Sublinear time algorithms for earth mover's distance

We study the problem of estimating the Earth Mover’s Distance (EMD) between probability distributions when given access only to samples of the distributions. We give closeness testers and additive-error estimators over domains in [0, 1][superscript d], with sample complexities independent of domain size – permitting the testability even of continuous distributions over infinite domains. Instead, our algorithms depend on other parameters, such as the diameter of the domain space, which may be significantly smaller. We also prove lower bounds showing the dependencies on these parameters to be essentially optimal. Additionally, we consider whether natural classes of distributions exist for which there are algorithms with better dependence on the dimension, and show that for highly clusterable data, this is indeed the case. Lastly, we consider a variant of the EMD, defined over tree metrics instead of the usual l 1 metric, and give tight upper and lower bounds

Sublinear algorithms for Earth Mover's Distance

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Includes bibliographical references (p. 14-15).We study the problem of estimating the Earth Mover's Distance (EMD) between probability distributions when given access only to samples. We give closeness testers and additive-error estimators over domains in [0, [delta]]d, with sample complexities independent of domain size - permitting the testability even of continuous distributions over infinite domains. Instead, our algorithms depend on other parameters, such as the diameter of the domain space, which may be significantly smaller. We also prove lower bounds showing our testers to be optimal in their dependence on these parameters. Additionally, we consider whether natural classes of distributions exist for which there are algorithms with better dependence on the dimension, and show that for highly clusterable data, this is indeed the case. Lastly, we consider a variant of the EMD, defined over tree metrics instead of the usual L₁ metric, and give optimal algorithms.by Khanh Do Ba.S.M

Average Sensitivity of Graph Algorithms

In modern applications of graphs algorithms, where the graphs of interest are large and dynamic, it is unrealistic to assume that an input representation contains the full information of a graph being studied. Hence, it is desirable to use algorithms that, even when only a (large) subgraph is available, output solutions that are close to the solutions output when the whole graph is available. We formalize this idea by introducing the notion of average sensitivity of graph algorithms, which is the average earth mover's distance between the output distributions of an algorithm on a graph and its subgraph obtained by removing an edge, where the average is over the edges removed and the distance between two outputs is the Hamming distance. In this work, we initiate a systematic study of average sensitivity. After deriving basic properties of average sensitivity such as composition, we provide efficient approximation algorithms with low average sensitivities for concrete graph problems, including the minimum spanning forest problem, the global minimum cut problem, the minimum $s$-$t$ cut problem, and the maximum matching problem. In addition, we prove that the average sensitivity of our global minimum cut algorithm is almost optimal, by showing a nearly matching lower bound. We also show that every algorithm for the 2-coloring problem has average sensitivity linear in the number of vertices. One of the main ideas involved in designing our algorithms with low average sensitivity is the following fact; if the presence of a vertex or an edge in the solution output by an algorithm can be decided locally, then the algorithm has a low average sensitivity, allowing us to reuse the analyses of known sublinear-time algorithms and local computation algorithms (LCAs). Using this connection, we show that every LCA for 2-coloring has linear query complexity, thereby answering an open question.Comment: 39 pages, 1 figur

Unsupervised Discovery and Representation of Subspace Trends in Massive Biomedical Datasets

The goal of this dissertation is to develop unsupervised algorithms for discovering previously unknown subspace trends in massive multivariate biomedical data sets without the benefit of prior information. A subspace trend is a sustained pattern of gradual/progressive changes within an unknown subset of feature dimensions. A fundamental challenge to subspace trend discovery is the presence of irrelevant data dimensions, noise, outliers, and confusion from multiple subspace trends driven by independent factors that are mixed in with each other. These factors can obscure the trends in traditional dimension reduction and projection based data visualizations. To overcome these limitations, we propose a novel graph-theoretic neighborhood similarity measure for sensing concordant progressive changes across data dimensions. Using this measure, we present an unsupervised algorithm for trend-relevant feature selection and visualization. Additionally, we propose to use an efficient online density-based representation to make the algorithm scalable for massive datasets. The representation not only assists in trend discovery, but also in cluster detection including rare populations. Our method has been successfully applied to diverse synthetic and real-world biomedical datasets, such as gene expression microarray and arbor morphology of neurons and microglia in brain tissue. Derived representations revealed biologically meaningful hidden subspace trend(s) that were obscured by irrelevant features and noise. Although our applications are mostly from the biomedical domain, the proposed algorithm is broadly applicable to exploratory analysis of high-dimensional data including visualization, hypothesis generation, knowledge discovery, and prediction in diverse other applications.Electrical and Computer Engineering, Department o

Learning task-specific similarity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2006.Includes bibliographical references (p. 139-147).The right measure of similarity between examples is important in many areas of computer science. In particular it is a critical component in example-based learning methods. Similarity is commonly defined in terms of a conventional distance function, but such a definition does not necessarily capture the inherent meaning of similarity, which tends to depend on the underlying task. We develop an algorithmic approach to learning similarity from examples of what objects are deemed similar according to the task-specific notion of similarity at hand, as well as optional negative examples. Our learning algorithm constructs, in a greedy fashion, an encoding of the data. This encoding can be seen as an embedding into a space, where a weighted Hamming distance is correlated with the unknown similarity. This allows us to predict when two previously unseen examples are similar and, importantly, to efficiently search a very large database for examples similar to a query. This approach is tested on a set of standard machine learning benchmark problems. The model of similarity learned with our algorithm provides and improvement over standard example-based classification and regression. We also apply this framework to problems in computer vision: articulated pose estimation of humans from single images, articulated tracking in video, and matching image regions subject to generic visual similarity.by Gregory Shakhnarovich.Ph.D

Underdetermined convolutive source separation using two dimensional non-negative factorization techniques

PhD ThesisIn this thesis the underdetermined audio source separation has been considered, that is, estimating the original audio sources from the observed mixture when the number of audio sources is greater than the number of channels. The separation has been carried out using two approaches; the blind audio source separation and the informed audio source separation. The blind audio source separation approach depends on the mixture signal only and it assumes that the separation has been accomplished without any prior information (or as little as possible) about the sources. The informed audio source separation uses the exemplar in addition to the mixture signal to emulate the targeted speech signal to be separated. Both approaches are based on the two dimensional factorization techniques that decompose the signal into two tensors that are convolved in both the temporal and spectral directions. Both approaches are applied on the convolutive mixture and the high-reverberant convolutive mixture which are more realistic than the instantaneous mixture. In this work a novel algorithm based on the nonnegative matrix factor two dimensional deconvolution (NMF2D) with adaptive sparsity has been proposed to separate the audio sources that have been mixed in an underdetermined convolutive mixture. Additionally, a novel Gamma Exponential Process has been proposed for estimating the convolutive parameters and number of components of the NMF2D/ NTF2D, and to initialize the NMF2D parameters. In addition, the effects of different window length have been investigated to determine the best fit model that suit the characteristics of the audio signal. Furthermore, a novel algorithm, namely the fusion K models of full-rank weighted nonnegative tensor factor two dimensional deconvolution (K-wNTF2D) has been proposed. The K-wNTF2D is developed for its ability in modelling both the spectral and temporal changes, and the spatial covariance matrix that addresses the high reverberation problem. Variable sparsity that derived from the Gibbs distribution is optimized under the Itakura-Saito divergence and adapted into the K-wNTF2D model. The tensors of this algorithm have been initialized by a novel initialization method, namely the SVD two-dimensional deconvolution (SVD2D). Finally, two novel informed source separation algorithms, namely, the semi-exemplar based algorithm and the exemplar-based algorithm, have been proposed. These algorithms based on the NMF2D model and the proposed two dimensional nonnegative matrix partial co-factorization (2DNMPCF) model. The idea of incorporating the exemplar is to inform the proposed separation algorithms about the targeted signal to be separated by initializing its parameters and guide the proposed separation algorithms. The adaptive sparsity is derived for both ii of the proposed algorithms. Also, a multistage of the proposed exemplar based algorithm has been proposed in order to further enhance the separation performance. Results have shown that the proposed separation algorithms are very promising, more flexible, and offer an alternative model to the conventional methods