arXiv subjects

Hariharan Narayanan

Publications and source records attributed to Hariharan Narayanan.

46 records · Page 3Linked to original sources

arXiv2013

Testing the Manifold Hypothesis

The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to an unknown probability distribution supported in a separable Hilbert space, only using i.i.d samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution $P$ supported on the unit ball of a separable Hilbert space $H$. Let $G(d, V, τ)$ be the set of submanifolds of the unit ball of $H$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $τ$. Let $L(M, P)$ denote mean-squared distance of a random point from the probability distribution $P$ to $M$. We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d random samples from $P$ as input, and determines which of the following two is true (at least one must be): (a) There exists $M \in G(d, CV, \fracτ{C})$ such that $L(M, P) \leq C ε.$ (b) There exists no $M \in G(d, V/C, Cτ)$ such that $L(M, P) \leq \fracε{C}.$ The answer is correct with probability at least $1-δ$.

math.ST↗

arXiv2013

Efficient Sampling from Time-Varying Log-Concave Distributions

We propose a computationally efficient random walk on a convex body which rapidly mixes and closely tracks a time-varying log-concave distribution. We develop general theoretical guarantees on the required number of steps; this number can be calculated on the fly according to the distance from and the shape of the next distribution. We then illustrate the technique on several examples. Within the context of exponential families, the proposed method produces samples from a posterior distribution which is updated as data arrive in a streaming fashion. The sampling technique can be used to track time-varying truncated distributions, as well as to obtain samples from a changing mixture model, fitted in a streaming fashion to data. In the setting of linear optimization, the proposed method has oracle complexity with best known dependence on the dimension for certain geometries. In the context of online learning and repeated games, the algorithm is an efficient method for implementing no-regret mixture forecasting strategies. Remarkably, in some of these examples, only one step of the random walk is needed to track the next distribution.

stat.ML↗

arXiv2013

Estimating Certain Non-Zero Littlewood-Richardson Coefficients

Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups ($GL_n$). The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for certain Littlewood-Richardson coefficients. 2. A proof of approximate log-concavity of certain Littlewood-Richardson coefficients.

math.CO↗

arXiv2012

Damped random walks and the characteristic polynomial of the weighted Laplacian on a graph

For $λ>0$, we define a $λ$-damped random walk to be a random walk that is started from a random vertex of a graph and stopped at each step with probability $\fracλ{1+λ}$, otherwise continued with probability $\frac{1}{1+λ}$. We use the Aldous-Broder algorithm (\cite{aldous, broder}) of generating a random spanning tree and the Matrix-tree theorem to relate the values of the characteristic polynomial of the Laplacian at $\pm λ$ and the stationary measures of the sets of nodes visited by $i$ independent $λ$-damped random walks for $i \in \N$. As a corollary, we obtain a new characterization of the non-zero eigenvalues of the Weighted Graph Laplacian.

math.PR↗

arXiv2012

Random Trees in Electrical Networks

This paper contains results relating currents and voltages in resistive networks to appropriate random trees or forests in those networks.

math.PR↗

arXiv2010

Learning with Spectral Kernels and Heavy-Tailed Data

Two ubiquitous aspects of large-scale data analysis are that the data often have heavy-tailed properties and that diffusion-based or spectral-based methods are often used to identify and extract structure of interest. Perhaps surprisingly, popular distribution-independent methods such as those based on the VC dimension fail to provide nontrivial results for even simple learning problems such as binary classification in these two settings. In this paper, we develop distribution-dependent learning methods that can be used to provide dimension-independent sample complexity bounds for the binary classification problem in these two popular settings. In particular, we provide bounds on the sample complexity of maximum margin classifiers when the magnitude of the entries in the feature vector decays according to a power law and also when learning is performed with the so-called Diffusion Maps kernel. Both of these results rely on bounding the annealed entropy of gap-tolerant classifiers in a Hilbert space. We provide such a bound, and we demonstrate that our proof technique generalizes to the case when the margin is measured with respect to more general Banach space norms. The latter result is of potential interest in cases where modeling the relationship between data elements as a dot product in a Hilbert space is too restrictive.

cs.LG↗

arXiv2009

A Geometric Interpretation of Half-Plane Capacity

Let A be a bounded, relatively closed subset of the upper half plane H whose complement C is simply connected. If B_t is a standard complex Brownian motion starting at iy and t_A = inf {t > 0: B_t not in C}, the half-plane capacity of A, hcap(A) is defined to be the limit as y goes to infinity of y E[Im(B_{t_A}]. This quantity arises naturally in the study of Schramm-Loewner Evolutions (SLE). In this note, we show that hcap(A) is comparable to a more geometric quantity hsiz(A) that we define to be the 2-dimensional Lebesgue measure of the union of all balls tangent to R whose centers belong to A. Our main result is that hsiz(A)/66 < hcap(A) leq 7 hsiz(A)/(2 pi).

math.PR↗

arXiv2008

Distributed Averaging in the presence of a Sparse Cut

We consider the question of averaging on a graph that has one sparse cut separating two subgraphs that are internally well connected. While there has been a large body of work devoted to algorithms for distributed averaging, nearly all algorithms involve only {\it convex} updates. In this paper, we suggest that {\it non-convex} updates can lead to significant improvements. We do so by exhibiting a decentralized algorithm for graphs with one sparse cut that uses non-convex averages and has an averaging time that can be significantly smaller than the averaging time of known distributed algorithms, such as those of \cite{tsitsiklis, Boyd}. We use stochastic dominance to prove this result in a way that may be of independent interest.

cs.DC↗

arXiv2006

Geographic Gossip on Geometric Random Graphs via Affine Combinations

In recent times, a considerable amount of work has been devoted to the development and analysis of gossip algorithms in Geometric Random Graphs. In a recently introduced model termed "Geographic Gossip," each node is aware of its position but possesses no further information. Traditionally, gossip protocols have always used convex linear combinations to achieve averaging. We develop a new protocol for Geographic Gossip, in which counter-intuitively, we use {\it non-convex affine combinations} as updates in addition to convex combinations to accelerate the averaging process. The dependence of the number of transmissions used by our algorithm on the number of sensors $n$ is $n \exp(O(\log \log n)^2) = n^{1 + o(1)}$. For the previous algorithm, this dependence was $\tilde{O}(n^{1.5})$. The exponent 1+ o(1) of our algorithm is asymptotically optimal. Our algorithm involves a hierarchical structure of $\log \log n$ depth and is not completely decentralized. However, the extent of control exercised by a sensor on another is restricted to switching the other on or off.

cs.MA↗

arXiv2005

The computation of Kostka Numbers and Littlewood-Richardson Coefficients is #P-complete

Kostka numbers and Littlewood-Richardson coefficients play an essential role in the representation theory of the symmetric groups and the special linear groups. There has been a significant amount of interest in their computation. The issue of their computational complexity has been a question of folklore, but was asked explicitly by E. Rassart. We prove that the computation of either quantity is #P-complete. The reduction to computing Kostka numbers, is from the #P-complete problem of counting the number of 2 x k contingency tables having given row and column sums. The main ingredient in this reduction is a correspondence discovered by D. E. Knuth. The reduction to the problem of computing Littlewood-Richardson coefficients is from that of computing Kostka numbers.

math.CO↗