25,213 research outputs found
PU Learning for Matrix Completion
In this paper, we consider the matrix completion problem when the
observations are one-bit measurements of some underlying matrix M, and in
particular the observed samples consist only of ones and no zeros. This problem
is motivated by modern applications such as recommender systems and social
networks where only "likes" or "friendships" are observed. The problem of
learning from only positive and unlabeled examples, called PU
(positive-unlabeled) learning, has been studied in the context of binary
classification. We consider the PU matrix completion problem, where an
underlying real-valued matrix M is first quantized to generate one-bit
observations and then a subset of positive entries is revealed. Under the
assumption that M has bounded nuclear norm, we provide recovery guarantees for
two different observation models: 1) M parameterizes a distribution that
generates a binary matrix, 2) M is thresholded to obtain a binary matrix. For
the first case, we propose a "shifted matrix completion" method that recovers M
using only a subset of indices corresponding to ones, while for the second
case, we propose a "biased matrix completion" method that recovers the
(thresholded) binary matrix. Both methods yield strong error bounds --- if M is
n by n, the Frobenius error is bounded as O(1/((1-rho)n), where 1-rho denotes
the fraction of ones observed. This implies a sample complexity of O(n\log n)
ones to achieve a small error, when M is dense and n is large. We extend our
methods and guarantees to the inductive matrix completion problem, where rows
and columns of M have associated features. We provide efficient and scalable
optimization procedures for both the methods and demonstrate the effectiveness
of the proposed methods for link prediction (on real-world networks consisting
of over 2 million nodes and 90 million links) and semi-supervised clustering
tasks
Adaptive Matrix Completion for the Users and the Items in Tail
Recommender systems are widely used to recommend the most appealing items to
users. These recommendations can be generated by applying collaborative
filtering methods. The low-rank matrix completion method is the
state-of-the-art collaborative filtering method. In this work, we show that the
skewed distribution of ratings in the user-item rating matrix of real-world
datasets affects the accuracy of matrix-completion-based approaches. Also, we
show that the number of ratings that an item or a user has positively
correlates with the ability of low-rank matrix-completion-based approaches to
predict the ratings for the item or the user accurately. Furthermore, we use
these insights to develop four matrix completion-based approaches, i.e.,
Frequency Adaptive Rating Prediction (FARP), Truncated Matrix Factorization
(TMF), Truncated Matrix Factorization with Dropout (TMF + Dropout) and Inverse
Frequency Weighted Matrix Factorization (IFWMF), that outperforms traditional
matrix-completion-based approaches for the users and the items with few ratings
in the user-item rating matrix.Comment: 7 pages, 3 figures, ACM WWW'1
Guarantees of Riemannian Optimization for Low Rank Matrix Completion
We study the Riemannian optimization methods on the embedded manifold of low
rank matrices for the problem of matrix completion, which is about recovering a
low rank matrix from its partial entries. Assume entries of an
rank matrix are sampled independently and uniformly with replacement. We
first prove that with high probability the Riemannian gradient descent and
conjugate gradient descent algorithms initialized by one step hard thresholding
are guaranteed to converge linearly to the measured matrix provided
\begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} where
is a numerical constant depending on the condition number of the
underlying matrix. The sampling complexity has been further improved to
\begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} via the resampled
Riemannian gradient descent initialization. The analysis of the new
initialization procedure relies on an asymmetric restricted isometry property
of the sampling operator and the curvature of the low rank matrix manifold.
Numerical simulation shows that the algorithms are able to recover a low rank
matrix from nearly the minimum number of measurements
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