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Researcher profile

Hariharan Narayanan

Hariharan Narayanan contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine LearningComputational GeometryData Structures and Algorithmsmath.PRComputationeess.SYmath.OCmath.STMethodologyNumerical AnalysisStatistics TheorySystems and Control

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Published work

7 published item(s)

preprint2026arXiv

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.

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preprint2022arXiv

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 $d$.

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preprint2022arXiv

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.

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preprint2020arXiv

Implicit Linear Algebra and Basic Circuit Theory

In this paper we derive some basic results of circuit theory using `Implicit Linear Algebra' (ILA). This approach has the advantage of simplicity and generality. Implicit linear algebra is outlined in [1]. We denote the space of all vectors on $S$ by $\mathcal{F}_S$ and the space containing only the zero vector on $S$ by $\mathbf{0}_S.$ The dual $\mathcal{V}_S^{\perp}$ of a vector space $\mathcal{V}_S$ is the collection of all vectors whose dot product with vectors in $\mathcal{V}_S$ is zero. The basic operation of ILA is a linking operation ('matched composition`) between vector spaces $\mathcal{V}_{SP},\mathcal{V}_{PQ}$ (regarded as collections of row vectors on column sets $S\cup P, P\cup Q,$ respectively with $S,P,Q$ disjoint) defined by $\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ}\equiv \{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (g_P,h_Q) \in \mathcal{V}_{PQ}\},$ and another ('skewed composition`) defined by $\mathcal{V}_{SP}\rightleftharpoons \mathcal{V}_{PQ}\equiv \{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (-g_P,h_Q) \in \mathcal{V}_{PQ}\}.$ The basic results of ILA are the Implicit Inversion Theorem (which states that $\mathcal{V}_{SP}\leftrightarrow(\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_S)= \mathcal{V}_S,$ iff $\mathcal{V}_{SP}\leftrightarrow \mathbf{0}_P\subseteq \mathcal{V}_S\subseteq \mathcal{V}_{SP}\leftrightarrow\mathcal{F}_S$) and Implicit Duality Theorem (which states that $(\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ})^{\perp}= (\mathcal{V}_{SP}^{\perp}\rightleftharpoons \mathcal{V}_{PQ}^{\perp}$). We show that the operations and results of ILA are useful in understanding basic circuit theory. We illustrate this by using ILA to present a generalization of Thevenin-Norton theorem where we compute multiport behaviour using adjoint multiport termination through a gyrator and a very general version of maximum power transfer theorem, which states that the port conditions that appear, during adjoint multiport termination through an ideal transformer, correspond to maximum power transfer.

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preprint2020arXiv

John's Walk

We present an affine-invariant random walk for drawing uniform random samples from a convex body $\mathcal{K} \subset \mathbb{R}^n$ that uses maximum volume inscribed ellipsoids, known as John's ellipsoids, for the proposal distribution. Our algorithm makes steps using uniform sampling from the John's ellipsoid of the symmetrization of $\mathcal{K}$ at the current point. We show that from a warm start, the random walk mixes in $\widetilde{O}(n^7)$ steps where the log factors depend only on constants associated with the warm start and desired total variation distance to uniformity. We also prove polynomial mixing bounds starting from any fixed point $x$ such that for any chord $pq$ of $\mathcal{K}$ containing $x$, $\left|\log \frac{|p-x|}{|q-x|}\right|$ is bounded above by a polynomial in $n$.

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preprint2020arXiv

Random concave functions on an equilateral lattice with periodic Hessians I: entropy and Laplacians

We show that a random concave function having a periodic hessian on an equilateral lattice has a quadratic scaling limit, if the average hessian of the function satisfies certain conditions. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$, incur addition by a linear function (this condition is equivalent to the periodicity of the hessian of $g$). We identify this set, up to addition by a constant, with a convex polytope $P_n(s)$, where $s$ corresponds to the average hessian. We show that 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, for any $ε_0 > 0$, the normalized Lebesgue measure of all points in $P_n(s)$ that are not contained in a $n^2$ dimensional cube $Q$ of sidelength $2 ε_0 n^2$, centered at the unique (up to addition of a linear term) quadratic polynomial with hessian $s$, tends to $0$ as $n$ tends to $\infty$.

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preprint2015arXiv

Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions

We consider the problem of optimizing an approximately convex function over a bounded convex set in $\mathbb{R}^n$ using only function evaluations. The problem is reduced to sampling from an \emph{approximately} log-concave distribution using the Hit-and-Run method, which is shown to have the same $\mathcal{O}^*$ complexity as sampling from log-concave distributions. In addition to extend the analysis for log-concave distributions to approximate log-concave distributions, the implementation of the 1-dimensional sampler of the Hit-and-Run walk requires new methods and analysis. The algorithm then is based on simulated annealing which does not relies on first order conditions which makes it essentially immune to local minima. We then apply the method to different motivating problems. In the context of zeroth order stochastic convex optimization, the proposed method produces an $ε$-minimizer after $\mathcal{O}^*(n^{7.5}ε^{-2})$ noisy function evaluations by inducing a $\mathcal{O}(ε/n)$-approximately log concave distribution. We also consider in detail the case when the "amount of non-convexity" decays towards the optimum of the function. Other applications of the method discussed in this work include private computation of empirical risk minimizers, two-stage stochastic programming, and approximate dynamic programming for online learning.

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