Parikshit Ram

Maximum inner-product search using cone trees

The problem of efficiently finding the best match for a query in a given set with respect to the Euclidean distance or the cosine similarity has been extensively studied. However, the closely related problem of efficiently finding the best match with respect to the inner-product has never been explored in the general setting to the best of our knowledge. In this paper we consider this problem and contrast it with the previous problems considered. First, we propose a general branch-and-bound algorithm based on a (single) tree data structure. Subsequently, we present a dual-tree algorithm for the case where there are multiple queries. Our proposed branch-and-bound algorithms are based on novel inner-product bounds. Finally we present a new data structure, the cone tree, for increasing the efficiency of the dual-tree algorithm. We evaluate our proposed algorithms on a variety of data sets from various applications, and exhibit up to five orders of magnitude improvement in query time over the naive search technique in some cases.

Efficient retrieval of recommendations in a matrix factorization framework

Low-rank Matrix Factorization (MF) methods provide one of the simplest and most effective approaches to collaborative filtering. This paper is the first to investigate the problem of efficient retrieval of recommendations in a MF framework. We reduce the retrieval in a MF model to an apparently simple task of finding the maximum dot-product for the user vector over the set of item vectors. However, to the best of our knowledge the problem of efficiently finding the maximum dot-product in the general case has never been studied. To this end, we propose two techniques for efficient search -- (i) We index the item vectors in a binary spatial-partitioning metric tree and use a simple branch and-bound algorithm with a novel bounding scheme to efficiently obtain exact solutions. (ii) We use spherical clustering to index the users on the basis of their preferences and pre-compute recommendations only for the representative user of each cluster to obtain extremely efficient approximate solutions. We obtain a theoretical error bound which determines the quality of any approximate result and use it to control the approximation. Both these simple techniques are fairly independent of each other and hence are easily combined to further improve recommendation retrieval efficiency. We evaluate our algorithms on real-world collaborative-filtering datasets, demonstrating more than ×7 speedup (with respect to the naive linear search) for the exact solution and over ×250 speedup for approximate solutions by combining both techniques.

Fast euclidean minimum spanning tree: algorithm, analysis, and applications

The Euclidean Minimum Spanning Tree problem has applications in a wide range of fields, and many efficient algorithms have been developed to solve it. We present a new, fast, general EMST algorithm, motivated by the clustering and analysis of astronomical data. Large-scale astronomical surveys, including the Sloan Digital Sky Survey, and large simulations of the early universe, such as the Millennium Simulation, can contain millions of points and fill terabytes of storage. Traditional EMST methods scale quadratically, and more advanced methods lack rigorous runtime guarantees. We present a new dual-tree algorithm for efficiently computing the EMST, use adaptive algorithm analysis to prove the tightest (and possibly optimal) runtime bound for the EMST problem to-date, and demonstrate the scalability of our method on astronomical data sets.

Density estimation trees

In this paper we develop density estimation trees (DETs), the natural analog of classification trees and regression trees, for the task of density estimation. We consider the estimation of a joint probability density function of a d-dimensional random vector X and define a piecewise constant estimator structured as a decision tree. The integrated squared error is minimized to learn the tree. We show that the method is nonparametric: under standard conditions of nonparametric density estimation, DETs are shown to be asymptotically consistent. In addition, being decision trees, DETs perform automatic feature selection. They empirically exhibit the interpretability, adaptability and feature selection properties of supervised decision trees while incurring slight loss in accuracy over other nonparametric density estimators. Hence they might be able to avoid the curse of dimensionality if the true density is sparse in dimensions. We believe that density estimation trees provide a new tool for exploratory data analysis with unique capabilities.

Dual-tree fast exact max-kernel search

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MLPACK: a scalable C++ machine learning library

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Linear-time Algorithms for Pairwise Statistical Problems

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Fast Exact Max-Kernel Search.

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Which Space Partitioning Tree to Use for Search

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Tree-Independent Dual-Tree Algorithms

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