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Register a new userLocality-Sensitive Hashing for f-Divergences: Mutual Information Loss and Beyond
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Lin Chen
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Computing approximate nearest neighbors in high dimensional spaces is a central problem in large-scale data mining with a wide range of applications in machine learning and data science. A popular and effective technique in computing nearest neighbors approximately is the locality-sensitive hashing (LSH) scheme. In this paper, we aim to develop LSH schemes for distance functions that measure the distance between two probability distributions, particularly for f-divergences as well as a generalization to capture mutual information loss. First, we provide a general framework to design LHS schemes for f-divergence distance functions and develop LSH schemes for the generalized Jensen-Shannon divergence and triangular discrimination in this framework. We show a two-sided approximation result for approximation of the generalized Jensen-Shannon divergence by the Hellinger distance, which may be of independent interest. Next, we show a general method of reducing the problem of designing an LSH scheme for a Krein kernel (which can be expressed as the difference of two positive definite kernels) to the problem of maximum inner product search. We exemplify this method by applying it to the mutual information loss, due to its several important applications such as model compression.
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Extended differential privacy, a generalization of standard differential privacy (DP) using a general metric, has been widely studied to provide rigorous privacy guarantees while keeping high utility. However, existing works on extended DP are limited to few metrics, such as the Euclidean metric. Consequently, they have only a small number of applications, such as location-based services and document processing. In this paper, we propose a couple of mechanisms providing extended DP with a different metric: angular distance (or cosine distance). Our mechanisms are based on locality sensitive hashing (LSH), which can be applied to the angular distance and work well for personal data in a high-dimensional space. We theoretically analyze the privacy properties of our mechanisms, and prove extended DP for input data by taking into account that LSH preserves the original metric only approximately. We apply our mechanisms to friend matching based on high-dimensional personal data with angular distance in the local model, and evaluate our mechanisms using two real datasets. We show that LDP requires a very large privacy budget and that RAPPOR does not work in this application. Then we show that our mechanisms enable friend matching with high utility and rigorous privacy guarantees based on extended DP.
Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval. Arkin et al. represent polygons by their turning function - a function which follows the angle between the polygons tangent and the $ x $-axis while traversing the perimeter of the polygon. They define the distance between polygons to be variations of the $ L_p $ (for $p=1,2$) distance between their turning functions. This metric is invariant under translation, rotation and scaling (and the selection of the initial point on the perimeter) and therefore models well the intuitive notion of shape resemblance. We develop and analyze LSH near neighbor data structures for several variations of the $ L_p $ distance for functions (for $p=1,2$). By applying our schemes to the turning functions of a collection of polygons we obtain efficient near neighbor LSH-based structures for polygons. To tune our structures to turning functions of polygons, we prove some new properties of these turning functions that may be of independent interest. As part of our analysis, we address the following problem which is of independent interest. Find the vertical translation of a function $ f $ that is closest in $ L_1 $ distance to a function $ g $. We prove tight bounds on the approximation guarantee obtained by the translation which is equal to the difference between the averages of $ g $ and $ f $.
Metagenomic binning is an essential task in analyzing metagenomic sequence datasets. To analyze structure or function of microbial communities from environmental samples, metagenomic sequence fragments are assigned to their taxonomic origins. Although sequence alignment algorithms can readily be used and usually provide high-resolution alignments and accurate binning results, the computational cost of such alignment-based methods becomes prohibitive as metagenomic datasets continue to grow. Alternative compositional-based methods, which exploit sequence composition by profiling local short k-mers in fragments, are often faster but less accurate than alignment-based methods. Inspired by the success of linear error correcting codes in noisy channel communication, we introduce Opal, a fast and accurate novel compositional-based binning method. It incorporates ideas from Gallagers low-density parity-check code to design a family of compact and discriminative locality-sensitive hashing functions that encode long-range compositional dependencies in long fragments. By incorporating the Gallager LSH functions as features in a simple linear SVM, Opal provides fast, accurate and robust binning for datasets consisting of a large number of species, even with mutations and sequencing errors. Opal not only performs up to two orders of magnitude faster than BWA, an alignment-based binning method, but also achieves improved binning accuracy and robustness to sequencing errors. Opal also outperforms models built on traditional k-mer profiles in terms of robustness and accuracy. Finally, we demonstrate that we can effectively use Opal in the coarse search stage of a compressive genomics pipeline to identify a much smaller candidate set of taxonomic origins for a subsequent alignment-based method to analyze, thus providing metagenomic binning with high scalability, high accuracy and high resolution.
We present the first provable Least-Squares Value Iteration (LSVI) algorithms that have runtime complexity sublinear in the number of actions. We formulate the value function estimation procedure in value iteration as an approximate maximum inner product search problem and propose a locality sensitive hashing (LSH) [Indyk and Motwani STOC98, Andoni and Razenshteyn STOC15, Andoni, Laarhoven, Razenshteyn and Waingarten SODA17] type data structure to solve this problem with sublinear time complexity. Moreover, we build the connections between the theory of approximate maximum inner product search and the regret analysis of reinforcement learning. We prove that, with our choice of approximation factor, our Sublinear LSVI algorithms maintain the same regret as the original LSVI algorithms while reducing the runtime complexity to sublinear in the number of actions. To the best of our knowledge, this is the first work that combines LSH with reinforcement learning resulting in provable improvements. We hope that our novel way of combining data-structures and iterative algorithm will open the door for further study into cost reduction in optimization.
Existing unsupervised document hashing methods are mostly established on generative models. Due to the difficulties of capturing long dependency structures, these methods rarely model the raw documents directly, but instead to model the features extracted from them (e.g. bag-of-words (BOW), TFIDF). In this paper, we propose to learn hash codes from BERT embeddings after observing their tremendous successes on downstream tasks. As a first try, we modify existing generative hashing models to accommodate the BERT embeddings. However, little improvement is observed over the codes learned from the old BOW or TFIDF features. We attribute this to the reconstruction requirement in the generative hashing, which will enforce irrelevant information that is abundant in the BERT embeddings also compressed into the codes. To remedy this issue, a new unsupervised hashing paradigm is further proposed based on the mutual information (MI) maximization principle. Specifically, the method first constructs appropriate global and local codes from the documents and then seeks to maximize their mutual information. Experimental results on three benchmark datasets demonstrate that the proposed method is able to generate hash codes that outperform existing ones learned from BOW features by a substantial margin.
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