arXiv subjects

Purnamrita Sarkar

Publications and source records attributed to Purnamrita Sarkar.

50 records · Page 3Linked to original sources

arXiv2017

Convergence Analysis of Gradient EM for Multi-component Gaussian Mixture

In this paper, we study convergence properties of the gradient Expectation-Maximization algorithm \cite{lange1995gradient} for Gaussian Mixture Models for general number of clusters and mixing coefficients. We derive the convergence rate depending on the mixing coefficients, minimum and maximum pairwise distances between the true centers and dimensionality and number of components; and obtain a near-optimal local contraction radius. While there have been some recent notable works that derive local convergence rates for EM in the two equal mixture symmetric GMM, in the more general case, the derivations need structurally different and non-trivial arguments. We use recent tools from learning theory and empirical processes to achieve our theoretical results.

math.ST↗

arXiv2017

On clustering network-valued data

Community detection, which focuses on clustering nodes or detecting communities in (mostly) a single network, is a problem of considerable practical interest and has received a great deal of attention in the research community. While being able to cluster within a network is important, there are emerging needs to be able to cluster multiple networks. This is largely motivated by the routine collection of network data that are generated from potentially different populations. These networks may or may not have node correspondence. When node correspondence is present, we cluster networks by summarizing a network by its graphon estimate, whereas when node correspondence is not present, we propose a novel solution for clustering such networks by associating a computationally feasible feature vector to each network based on trace of powers of the adjacency matrix. We illustrate our methods using both simulated and real data sets, and theoretical justifications are provided in terms of consistency.

stat.ML↗

arXiv2017

Two provably consistent divide and conquer clustering algorithms for large networks

In this article, we advance divide-and-conquer strategies for solving the community detection problem in networks. We propose two algorithms which perform clustering on a number of small subgraphs and finally patches the results into a single clustering. The main advantage of these algorithms is that they bring down significantly the computational cost of traditional algorithms, including spectral clustering, semi-definite programs, modularity based methods, likelihood based methods etc., without losing on accuracy and even improving accuracy at times. These algorithms are also, by nature, parallelizable. Thus, exploiting the facts that most traditional algorithms are accurate and the corresponding optimization problems are much simpler in small problems, our divide-and-conquer methods provide an omnibus recipe for scaling traditional algorithms up to large networks. We prove consistency of these algorithms under various subgraph selection procedures and perform extensive simulations and real-data analysis to understand the advantages of the divide-and-conquer approach in various settings.

stat.ML↗

arXiv2017

On Mixed Memberships and Symmetric Nonnegative Matrix Factorizations

The problem of finding overlapping communities in networks has gained much attention recently. Optimization-based approaches use non-negative matrix factorization (NMF) or variants, but the global optimum cannot be provably attained in general. Model-based approaches, such as the popular mixed-membership stochastic blockmodel or MMSB (Airoldi et al., 2008), use parameters for each node to specify the overlapping communities, but standard inference techniques cannot guarantee consistency. We link the two approaches, by (a) establishing sufficient conditions for the symmetric NMF optimization to have a unique solution under MMSB, and (b) proposing a computationally efficient algorithm called GeoNMF that is provably optimal and hence consistent for a broad parameter regime. We demonstrate its accuracy on both simulated and real-world datasets.

stat.ML↗

arXiv2016

On Robustness of Kernel Clustering

Clustering is one of the most important unsupervised problems in machine learning and statistics. Among many existing algorithms, kernel k-means has drawn much research attention due to its ability to find non-linear cluster boundaries and its inherent simplicity. There are two main approaches for kernel k-means: SVD of the kernel matrix and convex relaxations. Despite the attention kernel clustering has received both from theoretical and applied quarters, not much is known about robustness of the methods. In this paper we first introduce a semidefinite programming relaxation for the kernel clustering problem, then prove that under a suitable model specification, both the K-SVD and SDP approaches are consistent in the limit, albeit SDP is strongly consistent, i.e. achieves exact recovery, whereas K-SVD is weakly consistent, i.e. the fraction of misclassified nodes vanish.

stat.ML↗

arXiv2015

Role of normalization in spectral clustering for stochastic blockmodels

Spectral clustering is a technique that clusters elements using the top few eigenvectors of their (possibly normalized) similarity matrix. The quality of spectral clustering is closely tied to the convergence properties of these principal eigenvectors. This rate of convergence has been shown to be identical for both the normalized and unnormalized variants in recent random matrix theory literature. However, normalization for spectral clustering is commonly believed to be beneficial [Stat. Comput. 17 (2007) 395-416]. Indeed, our experiments show that normalization improves prediction accuracy. In this paper, for the popular stochastic blockmodel, we theoretically show that normalization shrinks the spread of points in a class by a constant fraction under a broad parameter regime. As a byproduct of our work, we also obtain sharp deviation bounds of empirical principal eigenvalues of graphs generated from a stochastic blockmodel.

stat.ML↗

arXiv2014

Active Learning for Crowd-Sourced Databases

Crowd-sourcing has become a popular means of acquiring labeled data for a wide variety of tasks where humans are more accurate than computers, e.g., labeling images, matching objects, or analyzing sentiment. However, relying solely on the crowd is often impractical even for data sets with thousands of items, due to time and cost constraints of acquiring human input (which cost pennies and minutes per label). In this paper, we propose algorithms for integrating machine learning into crowd-sourced databases, with the goal of allowing crowd-sourcing applications to scale, i.e., to handle larger datasets at lower costs. The key observation is that, in many of the above tasks, humans and machine learning algorithms can be complementary, as humans are often more accurate but slow and expensive, while algorithms are usually less accurate, but faster and cheaper. Based on this observation, we present two new active learning algorithms to combine humans and algorithms together in a crowd-sourced database. Our algorithms are based on the theory of non-parametric bootstrap, which makes our results applicable to a broad class of machine learning models. Our results, on three real-life datasets collected with Amazon's Mechanical Turk, and on 15 well-known UCI data sets, show that our methods on average ask humans to label one to two orders of magnitude fewer items to achieve the same accuracy as a baseline that labels random images, and two to eight times fewer questions than previous active learning schemes.

cs.LG↗

arXiv2013

Hypothesis Testing for Automated Community Detection in Networks

Community detection in networks is a key exploratory tool with applications in a diverse set of areas, ranging from finding communities in social and biological networks to identifying link farms in the World Wide Web. The problem of finding communities or clusters in a network has received much attention from statistics, physics and computer science. However, most clustering algorithms assume knowledge of the number of clusters k. In this paper we propose to automatically determine k in a graph generated from a Stochastic Blockmodel. Our main contribution is twofold; first, we theoretically establish the limiting distribution of the principal eigenvalue of the suitably centered and scaled adjacency matrix, and use that distribution for our hypothesis test. Secondly, we use this test to design a recursive bipartitioning algorithm. Using quantifiable classification tasks on real world networks with ground truth, we show that our algorithm outperforms existing probabilistic models for learning overlapping clusters, and on unlabeled networks, we show that we uncover nested community structure.

stat.ML↗

arXiv2013

Nonparametric Link Prediction in Large Scale Dynamic Networks

We propose a nonparametric approach to link prediction in large-scale dynamic networks. Our model uses graph-based features of pairs of nodes as well as those of their local neighborhoods to predict whether those nodes will be linked at each time step. The model allows for different types of evolution in different parts of the graph (e.g, growing or shrinking communities). We focus on large-scale graphs and present an implementation of our model that makes use of locality-sensitive hashing to allow it to be scaled to large problems. Experiments with simulated data as well as five real-world dynamic graphs show that we outperform the state of the art, especially when sharp fluctuations or nonlinearities are present. We also establish theoretical properties of our estimator, in particular consistency and weak convergence, the latter making use of an elaboration of Stein's method for dependency graphs.

stat.ML↗

arXiv2012

A Scalable Bootstrap for Massive Data

The bootstrap provides a simple and powerful means of assessing the quality of estimators. However, in settings involving large datasets---which are increasingly prevalent---the computation of bootstrap-based quantities can be prohibitively demanding computationally. While variants such as subsampling and the $m$ out of $n$ bootstrap can be used in principle to reduce the cost of bootstrap computations, we find that these methods are generally not robust to specification of hyperparameters (such as the number of subsampled data points), and they often require use of more prior information (such as rates of convergence of estimators) than the bootstrap. As an alternative, we introduce the Bag of Little Bootstraps (BLB), a new procedure which incorporates features of both the bootstrap and subsampling to yield a robust, computationally efficient means of assessing the quality of estimators. BLB is well suited to modern parallel and distributed computing architectures and furthermore retains the generic applicability and statistical efficiency of the bootstrap. We demonstrate BLB's favorable statistical performance via a theoretical analysis elucidating the procedure's properties, as well as a simulation study comparing BLB to the bootstrap, the $m$ out of $n$ bootstrap, and subsampling. In addition, we present results from a large-scale distributed implementation of BLB demonstrating its computational superiority on massive data, a method for adaptively selecting BLB's hyperparameters, an empirical study applying BLB to several real datasets, and an extension of BLB to time series data.

stat.ME↗

arXiv2012

Nonparametric Link Prediction in Dynamic Networks

We propose a non-parametric link prediction algorithm for a sequence of graph snapshots over time. The model predicts links based on the features of its endpoints, as well as those of the local neighborhood around the endpoints. This allows for different types of neighborhoods in a graph, each with its own dynamics (e.g, growing or shrinking communities). We prove the consistency of our estimator, and give a fast implementation based on locality-sensitive hashing. Experiments with simulated as well as five real-world dynamic graphs show that we outperform the state of the art, especially when sharp fluctuations or non-linearities are present.

cs.LG↗

arXiv2012

The Big Data Bootstrap

The bootstrap provides a simple and powerful means of assessing the quality of estimators. However, in settings involving large datasets, the computation of bootstrap-based quantities can be prohibitively demanding. As an alternative, we present the Bag of Little Bootstraps (BLB), a new procedure which incorporates features of both the bootstrap and subsampling to obtain a robust, computationally efficient means of assessing estimator quality. BLB is well suited to modern parallel and distributed computing architectures and retains the generic applicability, statistical efficiency, and favorable theoretical properties of the bootstrap. We provide the results of an extensive empirical and theoretical investigation of BLB's behavior, including a study of its statistical correctness, its large-scale implementation and performance, selection of hyperparameters, and performance on real data.

cs.LG↗

arXiv2012

A Tractable Approach to Finding Closest Truncated-commute-time Neighbors in Large Graphs

Recently there has been much interest in graph-based learning, with applications in collaborative filtering for recommender networks, link prediction for social networks and fraud detection. These networks can consist of millions of entities, and so it is very important to develop highly efficient techniques. We are especially interested in accelerating random walk approaches to compute some very interesting proximity measures of these kinds of graphs. These measures have been shown to do well empirically (Liben-Nowell & Kleinberg, 2003; Brand, 2005). We introduce a truncated variation on a well-known measure, namely commute times arising from random walks on graphs. We present a very novel algorithm to compute all interesting pairs of approximate nearest neighbors in truncated commute times, without computing it between all pairs. We show results on both simulated and real graphs of size up to 100; 000 entities, which indicate near-linear scaling in computation time.

cs.SI↗

arXiv2012

Getting It All from the Crowd

Hybrid human/computer systems promise to greatly expand the usefulness of query processing by incorporating the crowd for data gathering and other tasks. Such systems raise many database system implementation questions. Perhaps most fundamental is that the closed world assumption underlying relational query semantics does not hold in such systems. As a consequence the meaning of even simple queries can be called into question. Furthermore query progress monitoring becomes difficult due to non-uniformities in the arrival of crowdsourced data and peculiarities of how people work in crowdsourcing systems. To address these issues, we develop statistical tools that enable users and systems developers to reason about tradeoffs between time/cost and completeness. These tools can also help drive query execution and crowdsourcing strategies. We evaluate our techniques using experiments on a popular crowdsourcing platform.

cs.DB↗