# Geometric Matrix Completion with Deep Conditional Random Fields

## TL;DR

This paper introduces a deep conditional random field approach for geometric matrix completion that learns similarities and predicts missing entries effectively, especially in scenarios with very limited observations and no side information.

## Contribution

It presents a novel end-to-end deep model that jointly learns similarities, computes CRF potentials, and performs MAP inference for matrix completion without requiring fixed graphs or side information.

## Key findings

- Outperforms state-of-the-art models on benchmark datasets
- Handles highly incomplete matrices effectively
- Learns entry similarities during training

## Abstract

The problem of completing high-dimensional matrices from a limited set of observations arises in many big data applications, especially, recommender systems. Existing matrix completion models generally follow either a memory- or a model-based approach, whereas, geometric matrix completion models combine the best from both approaches. Existing deep-learning-based geometric models yield good performance, but, in order to operate, they require a fixed structure graph capturing the relationships among the users and items. This graph is typically constructed by evaluating a pre-defined similarity metric on the available observations or by using side information, e.g., user profiles. In contrast, Markov-random-fields-based models do not require a fixed structure graph but rely on handcrafted features to make predictions. When no side information is available and the number of available observations becomes very low, existing solutions are pushed to their limits. In this paper, we propose a geometric matrix completion approach that addresses these challenges. We consider matrix completion as a structured prediction problem in a conditional random field (CRF), which is characterized by a maximum a posterior (MAP) inference, and we propose a deep model that predicts the missing entries by solving the MAP inference problem. The proposed model simultaneously learns the similarities among matrix entries, computes the CRF potentials, and solves the inference problem. Its training is performed in an end-to-end manner, with a method to supervise the learning of entry similarities. Comprehensive experiments demonstrate the superior performance of the proposed model compared to various state-of-the-art models on popular benchmark datasets and underline its superior capacity to deal with highly incomplete matrices.