In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Q: Graph matching is a challenging problem with very important applications in a wide range of fields, from image and video analysis to biological and biomedical problems. We propose a robust graph matching algorithm inspired in sparsityrelated techniques. We cast the problem, resembling group or collaborative sparsity formulations, as a non-smooth convex optimization problem that can be efficiently solved using augmented Lagrangian techniques. The method can deal with weighted or unweighted graphs, as well as multimodal data, where different graphs represent different types of data. The proposed approach is also naturally integrated with collaborative graph inference techniques, solving general network inference problems where the observed variables, possibly coming from different modalities, are not in correspondence. The algorithm is tested and compared with state-of-the-art graph matching techniques in both synthetic and real graphs. We also present results on multimodal graphs and applications to collaborative inference of brain connectivity from alignment-free functional magnetic resonance imaging (fMRI) data. The code is publicly available.

A: Robust Multimodal Graph Matching: Sparse Coding Meets Graph Matching
****
Q: We consider the problem of repeatedly solving a variant of the same dynamic programming problem in successive trials. An instance of the type of problems we consider is to find the optimal binary search tree. At the beginning of each trial, the learner probabilistically chooses a tree with the n keys at the internal nodes and the n+ 1 gaps between keys at the leaves. It is then told the frequencies of the keys and gaps and is charged by the average search cost for the chosen tree. The problem is online because the frequencies can change between trials. The goal is to develop algorithms with the property that their total average search cost (loss) in all trials is close to the total loss of the best tree chosen in hind sight for all trials. The challenge, of course, is that the algorithm has to deal with exponential number of trees. We develop a methodology for tackling such problems for a wide class of dynamic programming algorithms. Our framework allows us to extend online learning algorithms like Hedge [9] and Component Hedge [15] to a significantly wider class of combinatorial objects than was possible before.

A: Online Dynamic Programming
****
Q: In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of arbitrary rank to be identifiable from a set of matrix entries, yielding theoretical constraints and new algorithms for the problem of matrix completion. We conclude by algorithmically evaluating the tightness of the given conditions and algorithms for practically relevant matrix sizes, showing that the algebraic-combinatorial approach can lead to improvements over stateof-the-art matrix completion methods.

A:
A Combinatorial Algebraic Approach for the Identifiability of Low-Rank Matrix Completion
****