Graph theoretical approaches for the characterization of damage in hierarchical materials

We discuss the relevance of methods of graph theory for the study of damage in simple model materials described by the random fuse model. While such methods are not commonly used when dealing with regular random lattices, which mimic disordered but statistically homogeneous materials, they become relevant in materials with microstructures that exhibit complex multi-scale patterns. We specifically address the case of hierarchical materials, whose failure, due to an uncommon fracture mode, is not well described in terms of either damage percolation or crack nucleation-and-growth. We show that in these systems, incipient failure is accompanied by an increase in eigenvector localization and a drop in topological dimension. We propose these two novel indicators as possible candidates to monitor a system in the approach to failure. As such, they provide alternatives to monitoring changes in the precursory avalanche activity, which is often invoked as a candidate for failure prediction in materials which exhibit critical-like behavior at failure, but may not work in the context of hierarchical materials which exhibit scale-free avalanche statistics even very far from the critical load.Comment: 12 pages, 6 figure

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning

Many problems in sequential decision making and stochastic control often have natural multiscale structure: sub-tasks are assembled together to accomplish complex goals. Systematically inferring and leveraging hierarchical structure, particularly beyond a single level of abstraction, has remained a longstanding challenge. We describe a fast multiscale procedure for repeatedly compressing, or homogenizing, Markov decision processes (MDPs), wherein a hierarchy of sub-problems at different scales is automatically determined. Coarsened MDPs are themselves independent, deterministic MDPs, and may be solved using existing algorithms. The multiscale representation delivered by this procedure decouples sub-tasks from each other and can lead to substantial improvements in convergence rates both locally within sub-problems and globally across sub-problems, yielding significant computational savings. A second fundamental aspect of this work is that these multiscale decompositions yield new transfer opportunities across different problems, where solutions of sub-tasks at different levels of the hierarchy may be amenable to transfer to new problems. Localized transfer of policies and potential operators at arbitrary scales is emphasized. Finally, we demonstrate compression and transfer in a collection of illustrative domains, including examples involving discrete and continuous statespaces.Comment: 86 pages, 15 figure

Image Co-localization by Mimicking a Good Detector's Confidence Score Distribution

Given a set of images containing objects from the same category, the task of image co-localization is to identify and localize each instance. This paper shows that this problem can be solved by a simple but intriguing idea, that is, a common object detector can be learnt by making its detection confidence scores distributed like those of a strongly supervised detector. More specifically, we observe that given a set of object proposals extracted from an image that contains the object of interest, an accurate strongly supervised object detector should give high scores to only a small minority of proposals, and low scores to most of them. Thus, we devise an entropy-based objective function to enforce the above property when learning the common object detector. Once the detector is learnt, we resort to a segmentation approach to refine the localization. We show that despite its simplicity, our approach outperforms state-of-the-art methods.Comment: Accepted to Proc. European Conf. Computer Vision 201

Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks