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Matrix Completion with Queries

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Matrix Completion with Queries Matrix Completion with Queries Natali Ruchansky Mark Crovella Evimaria Terzi Boston University Boston University Boston University [email protected] [email protected] [email protected] ABSTRACT the matrix contains the volumes of traffic among source- In many applications, e.g., recommender systems and traffic destination pairs), and computer vision (image matrices). monitoring, the data comes in the form of a matrix that is In many cases, only a small percentage of the matrix entries only partially observed and low rank. A fundamental data- are observed. For example, the data used in the Netflix prize analysis task for these datasets is matrix completion, where competition was a matrix of 480K users × 18K movies, but the goal is to accurately infer the entries missing from the only 1% of the entries were known. matrix. Even when the data satisfies the low-rank assump- A common approach for recovering such missing data is tion, classical matrix-completion methods may output com- called matrix completion. The goal of matrix-completion pletions with significant error { in that the reconstructed methods is to accurately infer the values of missing entries, matrix differs significantly from the true underlying ma- subject to certain assumptions about the complete matrix [2, trix. Often, this is due to the fact that the information 3, 5, 7, 8, 9, 19]. For a true matrix T with observed val- contained in the observed entries is insufficient. In this work, ues only in a set of positions Ω, matrix-completion methods we address this problem by proposing an active version of exploit the information in the observed entries in T (de- matrix completion, where queries can be made to the true noted TΩ) in order to produce a \good" estimate Tb of T . In underlying matrix. Subsequently, we design Order&Extend, practice, the estimate may differ significantly from the true which is the first algorithm to unify a matrix-completion matrix. In particular, this can happen when the observed approach and a querying strategy into a single algorithm. entries TΩ are not adequate to provide sufficient information Order&Extend is able identify and alleviate insufficient in- to produce a good estimate. formation by judiciously querying a small number of addi- In many cases, it is possible to address the insufficiency tional entries. In an extensive experimental evaluation on of TΩ by actively obtaining additional observations. For real-world datasets, we demonstrate that our algorithm is example, in recommender systems, users may be asked to efficient and is able to accurately reconstruct the true ma- rate certain items; in traffic analysis, additional monitoring trix while asking only a small number of queries. points may be installed. These additional observations can lead to an augmented Ω0 such that T carries more infor- Ω0 mation about T and can lead to more accurate estimates Categories and Subject Descriptors Tb. In this active setting, the data analyst can become an H.2.8 [Database Management]: Database Applications| active participant in data collection by posing queries to T . data mining Of course such active involvement will only be acceptable if the number of queries is small. In this paper, we present a method for generating a small Keywords number of queries so as to ensure that the combination of matrix completion; recommender systems; active querying observed and queried values provides sufficient information for an accurate completion; i.e., the Tb estimated using the entries T is significantly better than the one estimated 1. INTRODUCTION Ω0 using TΩ. We call the problem of generating a small num- In many applications the data comes in the form of a ber of queries that guarantee small reconstruction error the low-rank matrix. Examples of such applications include rec- ActiveCompletion problem. ommender systems (where entries of the matrix indicated The difference between the classical matrix-completion user preferences over items), network traffic analysis (where problem and our problem is that in the former, the set of observed entries is fixed and the algorithm needs to find the Permission to make digital or hard copies of all or part of this work for personal or best completion given these entries. In ActiveCompletion, classroom use is granted without fee provided that copies are not made or distributed we are asked to design both a completion and a querying for profit or commercial advantage and that copies bear this notice and the full cita- strategy in order to minimize the reconstruction error. On tion on the first page. Copyrights for components of this work owned by others than the one hand, this task is more complex than standard ma- ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re- trix completion { since we have the additional job of design- publish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. ing a good querying strategy. On the other hand, having the KDD’15, August 10-13, 2015, Sydney, NSW, Australia. flexibility to ask some additional entries of T to be revealed c 2015 ACM. ISBN 978-1-4503-3664-2/15/08 ...$15.00. should yield lower reconstruction error. DOI: http://dx.doi.org/10.1145/2783258.2783259. At a high level, ActiveCompletion is related to other a querying strategy, nor a mechanism to signal when the recently proposed methods for active matrix completion, information is insufficient. e.g., [4]. However, existing approaches identify entries to be queried independently of the method of completion. In Random sampling: Cand´esand Recht introduced a thresh- contrast, a strength of our algorithm is that it addresses old on the number of entries needed for accurate matrix completion and querying in an integrated fashion. completion [2]. Under the assumption of randomly sampled The main contribution of our work is Order&Extend, an al- locations of known entries, they prove that an n1 × n2 ma- 6 gorithm that simultaneously minimizes the number of queries trix of rank r should have at least m > Cn 5 r log(n) for to ask and produces an estimate matrix Tb that is very close their algorithm to succeed with high probability, where n = to the true matrix T . The design of Order&Extend is in- max(n1; n2). Different authors in the matrix-completion lit- spired from recent matrix-completion methods that view the erature develop slightly different thresholds, but all are es- completion process as solving a sequence of (not necessarily sentially O(nr log(n)) [9, 15]. We point out that achieving linear) systems [10, 11, 13, 16]. We adopt this general view, this bound in the real world often requires a significantly focusing on a formulation that involves only linear systems. large number of samples. For example, adopting the rank Although existing work uses this insight for simple matrix r ≈ 40 of top solutions to the Netflix Challenge, over 151 completion, we go one step further and observe that there million entries would need to be queried. is a relationship between the ordering in which systems are solved, and the number of additional queries that need to be Structural matrix completion: Recently, a class of ma- posed. Therefore, the first step of Order&Extend focuses on trix completion has been proposed, which we call structural. finding a good ordering of the set of linear systems. Inter- Rather than taking an optimization approach, the meth- estingly, this ordering step relies on the combinatorial prop- ods of structural completion explicitly analyze the informa- erties of the mask graph, a graph that is associated with the tion content of the visible entries and are capable of stat- positions (but not the values) of observed entries Ω. ing definitively that the observed entries are information- In the second step, Order&Extend considers the linear sys- theoretically sufficient for reconstruction [10, 11, 13, 16]. tems in the chosen order, and asks queries every time it Structural methods are implicitly concerned with the num- encounters a problematic linear system Ax = b. A linear ber of possible completions that are consistent with the par- system can problematic in two ways: (a) when there are not tially observed matrix; this could be infinite, a finite, one, or enough equations for the number of unknowns, so that the none. A key observation shared by all structural approaches system does not have a unique solution; (b) when solving is that the number of possible completions does not depend the system Ax = b is numerically unstable given the specific on the values of the observed entries, but rather only on b involved. Note that, as we explain in the paper, this is not their positions. This statement, proved by Kiraly et al. [10], the same as simply saying that A is ill-conditioned; part of means that in our search for good ordering of linear systems our contribution is the design of fast methods for detecting we can work solely with the locations of known entries. and ameliorating such systems. The common characteristic between our method and struc- Our extensive experiments with datasets from a variety tural methods is that they also view matrix completion as a of application domains demonstrate that Order&Extend re- task of solving a sequence of (not necessarily linear) systems quires significantly fewer queries than any other baseline of equations where the result of one is used as input to an- querying strategy, and compares very favorably to approaches other. In fact, Meka et al. [13] adopt the same view as ours. based on well-known matrix completion algorithms. In fact, However, the key difference between these works and ours our experiments indicate that Order&Extend is \almost op- is that we are concerned particularly with the active ver- timal" as it gives solutions where the number of entries it sion of the problem and we need to effectively design both queries is generally very close to the information-theoretic a reconstruction and a querying strategy simultaneously.
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