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Provable Inductive Matrix Completion

Prateek Jain, Inderjit S. Dhillon

arXiv:1306.0626v1cs.LGcs.ITstat.ML

TL;DR

The paper addresses exact recovery and inductive prediction when low-rank matrices are observed through sparse ratings or rank-one measurements. It analyzes initialized alternating minimization under operator conditions and applies the framework to matrix sensing, inductive matrix completion, and missing-label regression. The resulting guarantees include efficient recovery and improved sample or storage requirements in the stated regimes, while rank-one sensing must be resampled for each target matrix.

  • Problem

    Standard matrix completion cannot predict for new users or movies, while missing-label regression and low-rank sensing require exact recovery from incomplete or rank-one observations.

  • Method

    The paper generalizes these tasks to low-rank matrix estimation with rank-one measurements and proves global convergence of initialized alternating minimization when three operator properties hold.

  • Results

    The framework gives exact-recovery guarantees across matrix sensing, inductive matrix completion, and missing-label regression, including ˜O(d1·L) samples for the latter versus ˜O(n1 + L) for completing labels first.

  • Takeaways & Limitations

    Rank-one measurements can reduce storage and computational requirements, and feature-aware recovery improves sample complexity when feature dimensions are much smaller than users, movies, or training points.

  • Takeaways & Limitations

    Rank-one Gaussian measurement operators are not universal: they must be resampled for each target matrix, unlike RIP-based operators.

Abstract

from arXiv · show

Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matrix completion in the exact recovery setting. That is, we assume that the ratings matrix is generated by applying feature vectors to a low-rank matrix and the goal is to recover back the underlying matrix. Furthermore, we generalize the problem to that of low-rank matrix estimation using rank-1 measurements. We study this generic problem and provide conditions that the set of measurements should satisfy so that the alternating minimization method (which otherwise is a non-convex method with no convergence guarantees) is able to recover back the {\em exact} underlying low-rank matrix. In addition to inductive matrix completion, we show that two other low-rank estimation problems can be studied in our framework: a) general low-rank matrix sensing using rank-1 measurements, and b) multi-label regression with missing labels. For both the problems, we provide novel and interesting bounds on the number of measurements required by alternating minimization to provably converges to the {\em exact} low-rank matrix. In particular, our analysis for the general low rank matrix sensing problem significantly improves the required storage and computational cost than that required by the RIP-based matrix sensing methods \cite{RechtFP2007}. Finally, we provide empirical validation of our approach and demonstrate that alternating minimization is able to recover the true matrix for the above mentioned problems using a small number of measurements.

1 Introduction

The paper develops a rank-one measurement framework for exact low-rank recovery with alternating minimization, targeting inductive prediction from user and movie features. It applies the framework to matrix sensing, inductive matrix completion, and multi-label regression with missing labels, with guarantees that improve efficiency or sample complexity in relevant regimes.

  • Inductive Matrix Completion: Standard matrix completion is transductive, whereas inductive matrix completion uses user and movie features to predict ratings for new users and movies.The rating model uses feature vectors and a low-rank parameter matrix, with standard matrix completion recovered when features are identity vectors.
  • General Framework: The paper formulates low-rank matrix estimation using rank-one measurements and studies alternating minimization with appropriate initialization.Measurements use feature vectors supplied with the observations, and the recovered matrix is factorized as W = UV^T.
  • General Framework: Alternating minimization converges to the global optimum at a linear rate when the rank-one measurement operator satisfies three key properties.The analysis addresses the non-convex rank constraint while retaining least-squares subproblems rather than full-SVD convex optimization.
  • Inductive Matrix Completion: For incoherent features, inductive matrix completion requires m ≥ C·k^3β^2(d1·d2) log(d1 + d2) observations, improving over standard completion when d1·d2 ≪ n1 + n2.The bound is quadratic in the feature dimensions and yields global optimality through the generic analysis.
  • Multi-label Regression: For missing-label regression, ˜O(d1·L) observed entries suffice, versus ˜O(n1 + L) for completing the label matrix first.The unified approach exactly recovers W when entries are sampled uniformly, with an advantage when n1 is much larger than L.

2 Low-rank Matrix Estimation using Rank-one Measurements

The paper defines a general low-rank estimation problem from rank-one linear measurements and analyzes alternating minimization through operator properties that control its updates. Under these properties, the iterates contract toward the target and reach a specified accuracy after logarithmically many iterations.

  • Problem Formulation: LRROM recovers a rank-k matrix W* from linear measurements generated by rank-one operators Ai = xiy_i^T.The vectors xi and yi are feature vectors provided with the measurements.
  • Problem Formulation: The recovery problem is non-convex because W is factorized as W = UV^T and constrained to rank at most k.Alternating minimization solves least-squares updates over the factors U and V.
  • Convergence Analysis: Three operator properties control conditioning and perturbation terms so that alternating minimization can be shown to converge globally rather than only to a local minimum.The proof bounds near-isometry and cross-term effects for vectors independent of the measurement randomness.
  • Convergence Analysis: After H = O(log(σ*/ϵ)) iterations, the recovered factor product WH approaches the target at the prescribed accuracy.The rank-one proof tracks the correct and orthogonal components of successive updates before extending to general rank k.
  • Convergence Analysis: Measurement partitioning across iterations ensures independence of intermediate vectors from the randomness used in each update.This increases the total measurement complexity by a logarithmic factor in the target norm and accuracy.

3 Rank-one Matrix Sensing using Gaussian Measurements

The paper studies Gaussian rank-one measurements for recovering general low-rank matrices and proves that alternating minimization can achieve exact recovery without RIP. This approach uses substantially less storage and computation than dense RIP-based operators, but sacrifices universality.

  • The sensing problem designs measurements and a recovery algorithm to reconstruct a true low-rank matrix from linear measurements.
  • m = Ω(k^4β^2(d1+d2)log^2(d1+d2)) measurements suffice for AltMin-LRROM to exactly recover any fixed rank-k matrix with high probability.Here β is the condition number parameter of W∗, and the recovery guarantee is for Gaussian rank-one measurements.
  • Rank-one operators require O(m(d1+d2)) storage, compared with O(md1d2) for typical RIP operators.The lower storage requirement also reduces the associated signal-acquisition and recovery costs.
  • Rank-one Gaussian measurements fail RIP for rank-1 matrices with high probability when the number of measurements is only near-linear in d1+d2.The cited analysis states that RIP requires m = Ω(d1d2 / log(5d1d2)) in this setting.
  • The analysis distills three measurement-operator properties that imply global convergence of alternating minimization at a linear rate.For Gaussian rank-one measurements, these properties hold under the stated measurement bound.
  • Rank-one Gaussian measurements must be resampled for each target matrix, unlike universal RIP operators that can recover any rank-k matrix.The paper frames this as a trade-off between randomness and universality.

4 Inductive Matrix Completion

The paper formulates inductive matrix completion using user and movie features, enabling recovery beyond the observed users and movies. Under incoherence, alternating minimization achieves exact recovery with feature-dependent sample bounds.

  • Inductive matrix completion uses side information about users and movies rather than relying only on observed ratings.Standard matrix completion ignores such demographic or genre features and is restricted to the existing users and movies.
  • Standard matrix completion requires |Ω| ≥ Ck(n1+n2)log(n1+n2) samples, even though the feature-based matrix has O(k(d1+d2)) degrees of freedom.This gap arises when d1+d2 is much smaller than n1+n2.
  • O(kd1d2 log(d1d2)) random samples let AltMin-LRROM recover the true matrix and support inductive prediction when d1d2 ≪ n1+n2.The paper notes that several users or movies need not have any observed rating, while the bound remains above the information-theoretic target O(k(d1+d2)log(d1+d2)).
  • Under the same conditions, AltMin-LRROM converges to the global optimum in O(log(∥W∗∥F/ϵ)) iterations.The proof verifies the three properties required by the generic alternating-minimization theorem.
  • Assuming incoherent feature matrices, m ≥ Ck^3β^2d1d2log(d1+d2) supports the required measurement-operator properties with high probability.The corresponding operator is built from uniformly sampled observed user–movie pairs and their feature vectors.
  • The analysis normalizes the feature matrices and relates the condition number of the ratings matrix to that of the transformed low-rank parameter.This permits the proof to assume orthonormal feature matrices while preserving the relevant condition number.

5 Multi-label Learning

The paper studies multi-label regression with missing target entries as a rank-one low-rank estimation problem. Under incoherence and sampling conditions, alternating minimization can recover the parameter matrix exactly, with sample complexity improving over separate matrix completion and regression in some regimes.

  • Problem: Multi-label regression learns a low-rank parameter matrix W∗ from feature vectors and a target matrix R whose entries may be missing.The observations are rank-one measurements combining feature vectors with canonical label vectors.
  • Problem: The problem is inductive over data points but treats the labels as fixed, combining inductive and standard matrix-completion measurements.The left measurement vector uses the data-point features, while the right vector is a canonical basis vector for a label.
  • Theory: Under an incoherence assumption on X, the measurement operators satisfy Properties 1, 2, and 3 required by the alternating-minimization theorem.The stated lemma uses a uniformly random observation set and a sample-size condition involving β, d1, n2, and logarithmic factors.
  • Sample complexity: m = d1 · L samples suffice to recover W∗ exactly when β and k are constant and logarithmic factors are ignored.This improves over m = n1 + L samples required by matrix completion when the number of training points is significantly larger than d1 · L.

6 Experiments

Experiments compare rank-one and RIP-based matrix sensing operators and evaluate prediction error for multi-label regression. The rank-one operator is faster with similar recovery error, while prediction error remains fairly accurate for small rank and dimension and degrades gracefully as they increase.

  • Low-rank matrix sensing: The rank-one matrix-sensing operator is significantly faster than the RIP-based method while incurring similar recovery error.The comparison uses recovery error and running time as the number of measurements increases.
  • Multi-label regression: With L = 50, n1 = 100, and 200 observed entries, the method produces fairly accurate test predictions for small k and d.The experiment varies the rank k and feature dimension d and evaluates error on held-out test points.
  • Multi-label regression: Test error degrades gracefully as either the rank k or feature dimension d increases.The reported prediction-error trends are shown while varying k and d separately.

A Preliminaries

The preliminaries introduce the concentration result used to control random matrix quantities in the low-rank estimation analysis.

  • Concentration tool: The analysis uses a concentration theorem for independent, mean-zero random matrices with an almost-sure spectral-norm bound.The theorem defines a variance parameter from the associated matrix expectations.

B Proof of General Theorem for Low-rank Matrix Estimation

The proof generalizes alternating-minimization analysis from rank one to rank k. It tracks subspace distance, derives least-squares updates, and uses the three measurement properties to establish contraction from a suitable initialization.

  • Rank-k distance: The rank-k proof replaces a rank-one error measure with a principal-angle distance between k-dimensional subspaces.This distance compares the evolving and target singular-vector subspaces.
  • Alternating update: At each iteration, the updated factor is obtained by solving a least-squares objective and then applying QR decomposition.The proof writes the update as an error matrix around the target factor.
  • Convergence argument: The initialization guarantee bounds the initial subspace distance, after which singular-vector perturbation and the auxiliary lemmas yield the theorem.The argument uses the initialization property together with bounds on the relevant update operators.
  • Measurement conditions: The auxiliary lemmas control singular values and error terms using Properties 1, 2, and 3 of the rank-one measurement operator.These properties provide the conditioning and perturbation bounds needed for the rank-k update analysis.
  • Proof organization: The proof introduces block matrices B, C, D, and S to express the factor-update equations and bound their perturbation terms.These matrices organize the rank-k normal equations used in the contraction analysis.
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