Persistence bag-of-Words
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs are compact 2D representations formed by multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper, we introduce persistence bag-of-words, which is a novel, expressive and discriminative vectorized representation of PDs for topological data analysis. It represents PDs in a convenient way for machine learning and statistical analysis and has a number of favorable practical and theoretical properties like 1-Wasserstein stability. We evaluate our representation on several heterogeneous datasets and show its high discriminative power. Our approach achieves state-of-the-art performance and even beyond in much less time than alternative approaches. Thereby, it facilitates the topological analysis of large-scale data sets in future. …

Cluster Grap
In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers. A cluster graph is a graph whose every connected component is a complete graph. …

Equivariant Relational Layer (ERL)
Due to its extensive use in databases, the relational model is ubiquitous in representing big-data. We propose to apply deep learning to this type of relational data by introducing an Equivariant Relational Layer (ERL), a neural network layer derived from the entity-relationship model of the database. Our layer relies on identification of exchangeabilities in the relational data(base), and their expression as a permutation group. We prove that an ERL is an optimal parameter-sharing scheme under the given exchangeability constraints, and subsumes recently introduced deep models for sets, exchangeable tensors, and graphs. The proposed model has a linear complexity in the size of the relational data, and it can be used for both inductive and transductive reasoning in databases, including the prediction of missing records, and database embedding. This opens the door to the application of deep learning to one of the most abundant forms of data. …

meta-iNat Benchmark
Traditional recognition methods typically require large, artificially-balanced training classes, while few-shot learning methods are tested on artificially small ones. In contrast to both extremes, real world recognition problems exhibit heavy-tailed class distributions, with cluttered scenes and a mix of coarse and fine-grained class distinctions. We show that prior methods designed for few-shot learning do not work out of the box in these challenging conditions, based on a new ‘meta-iNat’ benchmark. We introduce three parameter-free improvements: (a) better training procedures based on adapting cross-validation to meta-learning, (b) novel architectures that localize objects using limited bounding box annotations before classification, and (c) simple parameter-free expansions of the feature space based on bilinear pooling. Together, these improvements double the accuracy of state-of-the-art models on meta-iNat while generalizing to prior benchmarks, complex neural architectures, and settings with substantial domain shift. …