46,721 research outputs found
Is Simple Better? Revisiting Non-linear Matrix Factorization for Learning Incomplete Ratings
Matrix factorization techniques have been widely used as a method for
collaborative filtering for recommender systems. In recent times, different
variants of deep learning algorithms have been explored in this setting to
improve the task of making a personalized recommendation with user-item
interaction data. The idea that the mapping between the latent user or item
factors and the original features is highly nonlinear suggest that classical
matrix factorization techniques are no longer sufficient. In this paper, we
propose a multilayer nonlinear semi-nonnegative matrix factorization method,
with the motivation that user-item interactions can be modeled more accurately
using a linear combination of non-linear item features. Firstly, we learn
latent factors for representations of users and items from the designed
multilayer nonlinear Semi-NMF approach using explicit ratings. Secondly, the
architecture built is compared with deep-learning algorithms like Restricted
Boltzmann Machine and state-of-the-art Deep Matrix factorization techniques. By
using both supervised rate prediction task and unsupervised clustering in
latent item space, we demonstrate that our proposed approach achieves better
generalization ability in prediction as well as comparable representation
ability as deep matrix factorization in the clustering task.Comment: version
Neural Graph Collaborative Filtering
Learning vector representations (aka. embeddings) of users and items lies at
the core of modern recommender systems. Ranging from early matrix factorization
to recently emerged deep learning based methods, existing efforts typically
obtain a user's (or an item's) embedding by mapping from pre-existing features
that describe the user (or the item), such as ID and attributes. We argue that
an inherent drawback of such methods is that, the collaborative signal, which
is latent in user-item interactions, is not encoded in the embedding process.
As such, the resultant embeddings may not be sufficient to capture the
collaborative filtering effect.
In this work, we propose to integrate the user-item interactions -- more
specifically the bipartite graph structure -- into the embedding process. We
develop a new recommendation framework Neural Graph Collaborative Filtering
(NGCF), which exploits the user-item graph structure by propagating embeddings
on it. This leads to the expressive modeling of high-order connectivity in
user-item graph, effectively injecting the collaborative signal into the
embedding process in an explicit manner. We conduct extensive experiments on
three public benchmarks, demonstrating significant improvements over several
state-of-the-art models like HOP-Rec and Collaborative Memory Network. Further
analysis verifies the importance of embedding propagation for learning better
user and item representations, justifying the rationality and effectiveness of
NGCF. Codes are available at
https://github.com/xiangwang1223/neural_graph_collaborative_filtering.Comment: SIGIR 2019; the latest version of NGCF paper, which is distinct from
the version published in ACM Digital Librar
The Maximal Positively Invariant Set: Polynomial Setting
This note considers the maximal positively invariant set for polynomial
discrete time dynamics subject to constraints specified by a basic
semialgebraic set. The note utilizes a relatively direct, but apparently
overlooked, fact stating that the related preimage map preserves basic
semialgebraic structure. In fact, this property propagates to underlying
set--dynamics induced by the associated restricted preimage map in general and
to its maximal trajectory in particular. The finite time convergence of the
corresponding maximal trajectory to the maximal positively invariant set is
verified under reasonably mild conditions. The analysis is complemented with a
discussion of computational aspects and a prototype implementation based on
existing toolboxes for polynomial optimization
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