58,428 research outputs found
A Non-monotone Alternating Updating Method for A Class of Matrix Factorization Problems
In this paper we consider a general matrix factorization model which covers a
large class of existing models with many applications in areas such as machine
learning and imaging sciences. To solve this possibly nonconvex, nonsmooth and
non-Lipschitz problem, we develop a non-monotone alternating updating method
based on a potential function. Our method essentially updates two blocks of
variables in turn by inexactly minimizing this potential function, and updates
another auxiliary block of variables using an explicit formula. The special
structure of our potential function allows us to take advantage of efficient
computational strategies for non-negative matrix factorization to perform the
alternating minimization over the two blocks of variables. A suitable line
search criterion is also incorporated to improve the numerical performance.
Under some mild conditions, we show that the line search criterion is well
defined, and establish that the sequence generated is bounded and any cluster
point of the sequence is a stationary point. Finally, we conduct some numerical
experiments using real datasets to compare our method with some existing
efficient methods for non-negative matrix factorization and matrix completion.
The numerical results show that our method can outperform these methods for
these specific applications
Towards Question-based Recommender Systems
Conversational and question-based recommender systems have gained increasing
attention in recent years, with users enabled to converse with the system and
better control recommendations. Nevertheless, research in the field is still
limited, compared to traditional recommender systems. In this work, we propose
a novel Question-based recommendation method, Qrec, to assist users to find
items interactively, by answering automatically constructed and algorithmically
chosen questions. Previous conversational recommender systems ask users to
express their preferences over items or item facets. Our model, instead, asks
users to express their preferences over descriptive item features. The model is
first trained offline by a novel matrix factorization algorithm, and then
iteratively updates the user and item latent factors online by a closed-form
solution based on the user answers. Meanwhile, our model infers the underlying
user belief and preferences over items to learn an optimal question-asking
strategy by using Generalized Binary Search, so as to ask a sequence of
questions to the user. Our experimental results demonstrate that our proposed
matrix factorization model outperforms the traditional Probabilistic Matrix
Factorization model. Further, our proposed Qrec model can greatly improve the
performance of state-of-the-art baselines, and it is also effective in the case
of cold-start user and item recommendations.Comment: accepted by SIGIR 202
Ward identities and combinatorics of rainbow tensor models
We discuss the notion of renormalization group (RG) completion of
non-Gaussian Lagrangians and its treatment within the framework of
Bogoliubov-Zimmermann theory in application to the matrix and tensor models.
With the example of the simplest non-trivial RGB tensor theory (Aristotelian
rainbow), we introduce a few methods, which allow one to connect calculations
in the tensor models to those in the matrix models. As a byproduct, we obtain
some new factorization formulas and sum rules for the Gaussian correlators in
the Hermitian and complex matrix theories, square and rectangular. These sum
rules describe correlators as solutions to finite linear systems, which are
much simpler than the bilinear Hirota equations and the infinite Virasoro
recursion. Search for such relations can be a way to solving the tensor models,
where an explicit integrability is still obscure.Comment: 48 page
Two Algorithms for Orthogonal Nonnegative Matrix Factorization with Application to Clustering
Approximate matrix factorization techniques with both nonnegativity and
orthogonality constraints, referred to as orthogonal nonnegative matrix
factorization (ONMF), have been recently introduced and shown to work
remarkably well for clustering tasks such as document classification. In this
paper, we introduce two new methods to solve ONMF. First, we show athematical
equivalence between ONMF and a weighted variant of spherical k-means, from
which we derive our first method, a simple EM-like algorithm. This also allows
us to determine when ONMF should be preferred to k-means and spherical k-means.
Our second method is based on an augmented Lagrangian approach. Standard ONMF
algorithms typically enforce nonnegativity for their iterates while trying to
achieve orthogonality at the limit (e.g., using a proper penalization term or a
suitably chosen search direction). Our method works the opposite way:
orthogonality is strictly imposed at each step while nonnegativity is
asymptotically obtained, using a quadratic penalty. Finally, we show that the
two proposed approaches compare favorably with standard ONMF algorithms on
synthetic, text and image data sets.Comment: 17 pages, 8 figures. New numerical experiments (document and
synthetic data sets
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