arXiv subjects

Hariharan Narayanan

Publications and source records attributed to Hariharan Narayanan.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2021

On Thevenin-Norton and Maximum power transfer theorems

In this paper we show how to compute port behaviour of multiports which have port equations of the general form $B v_P - Q i_P = s,$ which cannot even be put into the hybrid form, indeed may have number of equations ranging from $0$ to $2n,$ where $n$ is the number of ports. We do this through repeatedly solving with different source inputs, a larger network obtained by terminating the multiport by its adjoint through a gyrator. The method works for linear multiports which are consistent for arbitrary internal source values and further have the property that the port conditions uniquely determine internal conditions. We also present the most general version of maximum power transfer theorem possible. This version of the theorem states that `stationarity' (derivative zero condition) of power transfer occurs when the multiport is terminated by its adjoint, provided the resulting network has a solution. If this network does not have a solution there is no port condition for which stationarity holds. This theorem does not require that the multiport has a hybrid immittance matrix.

eess.SY↗

arXiv2020

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 $n$.

stat.ML↗

arXiv2020

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.

eess.SY↗

arXiv2020

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 $\infty$.

math.PR↗

arXiv2019

Reconstruction and interpolation of manifolds I: The geometric Whitney problem

We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold reconstruction where a smooth $n$-dimensional submanifold $S\subset {\mathbb R}^m$, $m>n$ needs to be constructed to approximate a point cloud in ${\mathbb R}^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results we give a new characterisation of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures.

math.DG↗

arXiv2019

On the linking of number lattices

In this paper we study ideas which have proved useful in topological network theory in the context of lattices of numbers. A number lattice $L_S$ is a collection of row vectors, over $\mathbb{Q}$ on a finite column set $S,$ generated by integral linear combination of a finite set of row vectors. A generalized number lattice $K_S$ is the sum of a number lattice $L_S$ and a vector space $V_S$ which has only the zero vector in common with it. The dual $K^d_S$ of a generalized number lattice is the collection of all vectors whose dot product with vectors in $K_S$ are integral and is another generalized number lattice. We consider a linking operation ('matched composition`) between generalized number lattices $K_{SP},K_{P}$ (regarded as collections of row vectors on column sets $S\cup P, P,$ respectively with $S,P,$ disjoint) defined by $K_{SP}\leftrightarrow K_{P}\equiv \{f_S:((f_S,g_P)\in K_{SP}, g_P \in K_{P}\}.$ We show that this operation together with contraction and restriction, and the results, the implicit inversion theorem (which gives simple conditions for the equality $K_{SP}\leftrightarrow (K_{SP}\leftrightarrow K_S)= K_S,$ to hold) and implicit duality theorem ($(K_{SP}\leftrightarrow K_{P})^d= K_{SP}^d\leftrightarrow K_{P}^d$)), are both relevant and useful in suggesting problems concerning number lattices and their solutions. Using the implicit duality theorem, we give simple methods of constructing new self dual lattices from old. We also give new and efficient algorithms for the following. Given $V_{SP},K_P,$ such that $V_{SP}\leftrightarrow (V_{SP}\leftrightarrow K_P)= K_P,$ where $V_{SP}$ is a vector space with a totally unimodular basis matrix, to construct reduced bases for the number lattice part of $V_{SP}\leftrightarrow K_P, K_P^d, (V_{SP}\leftrightarrow K_P)^d,$ from a reduced basis for the number lattice part of $K_P.$

math.NT↗

arXiv2019

Reconstruction of a Riemannian manifold from noisy intrinsic distances

We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian manifold $(M,g)$ is considered as an abstract metric space with intrinsic distances, not as an embedded submanifold of an ambient Euclidean space. Let $\{X_1,X_2,\dots,X_N\}$ bea set of $N$ sample points sampled randomly from an unknown Riemannian $M$ manifold. We assume that we are given the numbers $D_{jk}=d_M(X_j,X_k)+η_{jk}$, where $j,k\in \{1,2,\dots,N\}$. Here, $d_M(X_j,X_k)$ are geodesic distances, $η_{jk}$ are independent, identically distributed random variables such that $\mathbb E e^{|η_{jk}|}$ is finite. We show that when $N$ is large enough, it is possible to construct an approximation of the Riemannian manifold $(M,g)$ with a large probability. This problem is a generalization of the geometric Whitney problem with random measurement errors. We consider also the case when the information on noisy distance $D_{jk}$ of points $X_j$ and $X_k$ is missing with some probability. In particular, we consider the case when we have no information on points that are far away.

math.PR↗

arXiv2018

Generating an equidistributed net on a unit n-sphere using random rotations

We develop a randomized algorithm (that succeeds with high probability) for generating an $ε$-net in a sphere of dimension n. The basic scheme is to pick $O(n \ln(1/n) + \ln(1/δ))$ random rotations and take all possible words of length $O(n \ln(1/ε))$ in the same alphabet and act them on a fixed point. We show this set of points is equidistributed at a scale of $ε$. Our main application is to approximate integration of Lipschitz functions over an n-sphere.

math.PR↗

arXiv2018

On the distribution of random words in a compact Lie group

Let $G$ be a compact Lie group. Suppose $g_1, \dots, g_k$ are chosen independently from the Haar measure on $G$. Let $\mathcal{A} = \cup_{i \in [k]} \mathcal{A}_i$, where, $\mathcal{A}_i := \{g_i\} \cup \{g_i^{-1}\}$. Let $μ_{\mathcal{A}}^\ell$ be the uniform measure over all words of length $\ell$ whose alphabets belong to $\mathcal{A}$. We give probabilistic bounds on the nearness of a heat kernel smoothening of $μ_{\mathcal{A}}^\ell$ to a constant function on $G$ in $\mathcal{L}^2(G)$. We also give probabilistic bounds on the maximum distance of a point in $G$ to the support of $μ_{\mathcal{A}}^\ell$. Lastly, we show that these bounds cannot in general be significantly improved by analyzing the case when $G$ is the $n-$dimensional torus. The question of a spectral gap of a natural Markov operator associated with $\mathcal{A}$ when $G$ is $SU_2$ was reiterated by Bourgain and Gamburd, being first raised by Lubotzky, Philips and Sarnak in 1987 and is still open. In the setting of $SU_2$, our results can be viewed as addressing a quantitative version of a weak variant of this question.

math.PR↗

arXiv2018

On the linear static output feedback problem: the annihilating polynomial approach

One of the fundamental open problems in control theory is that of the stabilization of a linear time invariant dynamical system through static output feedback. We are given a linear dynamical system defined through \begin{align*} \mydot{w} &= Aw + Bu y &= Cw . \end{align*} The problem is to find, if it exists, a feedback $u=Ky$ such that the matrix $A+BKC$ has all its eigenvalues in the complex left half plane and, if such a feedback does not exist, to prove that it does not. Substantial progress has not been made on the computational aspect of the solution to this problem. In this paper we consider instead `which annihilating polynomials can a matrix of the form $A+BKC$ possess?'. We give a simple solution to this problem when the system has either a single input or a single output. For the multi input - multi output case, we use these ideas to characterize the annihilating polynomials when $K$ has rank one, and suggest possible computational solutions for general $K.$ We also present some numerical evidence for the plausibility of this approach for the general case as well as for the problem of shifting the eigenvalues of the system.

eess.SP↗

arXiv2018

Structural Risk Minimization for $C^{1,1}(\mathbb{R}^d)$ Regression

One means of fitting functions to high-dimensional data is by providing smoothness constraints. Recently, the following smooth function approximation problem was proposed: given a finite set $E \subset \mathbb{R}^d$ and a function $f: E \rightarrow \mathbb{R}$, interpolate the given information with a function $\widehat{f} \in \dot{C}^{1, 1}(\mathbb{R}^d)$ (the class of first-order differentiable functions with Lipschitz gradients) such that $\widehat{f}(a) = f(a)$ for all $a \in E$, and the value of $\mathrm{Lip}(\nabla \widehat{f})$ is minimal. An algorithm is provided that constructs such an approximating function $\widehat{f}$ and estimates the optimal Lipschitz constant $\mathrm{Lip}(\nabla \widehat{f})$ in the noiseless setting. We address statistical aspects of reconstructing the approximating function $\widehat{f}$ from a closely-related class $C^{1, 1}(\mathbb{R}^d)$ given samples from noisy data. We observe independent and identically distributed samples $y(a) = f(a) + ξ(a)$ for $a \in E$, where $ξ(a)$ is a noise term and the set $E \subset \mathbb{R}^d$ is fixed and known. We obtain uniform bounds relating the empirical risk and true risk over the class $\mathcal{F}_{\widetilde{M}} = \{f \in C^{1, 1}(\mathbb{R}^d) \mid \mathrm{Lip}(\nabla f) \leq \widetilde{M}\}$, where the quantity $\widetilde{M}$ grows with the number of samples at a rate governed by the metric entropy of the class $C^{1, 1}(\mathbb{R}^d)$. Finally, we provide an implementation using Vaidya's algorithm, supporting our results via numerical experiments on simulated data.

stat.ML↗

arXiv2017

Higher dimensional electrical circuits and the matroid dual of a nonplanar graph

In this paper we describe a physical problem, based on electromagnetic fields, whose topological constraints are higher dimensional versions of Kirchhoff's laws, involving $2-$ simplicial complexes embedded in $\mathbb{R} ^3$ rather than graphs. However, we show that, for the skeleton of this complex, involving only triangles and edges, we can build a matroid dual which is a graph. On this graph we build an `ordinary' electrical circuit, solving which we obtain the solution to our original problem. Construction of this graph is through a `sliding' algorithm which simulates sliding on the surfaces of the triangles, moving from one triangle to another which shares an edge with it but which also is adjacent with respect to the embedding of the complex in $\mathbb{R} ^3.$ For this purpose, the only information needed is the order in which we encounter the triangles incident at an edge, when we rotate say clockwise with respect to the orientation of the edge. The dual graph construction is linear time on the size of the $2-$ complex.

math.CO↗

arXiv2017

Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis

We consider the problem of recovering a $d-$dimensional manifold $\mathcal{M} \subset \mathbb{R}^n$ when provided with noiseless samples from $\mathcal{M}$. There are many algorithms (e.g., Isomap) that are used in practice to fit manifolds and thus reduce the dimensionality of a given data set. Ideally, the estimate $\mathcal{M}_\mathrm{put}$ of $\mathcal{M}$ should be an actual manifold of a certain smoothness; furthermore, $\mathcal{M}_\mathrm{put}$ should be arbitrarily close to $\mathcal{M}$ in Hausdorff distance given a large enough sample. Generally speaking, existing manifold learning algorithms do not meet these criteria. Fefferman, Mitter, and Narayanan (2016) have developed an algorithm whose output is provably a manifold. The key idea is to define an approximate squared-distance function (asdf) to $\mathcal{M}$. Then, $\mathcal{M}_\mathrm{put}$ is given by the set of points where the gradient of the asdf is orthogonal to the subspace spanned by the largest $n - d$ eigenvectors of the Hessian of the asdf. As long as the asdf meets certain regularity conditions, $\mathcal{M}_\mathrm{put}$ is a manifold that is arbitrarily close in Hausdorff distance to $\mathcal{M}$. In this paper, we define two asdfs that can be calculated from the data and show that they meet the required regularity conditions. The first asdf is based on kernel density estimation, and the second is based on estimation of tangent spaces using local principal components analysis.

math.ST↗

arXiv2015

Randomized Interior Point methods for Sampling and Optimization

We present a Markov chain (Dikin walk) for sampling from a convex body equipped with a self-concordant barrier, whose mixing time from a "central point" is strongly polynomial in the description of the convex set. The mixing time of this chain is invariant under affine transformations of the convex set, thus eliminating the need for first placing the body in an isotropic position. This recovers and extends previous results of from polytopes to more general convex sets. On every convex set of dimension $n$, there exists a self-concordant barrier whose "complexity" is polynomially bounded. Consequently, a rapidly mixing Markov chain of the kind we describe can be defined on any convex set. We use these results to design an algorithm consisting of a single random walk for optimizing a linear function on a convex set. We show that this random walk reaches an approximately optimal point in polynomial time with high probability and that the corresponding objective values converge with probability 1 to the optimal objective value as the number of steps tends to infinity. One technical contribution is a family of lower bounds for the isoperimetric constants of (weighted) Riemannian manifolds on which, interior point methods perform a kind of steepest descent. Using results of Barthe \cite{barthe} and Bobkov and Houdré, on the isoperimetry of products of (weighted) Riemannian manifolds, we obtain sharper upper bounds on the mixing time of Dikin walk on products of convex sets than the bounds obtained from a direct application of the Localization Lemma, on which, since (Lovász and Simonovits), the analyses of all random walks on convex sets have relied.

cs.DS↗

arXiv2015

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.

math.NA↗

arXiv2014

On Zeroth-Order Stochastic Convex Optimization via Random Walks

We propose a method for zeroth order stochastic convex optimization that attains the suboptimality rate of $\tilde{\mathcal{O}}(n^{7}T^{-1/2})$ after $T$ queries for a convex bounded function $f:{\mathbb R}^n\to{\mathbb R}$. The method is based on a random walk (the \emph{Ball Walk}) on the epigraph of the function. The randomized approach circumvents the problem of gradient estimation, and appears to be less sensitive to noisy function evaluations compared to noiseless zeroth order methods.

cs.LG↗