arXiv subjects
Publications and source records attributed to Peter Bloem.
This document provides a tutorial description of the use of the MDL principle in complex graph analysis. We give a brief summary of the preliminary subjects, and describe the basic principle, using the example of analysing the size of the largest clique in a graph. We also provide a discussion of how to interpret the results of such an analysis, making note of several common pitfalls.
Many transformations in deep learning architectures are sparsely connected. When such transformations cannot be designed by hand, they can be learned, even through plain backpropagation, for instance in attention mechanisms. However, during learning, such sparse structures are often represented in a dense form, as we do not know beforehand which elements will eventually become non-zero. We introduce the adaptive, sparse hyperlayer, a method for learning a sparse transformation, paramatrized sparsely: as index-tuples with associated values. To overcome the lack of gradients from such a discrete structure, we introduce a method of randomly sampling connections, and backpropagating over the randomly wired computation graph. To show that this approach allows us to train a model to competitive performance on real data, we use it to build two architectures. First, an attention mechanism for visual classification. Second, we implement a method for differentiable sorting: specifically, learning to sort unlabeled MNIST digits, given only the correct order.
Knowledge graphs enable a wide variety of applications, including question answering and information retrieval. Despite the great effort invested in their creation and maintenance, even the largest (e.g., Yago, DBPedia or Wikidata) remain incomplete. We introduce Relational Graph Convolutional Networks (R-GCNs) and apply them to two standard knowledge base completion tasks: Link prediction (recovery of missing facts, i.e. subject-predicate-object triples) and entity classification (recovery of missing entity attributes). R-GCNs are related to a recent class of neural networks operating on graphs, and are developed specifically to deal with the highly multi-relational data characteristic of realistic knowledge bases. We demonstrate the effectiveness of R-GCNs as a stand-alone model for entity classification. We further show that factorization models for link prediction such as DistMult can be significantly improved by enriching them with an encoder model to accumulate evidence over multiple inference steps in the relational graph, demonstrating a large improvement of 29.8% on FB15k-237 over a decoder-only baseline.
We present an Expectation-Maximization algorithm for the fractal inverse problem: the problem of fitting a fractal model to data. In our setting the fractals are Iterated Function Systems (IFS), with similitudes as the family of transformations. The data is a point cloud in ${\mathbb R}^H$ with arbitrary dimension $H$. Each IFS defines a probability distribution on ${\mathbb R}^H$, so that the fractal inverse problem can be cast as a problem of parameter estimation. We show that the algorithm reconstructs well-known fractals from data, with the model converging to high precision parameters. We also show the utility of the model as an approximation for datasources outside the IFS model class.