study the long-time behavior of the $κ$-Markov local-field equation ($κ$-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the $κ$-regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the $κ$-MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the $κ$-MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the $κ$-MLFE and are also marginals of splitting Gibbs measures on the $κ$-regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case $κ= 2$ we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the $2$-MLFE measure flow to its unique stationary distribution.

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arXiv2024

Hydrodynamic Limits of non-Markovian Interacting Particle Systems on Sparse Graphs

Consider an interacting particle system indexed by the vertices of a (possibly random) locally finite graph whose vertices and edges are equipped with marks representing parameters of the model such as the environment and initial conditions. Each particle takes values in a countable state space and evolves according to a pure jump process whose jump intensities depend on only the states (or histories) and marks of itself and particles and edges in its neighborhood. Under mild conditions, it is shown that if the sequence of (marked) interaction graphs converges locally in probability to a limit (marked) graph that satisfies a certain finite dissociability property, then the corresponding sequence of empirical measures of the particle trajectories converges weakly to the law of the marginal dynamics at the root vertex of the limit graph. The proof of this limit relies on several results of independent interest. First, such interacting particle systems are shown to be well-posed on almost surely finitely dissociable graphs, which include graphs of maximal bounded degree and any Galton-Watson tree whose offspring distribution has a finite first moment. A counterexample is provided to show that well-posedness can fail for dynamics on graphs outside this class. Second, the dynamics on a locally convergent sequence of graphs are shown to converge in the local weak sense to the dynamics on the limit graph when the latter is finitely dissociable. Finally, the dynamics are also shown to exhibit an (annealed) asymptotic correlation decay property. These results complement recent work that establishes hydrodynamic limits of locally interacting probabilistic cellular automata and diffusions on sparse random graphs. However, the analysis of jump processes requires very different techniques, including percolation arguments and notions such as (consistent) spatial localization and causal chains.

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arXiv2024

The fundamental martingale with applications to Markov Random Fields

We consider collections of SDEs indexed by a graph. Each SDE is driven by an additive Gaussian noise and each drift term interacts with all other SDEs within the graph neighbourhood. We derive the fundamental martingale for a class of Gaussian processes and use this to prove a Girsanov type theorem. Further, we use this to construct a clique factorisation to prove that the law of the interacting SDEs forms a 2-Markov Random Field.

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arXiv2024

A Mimicking Theorem for processes driven by fractional Brownian motion

In this paper, we prove a mimicking theorem for stochastic processes with an additive Gaussian noise along with some entropy and transport type estimates. As an application of these results, we prove sharp quantitative propagation of chaos result and derive a formula for the marginal dynamics of collections of locally interacting stochastic differential equations with additive Gaussian noise.

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arXiv2024

The $\ell_r$-Levy-Grothendieck problem and $r\rightarrow p$ norms of Levy matrices

Given an $n\times n$ matrix $A_n$ and $1\leq r, p \leq\infty$, consider the following quadratic optimization problem referred to as the $\ell_r$-Grothendieck problem: \begin{align}M_r(A_n)\coloneqq\max_{\boldsymbol{x}\in\mathbb{R}^n:\|\boldsymbol{x}\|_r\leq1}\boldsymbol{x}^{\top} A_n \boldsymbol{x},\end{align} as well as the $r\rightarrow p$ operator norm of the matrix $A_n$, defined as \begin{align}\|A_n\|_{r \rightarrow p}\coloneqq \sup _{\boldsymbol{x}\in\mathbb{R}^n:\|\boldsymbol{x}\|_r \leq 1}\|A_n \boldsymbol{x}\|_p,\end{align} where $\|\boldsymbol{x}\|_r$ denotes the $\ell_r$-norm of the vector $\boldsymbol{x}$. This work analyzes high-dimensional asymptotics of these quantities when $A_n$ are symmetric random matrices with independent and identically distributed heavy-tailed upper-triangular entries with index $α$. When $1\leq r\leq 2$ (respectively, $1\leq r\leq p$) and $α\in(0,2)$, suitably scaled versions of $M_r(A_n)$ and $\|A_n\|_{r\rightarrow p}$ are shown to converge to a Fréchet distribution as $n\rightarrow\infty$. In contrast, when $2< r<\infty$ (respectively, $1\leq p< r$), it is shown that there exists $α_*\in(1,2)$ such that for every $α\in(0,α_*)$, suitably scaled versions of $M_r(A_n)$ and $\|A_n\|_{r\rightarrow p}$ converge to the power of a stable distribution. Furthermore, it is shown that there exists $\barα_*>α_*$ such that when $α\in(α_*,\barα_*)$, the latter convergence result holds only when the matrix entries are centered; when the entries have non-zero mean, a different limit arises after additional centering and scaling. As a corollary, these results yield a characterization of the limiting ground state of the Levy spin glass when $α\in (0,1)$. The analysis uses a combination of tools from the theory of heavy-tailed distributions, the nonlinear power method and concentration inequalities.

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arXiv2024

On $r$-to-$p$ norms of random matrices with nonnegative entries: Asymptotic normality and $\ell_\infty$-bounds for the maximizer

For an $n\times n$ matrix $A_n$, the $r\to p$ operator norm is defined as $$\|A_n\|_{r\to p}:= \sup_{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x} \|_r\leq 1 } \|A_n\mathbf{x} \|_p\quad\text{for}\quad r,p\geq 1.$$ For different choices of $r$ and $p$, this norm corresponds to key quantities that arise in diverse applications including matrix condition number estimation, clustering of data, and construction of oblivious routing schemes in transportation networks. This article considers $r\to p$ norms of symmetric random matrices with nonnegative entries, including adjacency matrices of Erdős-Rényi random graphs, matrices with positive sub-Gaussian entries, and certain sparse matrices. For $1<p\leq r<\infty$, the asymptotic normality, as $n\to\infty$, of the appropriately centered and scaled norm $\|A_n\|_{r\to p}$ is established. When $p \geq 2$, this is shown to imply asymptotic normality of the solution to the $\ell_p$ quadratic maximization problem, also known as the $\ell_p$ Grothendieck problem. Furthermore, a sharp $\ell_\infty$-approximation bound for the unique maximizing vector in the definition of $\|A_n\|_{r\to p}$ is obtained, and may be viewed as an $\ell_\infty$-stability result of the maximizer under random perturbations of the matrix with mean entries. This result is in fact shown to hold for a broad class of deterministic sequences of matrices having certain asymptotic expansion properties. The results obtained can be viewed as a generalization of the seminal results of Füredi and Komlós (1981) on asymptotic normality of the largest singular value of a class of symmetric random matrices. In the general case with $1<p\leq r< \infty$, spectral methods are no longer applicable, and so a new approach is developed involving a refined convergence analysis of a nonlinear power method and a perturbation bound on the maximizing vector, which may be of independent interest.

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arXiv2023

Interacting stochastic processes on sparse random graphs

Large ensembles of stochastically evolving interacting particles describe phenomena in diverse fields including statistical physics, neuroscience, biology, and engineering. In such systems, the infinitesimal evolution of each particle depends only on its own state (or history) and the states (or histories) of neighboring particles with respect to an underlying, possibly random, interaction graph. While these high-dimensional processes are typically too complex to be amenable to exact analysis, their dynamics are quite well understood when the interaction graph is the complete graph. In this case, classical theorems show that in the limit as the number of particles goes to infinity, the dynamics of the empirical measure and the law of a typical particle coincide and can be characterized in terms of a much more tractable dynamical system of reduced dimension called the mean-field limit. In contrast, until recently not much was known about corresponding convergence results in the complementary case when the interaction graph is sparse (i.e., with uniformly bounded average degree). This article provides a brief survey of classical work and then describes recent progress on the sparse regime that relies on a combination of techniques from random graph theory, Markov random fields, and stochastic analysis. The article concludes by discussing ramifications for applications and posing several open problems.

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arXiv2023

On the large deviation rate function for marked sparse random graphs

We consider (annealed) large deviation principles for component empirical measures of several families of marked sparse random graphs, including (i) uniform graphs on $n$ vertices with a fixed degree distribution; (ii) uniform graphs on $n$ vertices with a fixed number of edges; (iii) Erdős-Rényi $G(n, c/n)$ random graphs. Assuming that edge and vertex marks are independent, identically distributed, and take values in a finite state space, we show that the large deviation rate function admits a concise representation as a sum of relative entropies that quantify the cost of deviation of a probability measure on marked rooted graphs from certain auxiliary independent and conditionally independent versions. The proof exploits unimodularity, the consequent mass transport principle, and random tree labelings to express certain combinatorial quantities as expectations with respect to size-biased distributions, and to identify unimodular extensions with suitable conditional laws. We also illustrate how this representation can be used to establish Gibbs conditioning principles that provide insight into the structure of marked random graphs conditioned on a rare event. Additional motivation for this work arises from the fact that such a representation is also useful for characterizing the annealed pressure of statistical physics models with general spins, and large deviations of evolving interacting particle systems on sparse random graphs.

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arXiv2023

Exact description of limiting SIR and SEIR dynamics on locally tree-like graphs

We study the Susceptible-Infected-Recovered (SIR) and the Susceptible-Exposed-Infected-Recovered (SEIR) models of epidemics, with possibly time-varying rates, on a class of networks that are locally tree-like, which includes sparse Erdős-Rènyi random graphs, random regular graphs, and other configuration models. We identify tractable systems of ODEs that exactly describe the dynamics of the SIR and SEIR processes in a suitable asymptotic regime in which the population size goes to infinity. Moreover, in the case of constant recovery and infection rates, we characterize the outbreak size as the unique zero of an explicit functional. We use this to show that a (suitably defined) mean-field prediction always overestimates the outbreak size, and that the outbreak sizes for SIR and SEIR processes with the same initial condition and constant infection and recovery rates coincide. In contrast, we show that the outbreak sizes for SIR and SEIR processes with the same time-varying infection and recovery rates can in general be quite different. We also demonstrate via simulations the efficacy of our approximations for populations of moderate size.

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arXiv2023

Quenched large deviation principles for random projections of $\ell_p^n$ balls

Let $(k_n)_{n \in \mathbb{N}}$ be a sequence of positive integers growing to infinity at a sublinear rate, $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$. Given a sequence of $n$-dimensional random vectors $\{Y^{(n)}\}_{n \in \mathbb{N}}$ belonging to a certain class, which includes uniform distributions on suitably scaled $\ell_p^n$-balls or $\ell_p^n$-spheres, $p \geq 2$, and product distributions with sub-Gaussian marginals, we study the large deviations behavior of the corresponding sequence of $k_n$-dimensional orthogonal projections $n^{-1/2} \boldsymbol{a}_{n,k_n} Y^{(n)}$, where $\boldsymbol{a}_{n,k_n}$ is an $(n \times k_n)$-dimensional projection matrix lying in the Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$. For almost every sequence of projection matrices, we establish a large deviation principle (LDP) for the corresponding sequence of projections, with a fairly explicit rate function that does not depend on the sequence of projection matrices. As corollaries, we also obtain quenched LDPs for sequences of $\ell_2$-norms and $\ell_\infty$-norms of the coordinates of the projections. Past work on LDPs for projections with growing dimension has mainly focused on the annealed setting, where one also averages over the random projection matrix, chosen from the Haar measure, in which case the coordinates of the projection are exchangeable. The quenched setting lacks such symmetry properties, and gives rise to significant new challenges in the setting of growing projection dimension. Along the way, we establish new Gaussian approximation results on the Stiefel manifold that may be of independent interest. Such LDPs are of relevance in asymptotic convex geometry, statistical physics and high-dimensional statistics.

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arXiv2023

Geometric sharp large deviations for random projections of $\ell_p^n$ spheres and balls

Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the "prefactor" in front of the exponentially decaying term. For fixed $p \in (1,\infty)$, consider independent sequences $(X^{(n,p)})_{n \in \mathbb{N}}$ and $(Θ^n)_{n \in \mathbb{N}}$ of random vectors with $Θ^n$ distributed according to the normalized cone measure on the unit $\ell_2^n$ sphere, and $X^{(n,p)}$ distributed according to the normalized cone measure on the unit $\ell_p^n$ sphere. For almost every realization $(θ^n)_{n\in\mathbb{N}}$ of $(Θ^n)_{n\in\mathbb{N}}$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{(n,p)}$ onto $θ^n$, that are asymptotically exact (as the dimension $n$ tends to infinity). Furthermore, the case when $(X^{(n,p)})_{n \in \mathbb{N}}$ is replaced with $(\mathscr{X}^{(n,p)})_{n \in \mathbb{N}}$, where $\mathscr{X}^{(n,p)}$ is distributed according to the uniform (or normalized volume) measure on the unit $\ell_p^n$ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions $(θ^n)_{n \in\mathbb{N}}$ that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of $n$.

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arXiv2022

Interacting Jump Processes Preserve Semi-Global Markov Random Fields on Path Space

Consider a system of interacting particles indexed by the nodes of a graph whose vertices are equipped with marks representing parameters of the model such as the environment or initial data. Each particle takes values in a countable state space and evolves according to a (possibly non-Markovian) continuous-time pure jump process whose jump intensities depend only on its own state (or history) and marks as well as the states (or histories) and marks of particles and edges in its neighborhood in the graph. Under mild conditions on the jump intensities, it is shown that the trajectories of the interacting particle system exhibit a certain local or semi-global Markov random field property whenever the initial condition satisfies the same property. Our results complement recent works that establish the preservation of a local second-order Markov random field property for interacting diffusions. Our proof methodology in the context of jump processes is different, and works directly on infinite graphs, thereby bypassing any limiting arguments. Our results apply to models arising in diverse fields including statistical physics, neuroscience, epidemiology and opinion dynamics, and have direct applications to the study of marginal distributions of interacting particle systems on Cayley trees.

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arXiv2022

Local weak convergence for sparse networks of interacting processes

We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles interact directly only with their nearest neighbors in the graph. To encode sparsity we work in the framework of local weak convergence of marked (random) graphs. We show that the joint law of the particle system varies continuously with respect to local weak convergence of the underlying graph marked with the initial conditions. In addition, we show that the global empirical measure converges to a non-random limit for a large class of graph sequences including sparse Erdös-Rényi graphs and configuration models, whereas the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Along the way, we develop some related results on the time-propagation of ergodicity and empirical field convergence, as well as some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new. The results obtained here are also useful for obtaining autonomous descriptions of marginal dynamics of interacting diffusions and Markov chains on sparse graphs. While limits of interacting particle systems on dense graphs have been extensively studied, there are relatively few works that have studied the sparse regime in generality.

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arXiv2021

An asymptotic thin shell condition and large deviations for random multidimensional projections

Consider the projection of an $n$-dimensional random vector onto a random $k_n$-dimensional basis, $k_n \leq n$, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$, in three different asymptotic regimes as $n \rightarrow \infty$: "constant" ($k_n=k$), "sublinear" ($k_n \rightarrow \infty$ but $k_n/n \rightarrow 0$) and "linear" $k_n/n \rightarrow λ$ with $0 < λ\le 1$). When the sequence of random vectors satisfies a certain "asymptotic thin shell condition", we establish annealed large deviation principles (LDPs) for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish LDPs for certain scaled $\ell_q$ norms of the random projections in these different regimes. Moreover, we verify our assumptions for various sequences of random vectors of interest, including those distributed according to Gibbs measures with superquadratic interaction potential, or the uniform measure on suitably scaled $\ell_p^n$ balls, for $p \in [1,\infty)$, and generalized Orlicz balls defined via a superquadratic function. Our results complement the central limit theorem for convex sets and related results which are known to hold under a "thin shell" condition. These results also substantially extend existing large deviation results for random projections, which are first, restricted to the setting of measures on $\ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e., in $\mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.

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