Lemp

Abstract

We study the problem of efficiently retrieving large entries in the product of two given matrices, which arises in a number of data mining and information retrieval tasks. We focus on the setting where the two input matrices are tall and skinny, i.e., with millions of rows and tens to hundreds of columns. In such settings, the product matrix is large and its complete computation is generally infeasible in practice. To address this problem, we propose the LEMP algorithm, which efficiently retrieves only the large entries in the product matrix without actually computing it. LEMP maps the large-entry retrieval problem to a set of smaller cosine similarity search problems, for which existing methods can be used. We also propose novel algorithms for cosine similarity search, which are tailored to our setting. Our experimental study on large real-world datasets indicates that LEMP is up to an order of magnitude faster than state-of-the-art approaches.

Estimating the Largest Elements of a Matrix

“…For example, the use of factor models in recommender systems leads to matrix products B T C with B, C ∈ R m×n , m n, and n very large [29]. Another application is link prediction in graphs [26].…”

“…An important feature of our algorithms is that they can be treated as black boxes that can be applied to many different problems, in contrast to the more specialized algorithms designed for products of two matrices, such as those in [3], [29]. Since our algorithms only require the computation of matrix-vector products, they are relatively simple to implement and can serve as a benchmark for testing more specialized algorithms in multiple application areas.…”

Simpler is Much Faster: Fair and Independent Inner Product Search

“…The inner product search (IPS) problem is important in many fields, e.g., information retrieval [18,32,35], recommender systems [6,7,30], data mining [19,27], databases [22,24,29], artificial intelligence [13,36], and machine learning [14,23,34]. These fields usually consider the 𝑘 maximum inner product search (𝑘-MIPS) problem, which, given a query vector and an output size 𝑘, returns the 𝑘 vectors having the maximum inner product with the query.…”

“…○ We iterate the above operation until we have |Q 𝑘 | = 𝑘 2. State-of-the-art exact IPS algorithms are based on linear scan[22,29]. Trivially, approximation algorithms which do not guarantee that all vectors such that x • q ≥ 𝜏 are included in the search result cannot solve the FI-IPS problem correctly.…”

Approximate Top-k Inner Product Join with a Proximity Graph

“…This paper considers the top-k inner product join problem, which is defined as follows: Given two sets X and Y of high-dimensional vectors and a result size k, top-k inner product join between X and Y retrieves k pairs of vectors x, y , where x ∈ X and y ∈ Y, with the largest inner product among X × Y. This problem has important applications, such as recommendation [1]- [3], information extraction [4], and finding outlier correlations [5]. More specifically, Figure 1 depicts a histogram of inner products of 1 million randomly sampled vector pairs in a Yahoo!…”