Classification in the dissimilarity space has become a very active research area since it provides a possibility to learn from data given in the form of pairwise non-metric dissimilarities, which otherwise would be difficult to cope with. The selection of prototypes is a key step for the further creation of the space. However, despite previous efforts to find good prototypes, how to select the best representation set remains an open issue. In this paper we proposed scalable methods to select the set of prototypes out of very large datasets. The methods are based on genetic algorithms, dissimilarity-based hashing, and two different unsupervised and supervised scalable criteria. The unsupervised criterion is based on the Minimum Spanning Tree of the graph created by the prototypes as nodes and the dissimilarities as edges. The supervised criterion is based on counting matching labels of objects and their closest prototypes. The suitability of these type of algorithms is analyzed for the specific case of dissimilarity representations. The experimental results showed that the methods select good prototypes taking advantage of the large datasets, and they do so at low runtimes. Preprint submitted to Elsevier December 27, 2017 1. Introduction The vector space representation is a common option to represent the data for learning tasks since many statistical techniques are applicable for this kind of representation. However, there is an increasing number of real-world problems which are not vectorial. Instead, the data are given in terms of pairwise dissimilarities which may be non-Euclidean and even non-metric.

A significantly faster algorithm is presented for the original kNN mode seeking procedure. It has the advantages over the well-known mean shift algorithm that it is feasible in high-dimensional vector spaces and results in uniquely, well defined modes. Moreover, without any additional computational effort it may yield a multi-scale hierarchy of clusterings. The time complexity is just O(n^1.5). resulting computing times range from seconds for 10^4 objects to minutes for 10^5 objects and to less than an hour for 10^6 objects. The space complexity is just O(n). The procedure is well suited for finding large sets of small clusters and is thereby a candidate to analyze thousands of clusters in millions of objects. The kNN mode seeking procedure can be used for active learning by assigning the clusters to the class of the modal objects of the clusters. Its feasibility is shown by some examples with up to 1.5 million handwritten digits. The obtained classification results based on the clusterings are compared with those obtained by the nearest neighbor rule and the support vector classifier based on the same labeled objects for training. It can be concluded that using the clustering structure for classification can be significantly better than using the trained classifiers. A drawback of using the clustering for classification, however, is that no classifier is obtained that may be used for out-of-sample objects.

Graphs (networks) are ubiquitous and allow us to model entities (nodes) and the dependencies (edges) between them. Learning a useful feature representation from graph data lies at the heart and success of many machine learning tasks such as classification, anomaly detection, link prediction, among many others. Many existing techniques use random walks as a basis for learning features or estimating the parameters of a graph model for a downstream prediction task. Examples include recent node embedding methods such as DeepWalk, node2vec, as well as graph-based deep learning algorithms. However, the simple random walk used by these methods is fundamentally tied to the identity of the node. This has three main disadvantages. First, these approaches are inherently transductive and do not generalize to unseen nodes and other graphs. Second, they are not space-efficient as a feature vector is learned for each node which is impractical for large graphs. Third, most of these approaches lack support for attributed graphs. To make these methods more generally applicable, we propose a framework for inductive network representation learning based on the notion of attributed random walk that is not tied to node identity and is instead based on learning a function $\Phi : \mathrm{\rm \bf x} \rightarrow w$ that maps a node attribute vector $\mathrm{\rm \bf x}$ to a type $w$. This framework serves as a basis for generalizing existing methods such as DeepWalk, node2vec, and many other previous methods that leverage traditional random walks.

Recent studies on knowledge base completion, the task of recovering missing facts based on observed facts, demonstrate the importance of learning embeddings from multi-step relations. Due to the size of knowledge bases, previous works manually design relation paths of observed triplets in symbolic space (e.g. random walk) to learn multi-step relations during training. However, these approaches suffer some limitations as most paths are not informative, and it is prohibitively expensive to consider all possible paths. To address the limitations, we propose learning to traverse in vector space directly without the need of symbolic space guidance. To remember the connections between related observed triplets and be able to adaptively change relation paths in vector space, we propose Implicit ReasoNets (IRNs), that is composed of a global memory and a controller module to learn multi-step relation paths in vector space and infer missing facts jointly without any human-designed procedure. Without using any axillary information, our proposed model achieves state-of-the-art results on popular knowledge base completion benchmarks.

Conceptual spaces are geometric representations of conceptual knowledge, in which entities correspond to points, natural properties correspond to convex regions, and the dimensions of the space correspond to salient features. While conceptual spaces enable elegant models of various cognitive phenomena, the lack of automated methods for constructing such representations have so far limited their application in artificial intelligence. To address this issue, we propose a method which learns a vector-space embedding of entities from Wikipedia and constrains this embedding such that entities of the same semantic type are located in some lower-dimensional subspace. We experimentally demonstrate the usefulness of these subspaces as (approximate) conceptual space representations by showing, among others, that important features can be modelled as directions and that natural properties tend to correspond to convex regions.

When each data point is a large graph, graph statistics such as densities of certain subgraphs (motifs) can be used as feature vectors for machine learning. While intuitive, motif counts are expensive to compute and difficult to work with theoretically. Via graphon theory, we give an explicit quantitative bound for the ability of motif homomorphisms to distinguish large networks under both generative and sampling noise. Furthermore, we give similar bounds for the graph spectrum and connect it to homomorphism densities of cycles. This results in an easily computable classifier on graph data with theoretical performance guarantee. Our method yields competitive results on classification tasks for the autoimmune disease Lupus Erythematosus.

The characteristics (or numerical patterns) of a feature vector in the transform domain of a perturbation model differ significantly from those of its corresponding feature vector in the input domain. These differences - caused by the perturbation techniques used for the transformation of feature patterns - degrade the performance of machine learning techniques in the transform domain. In this paper, we proposed a nonlinear parametric perturbation model that transforms the input feature patterns to a set of elliptical patterns, and studied the performance degradation issues associated with random forest classification technique using both the input and transform domain features. Compared with the linear transformation such as Principal Component Analysis (PCA), the proposed method requires less statistical assumptions and is highly suitable for the applications such as data privacy and security due to the difficulty of inverting the elliptical patterns from the transform domain to the input domain. In addition, we adopted a flexible block-wise dimensionality reduction step in the proposed method to accommodate the possible high-dimensional data in modern applications. We evaluated the empirical performance of the proposed method on a network intrusion data set and a biological data set, and compared the results with PCA in terms of classification performance and data privacy protection (measured by the blind source separation attack and signal interference ratio). Both results confirmed the superior performance of the proposed elliptical transformation. 1 1. INTRODUCTION Feature vectors carry useful numerical patterns that characterize the original domain (or a sub original domain - input domain) formed by the feature vectors themselves. Machine learning algorithms generally utilize these patterns to generate classifiers, that can help make decisions from data, by using supervised or unsupervised learning techniques (Suthaharan, 2015).

This paper presents a general graph representation learning framework called DeepGL for learning deep node and edge representations from large (attributed) graphs. In particular, DeepGL begins by deriving a set of base features (e.g., graphlet features) and automatically learns a multi-layered hierarchical graph representation where each successive layer leverages the output from the previous layer to learn features of a higher-order. Contrary to previous work, DeepGL learns relational functions (each representing a feature) that generalize across-networks and therefore useful for graph-based transfer learning tasks. Moreover, DeepGL naturally supports attributed graphs, learns interpretable features, and is space-efficient (by learning sparse feature vectors). In addition, DeepGL is expressive, flexible with many interchangeable components, efficient with a time complexity of $\mathcal{O}(|E|)$, and scalable for large networks via an efficient parallel implementation. Compared with the state-of-the-art method, DeepGL is (1) effective for across-network transfer learning tasks and attributed graph representation learning, (2) space-efficient requiring up to 6x less memory, (3) fast with up to 182x speedup in runtime performance, and (4) accurate with an average improvement of 20% or more on many learning tasks.

Topological Data Analysis (tda) is a recent and fast growing eld providing a set of new topological and geometric tools to infer relevant features for possibly complex data. This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for non experts. 1 Introduction and motivation Topological Data Analysis (tda) is a recent eld that emerged from various works in applied (algebraic) topology and computational geometry during the rst decade of the century. Although one can trace back geometric approaches for data analysis quite far in the past, tda really started as a eld with the pioneering works of Edelsbrunner et al. (2002) and Zomorodian and Carlsson (2005) in persistent homology and was popularized in a landmark paper in 2009 Carlsson (2009). tda is mainly motivated by the idea that topology and geometry provide a powerful approach to infer robust qualitative, and sometimes quantitative, information about the structure of data-see, e.g. Chazal (2017). tda aims at providing well-founded mathematical, statistical and algorithmic methods to infer, analyze and exploit the complex topological and geometric structures underlying data that are often represented as point clouds in Euclidean or more general metric spaces. During the last few years, a considerable eort has been made to provide robust and ecient data structures and algorithms for tda that are now implemented and available and easy to use through standard libraries such as the Gudhi library (C++ and Python) Maria et al. (2014) and its R software interface Fasy et al. (2014a). Although it is still rapidly evolving, tda now provides a set of mature and ecient tools that can be used in combination or complementary to other data sciences tools. The tdapipeline. tda has recently known developments in various directions and application elds. There now exist a large variety of methods inspired by topological and geometric approaches. Providing a complete overview of all these existing approaches is beyond the scope of this introductory survey. However, most of them rely on the following basic and standard pipeline that will serve as the backbone of this paper: 1. The input is assumed to be a nite set of points coming with a notion of distance-or similarity between them. This distance can be induced by the metric in the ambient space (e.g. the Euclidean metric when the data are embedded in R d) or come as an intrinsic metric dened by a pairwise distance matrix. The denition of the metric on the data is usually given as an input or guided by the application. It is however important to notice that the choice of the metric may be critical to reveal interesting topological and geometric features of the data.

Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the study of the shape of the data. In this work we investigate the predictive power of TDA in the context of supervised learning. Since topological summaries, most noticeably the Persistence Diagram, are typically defined in complex spaces, we adopt a kernel approach to translate them into more familiar vector spaces. We define a topological exponential kernel, we characterize it, and we show that, despite not being positive semi-definite, it can be successfully used in regression and classification tasks.