In many machine learning applications, it is necessary to meaningfully aggregate, through alignment, different but related datasets. Optimal transport (OT)-based approaches pose alignment as a divergence minimization problem: the aim is to transform a source dataset to match a target dataset using the Wasserstein distance as a divergence measure. We introduce a hierarchical formulation of OT which leverages clustered structure in data to improve alignment in noisy, ambiguous, or multimodal settings. To solve this numerically, we propose a distributed ADMM algorithm that also exploits the Sinkhorn distance, thus it has an efficient computational complexity that scales quadratically with the size of the largest cluster. When the transformation between two datasets is unitary, we provide performance guarantees that describe when and how well aligned cluster correspondences can be recovered with our formulation, as well as provide worst-case dataset geometry for such a strategy. We apply this method to synthetic datasets that model data as mixtures of low-rank Gaussians and study the impact that different geometric properties of the data have on alignment. Next, we applied our approach to a neural decoding application where the goal is to predict movement directions and instantaneous velocities from populations of neurons in the macaque primary motor cortex. Our results demonstrate that when clustered structure exists in datasets, and is consistent across trials or time points, a hierarchical alignment strategy that leverages such structure can provide significant improvements in cross-domain alignment.
Various probability divergences have been proposed in the literature, such as Euclidean least-squares (when data ordering is known) @cite_33 @cite_24 @cite_17 , Kullbeck-Liebler (KL) @cite_16 , maximum mean discrepancy (MMD) @cite_40 @cite_20 @cite_14 @cite_7 , and the Wasserstein distance @cite_45 , where trade-offs are often statistical (e.g., consistency, sample complexity) versus computational. Alignment problems are ill-posed since the space of @math is large, so structure is often necessary to constrain @math based on geometric assumptions. Compact manifolds like the Grassmann or Stiefel @cite_15 @cite_18 are primary choices when little information is present, as they preserve isometry. Non-isometric transformations, though richer, demand much more structure (e.g., manifold or graph structure) @cite_38 @cite_31 @cite_19 @cite_21 @cite_45 .
@cite_38 @cite_18 @cite_14 @cite_31 @cite_33 @cite_7 @cite_21 @cite_24 @cite_19 @cite_40 @cite_45 @cite_15 @cite_16 @cite_20 @cite_17
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The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). Let D and V denote respectively Information Divergence and Total Variation Distance. Pinsker's and Vajda's inequalities are respectively D ≥ [ 1 2] V2 and D ≥ log[( 2+V) ( 2-V)] - [( 2V) ( 2+V)]. In this paper, several generalizations and improvements of these inequalities are established for wide classes of f -divergences. First, conditions on f are determined under which an f-divergence Df will satisfy Df ≥ cf V2 or Df ≥ c2,f V2 + c4,f V4, where the constants cf, c2,f and c4,f are best possible. As a consequence, lower bounds in terms of V are obtained for many well known distance and divergence measures, including the χ2 and Hellinger's discrimination and the families of Tsallis' and Renyi's divergences. For instance, if D(α) (P||Q) = [α(α-1)]-1 [∫pαq1-αdμ-1] and ℑα (P||Q) = (α-1)-1 log[∫pαq1-αdμ] are respectively the relative information of type α and the Renyi's information gain of order α, it is shown that D(α) ≥ [ 1 2] V2 + [ 1 72] (α+1)(2-α) V4 whenever -1 ≤ α ≤ 2, α ≠ 0,1 and that ℑα ≥ [( α) 2] V2 + [ 1 36] α(1 + 5 α- 5 α2 ) V4 for 0 <; α <; 1. In a somewhat different direction, and motivated by the fact that these Pinsker's type lower bounds are accurate only for small variation (V close to zero), lower bounds for Df which are accurate for both small and large variation (V close to two) are also obtained. In the special case of the information divergence they imply that D ≥ log[ 2 ( 2-V)] - [( 2-V) 2] log[( 2+V) 2], which uniformly improves Vajda's inequality. This tutorial is concerned with applications of information theory concepts in statistics, in the finite alphabet setting. The information measure known as information divergence or Kullback-Leibler distance or relative entropy plays a key role, often with a geometric flavor as an analogue of squared Euclidean distance, as in the concepts of I-projection, I-radius and I-centroid. The topics covered include large deviations, hypothesis testing, maximum likelihood estimation in exponential families, analysis of contingency tables, and iterative algorithms with an "information geometry" background. Also, an introduction is provided to the theory of universal coding, and to statistical inference via the minimum description length principle motivated by that theory. A new coding problem is introduced for a correlated source (X sup n ,Y sup n ) sub n=1 sup spl infin . The observer of X sup n can transmit data depending on X sup n at a prescribed rate R. Based on these data the observer of Y sup n tries to identify whether for some distortion measure spl rho (like the Hamming distance) n sup -1 spl rho (X sup n ,Y sup n ) spl les d, a prescribed fidelity criterion. We investigate as functions of R and d the exponents of two error probabilities, the probabilities for misacceptance, and the probabilities for misrejection. In the case where X sup n and Y sup n are independent, we completely characterize the achievable region for the rate R and the exponents of two error probabilities; in the case where X sup n and Y sup n are correlated, we get some interesting partial results for the achievable region. During the process, we develop a new method for proving converses, which is called "the inherently typical subset lemma". This new method goes considerably beyond the "entropy characterization" the "image size characterization," and its extensions. It is conceivable that this new method has a strong impact on multiuser information theory.