Choosing a diverse subset of a large collection of points in a metric space is a fundamental problem, with applications in feature selection, recommender systems, web search, data summarization, etc. Various notions of diversity have been proposed, tailored to different applications. The general algorithmic goal is to find a subset of points that maximize diversity, while obeying a cardinality (or more generally, matroid) constraint. The goal of this paper is to develop a novel linear programming (LP) framework that allows us to design approximation algorithms for such problems. We study an objective known as {\em sum-min} diversity, which is known to be effective in many applications, and give the first constant factor approximation algorithm. Our LP framework allows us to easily incorporate additional constraints, as well as secondary objectives.

Estimating optimal phylogenetic trees or hierarchical clustering trees from metric data is an important problem in evolutionary biology and data analysis. Intuitively, the goodness-of-fit of a metric space to a tree depends on its inherent treeness, as well as other metric properties such as intrinsic dimension. Existing algorithms for embedding metric spaces into tree metrics provide distortion bounds depending on cardinality. Because cardinality is a simple property of any set, we argue that such bounds do not fully capture the rich structure endowed by the metric. We consider an embedding of a metric space into a tree proposed by Gromov.

We propose a pool-based non-parametric active learning algorithm for general metric spaces, called MArgin Regularized Metric Active Nearest Neighbor (MARMANN), which outputs a nearest-neighbor classifier. We give prediction error guarantees that depend on the noisy-margin properties of the input sample, and are competitive with those obtained by previously proposed passive learners. We prove that the label complexity of MARMANN is significantly lower than that of any passive learner with similar error guarantees. Our algorithm is based on a generalized sample compression scheme and a new label-efficient active model-selection procedure. Papers published at the Neural Information Processing Systems Conference.

The problem of prediction with expert advice [ Cesa-Bianchi and Lugosi, 2006 ] is a by now standard model of online learning. Traditionally studied for outcom es taking values in a vector space, less seems to be known when the outcome space is a more general metr ic space. This paper partly addresses the problem by focusing on the case of NPC spaces, i .e., geodesic metric spaces with non-positive curvature in the sense of Alexandrov. The class of NPC spaces includes many metric spaces of partic ular interest in the data sciences. Apart from Hilbert spaces, interesting examples are hyperb olic spaces [ Nickel and Kiela, 2017 ], the space of real symmetric positive-definite matrices with Log -Euclidean [ Arsigny et al., 2007 ] or Log-Cholesky [ Lin, 2019 ] Riemannian metrics and more generally all complete and sim ply connected Riemannian manifolds with non-positive sectional curvatu re.

Graph autoencoders (AE) and variational autoencoders (VAE) recently emerged as powerful node embedding methods, with promising performances on challenging tasks such as link prediction and node clustering. Graph AE, VAE and most of their extensions rely on graph convolutional networks (GCN) encoders to learn vector space representations of nodes. In this paper, we propose to replace the GCN encoder by a significantly simpler linear model w.r.t. the direct neighborhood (one-hop) adjacency matrix of the graph. For the two aforementioned tasks, we show that this approach consistently reaches competitive performances w.r.t. GCN-based models for numerous real-world graphs, including all benchmark datasets commonly used to evaluate graph AE and VAE. We question the relevance of repeatedly using these datasets to compare complex graph AE and VAE. We also emphasize the effectiveness of the proposed encoding scheme, that appears as a simpler and faster alternative to GCN encoders for many real-world applications.

We present a unifying framework encompassing many social choice settings. Viewing each social choice setting as voting in a suitable metric space, we consider a general model of social choice over metric spaces, in which---similarly to the spatial model of elections---each voter specifies an ideal element of the metric space. The ideal element functions as a vote, where each voter prefers elements that are closer to her ideal element. But it also functions as a proposal, thus making all participants equal not only as voters but also as proposers. We consider Condorcet aggregation and a continuum of solution concepts, ranging from minimizing the sum of distances to minimizing the maximum distance. We study applications of the abstract model to various social choice settings, including single-winner elections, committee elections, participatory budgeting, and participatory legislation. For each setting, we compare each solution concept to known voting rules and study various properties of the resulting voting rules. Our framework provides expressive aggregation for a broad range of social choice settings while remaining simple for voters, and may enable a unified and integrated implementation for all these settings, as well as unified extensions such as sybil-resiliency, proxy voting, and deliberative decision making.

Relation Extraction (RE), the task of detecting and characterizing semantic relations between entities in text, has gained much importance in the last two decades, mainly in the biomedical domain. Many papers have been published on Relation Extraction using supervised machine learning techniques. Most of these techniques rely on statistical methods, such as feature-based and tree-kernels-based methods. Such statistical learning techniques are usually based on a propositional hypothesis space for representing examples, i.e., they employ an attribute-value representation of features. This kind of representation has some drawbacks, particularly in the extraction of complex relations which demand more contextual information about the involving instances, i.e., it is not able to effectively capture structural information from parse trees without loss of information. In this work, we present OntoILPER, a logic-based relational learning approach to Relation Extraction that uses Inductive Logic Programming for generating extraction models in the form of symbolic extraction rules. OntoILPER takes profit of a rich relational representation of examples, which can alleviate the aforementioned drawbacks. The proposed relational approach seems to be more suitable for Relation Extraction than statistical ones for several reasons that we argue. Moreover, OntoILPER uses a domain ontology that guides the background knowledge generation process and is used for storing the extracted relation instances. The induced extraction rules were evaluated on three protein-protein interaction datasets from the biomedical domain. The performance of OntoILPER extraction models was compared with other state-of-the-art RE systems. The encouraging results seem to demonstrate the effectiveness of the proposed solution.

Prior work inspired by compression algorithms has described how the Burrows Wheeler Transform can be used to create a distance measure for bioinformatics problems. We describe issues with this approach that were not widely known, and introduce our new Burrows Wheeler Markov Distance (BWMD) as an alternative. The BWMD avoids the shortcomings of earlier efforts, and allows us to tackle problems in variable length DNA sequence clustering. BWMD is also more adaptable to other domains, which we demonstrate on malware classification tasks. Unlike other compression-based distance metrics known to us, BWMD works by embedding sequences into a fixed-length feature vector. This allows us to provide significantly improved clustering performance on larger malware corpora, a weakness of prior methods.

Reeb spaces, as well as their discretized versions called Mappers, are common descriptors used in Topological Data Analysis, with plenty of applications in various fields of science, such as computational biology and data visualization, among others. The stability and quantification of the rate of convergence of the Mapper to the Reeb space has been studied a lot in recent works~\cite{Brown2019, Carriere2018a, Carriere2018, Munch2016}, focusing on the case where a scalar-valued filter is used for the computation of Mapper. On the other hand, much less is known in the multivariate case, where the domain of the filter is in $\mathbb R^d$ instead of $\mathbb R$. The only available result in this setting~\cite{Munch2016} only works for topological spaces and cannot be used as is for finite metric spaces representing data, such as point clouds and distance matrices. In this article, we present an approximation result for the Reeb space in the multivariate case using a Mapper-based estimator, which is a slight modification of the usual Mapper construction. Moreover, our approximation is stated with respect to a pseudometric that is an extension of the usual {\em interleaving distance} between persistence modules~\cite{Chazal2016}. Finally, we apply our results to the case where the filter function used to compute the Mapper is estimated from the data. We provide applications of this setting in statistics and machine learning and probability for different kinds of target filters, as well as numerical experiments that demonstrate the relevance of our approach.

LatticeNet: Fast Point Cloud Segmentation Using Permutohedral Lattices Radu Alexandru Rosu Peer Sch utt Jan Quenzel Sven Behnke Abstract -- Deep convolutional neural networks (CNNs) have shown outstanding performance in the task of semantically segmenting images. However, applying the same methods on 3D data still poses challenges due to the heavy memory requirements and the lack of structured data. Here, we propose LatticeNet, a novel approach for 3D semantic segmentation, which takes as input raw point clouds. A PointNet describes the local geometry which we embed into a sparse permutohedral lattice. The lattice allows for fast convolutions while keeping a low memory footprint. Further, we introduce DeformSlice, a novel learned data-dependent interpolation for projecting lattice features back onto the point cloud. We present results of 3D segmentation on various datasets where our method achieves state-of-the-art performance. I NTRODUCTION Environment understanding is a crucial ability for autonomous agents.