Maria Schmidt, M.Sc. Thesis Seminar
Advisor: Prof. A. Bronstein
Different methods where presented through the years to find good solutions for the Matrix Completion Problem. This problem appears in many tasks in life where a sparse signal, which lies on a grid of two non- Euclidean domains (Graphs or manifolds), should be predicted or completed. A classic example of this problem is the “Netflix Problem” which appears in the field of “Recommendation Systems” or “recommender systems”. In those systems recommendations of specific items are given to users (like friends in Facebook, products on Amazon, links in Google, movies in YouTube, songs on Spotify, twits on Twitter etc.).
In this work, we present a novel Learning Approach towards Geometric Matrix Completion on Non-Euclidean domains. Our approach towards the matrix completion problem suggests that when we are looking at the problem from the geometric point of view, neural networks can use a very strong prior for that problem. Hence, non-trained neural networks and learning methods that can be good for a single matrix completion tasks in the Euclidean domains (like image completion tasks), after re-definition, can be used also for the Matrix Completion problem on Non-Euclidean domains.
This re-definition is possible using the building blocks and operators coming from the Non-Euclidean geometry suitable for Non-Euclidean domains like graphs and manifolds and redefinition of the learning network layers (like convolution and pooling) accordingly. Following that approach, we present a novel, fast, intuitive learning method for the “Matrix Completion Problem”: the Matrix Data Deep Decoder - the “MDDD”, which is parallel to the newest state of art method for Euclidean domains like images - the ‘Deep Decoder’, and get a state of the art result for that problem with a very compact network within minutes.
Our suggested learning method implementation for the Matrix Completion Problem solution is a great method and the current state of art. However, our real contribution in this work is the proposition that neural networks for Non-Euclidean Data, when looking at that data from the geometric point of view, can be a very strong prior for this problem. This approach can be a basis for many future Non-Euclidean data completion methods and applications.