Fat Polygonal Partitions with Applications to Visualization and Embeddings

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in $\mathcal{T}$ is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in $\mathbb{R}^d$. We use these partitions with slack for embedding ultrametrics into $d$-dimensional Euclidean space: we give a $\mathop{\rm polylog}(\Delta)$-approximation algorithm for embedding $n$-point ultrametrics into $\mathbb{R}^d$ with minimum distortion, where $\Delta$ denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in $n$. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.Comment: 26 page

Learning Edge Representations via Low-Rank Asymmetric Projections

We propose a new method for embedding graphs while preserving directed edge information. Learning such continuous-space vector representations (or embeddings) of nodes in a graph is an important first step for using network information (from social networks, user-item graphs, knowledge bases, etc.) in many machine learning tasks. Unlike previous work, we (1) explicitly model an edge as a function of node embeddings, and we (2) propose a novel objective, the "graph likelihood", which contrasts information from sampled random walks with non-existent edges. Individually, both of these contributions improve the learned representations, especially when there are memory constraints on the total size of the embeddings. When combined, our contributions enable us to significantly improve the state-of-the-art by learning more concise representations that better preserve the graph structure. We evaluate our method on a variety of link-prediction task including social networks, collaboration networks, and protein interactions, showing that our proposed method learn representations with error reductions of up to 76% and 55%, on directed and undirected graphs. In addition, we show that the representations learned by our method are quite space efficient, producing embeddings which have higher structure-preserving accuracy but are 10 times smaller

Fat polygonal partitions with applications to visualization and embeddings

Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in Rd. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a polylog(¿)-approximation algorithm for embedding n-point ultrametrics into Rd with minimum distortion, where ¿ denotes the spread of the metric. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio

Deep Reinforcement Learning for Swarm Systems

Recently, deep reinforcement learning (RL) methods have been applied successfully to multi-agent scenarios. Typically, these methods rely on a concatenation of agent states to represent the information content required for decentralized decision making. However, concatenation scales poorly to swarm systems with a large number of homogeneous agents as it does not exploit the fundamental properties inherent to these systems: (i) the agents in the swarm are interchangeable and (ii) the exact number of agents in the swarm is irrelevant. Therefore, we propose a new state representation for deep multi-agent RL based on mean embeddings of distributions. We treat the agents as samples of a distribution and use the empirical mean embedding as input for a decentralized policy. We define different feature spaces of the mean embedding using histograms, radial basis functions and a neural network learned end-to-end. We evaluate the representation on two well known problems from the swarm literature (rendezvous and pursuit evasion), in a globally and locally observable setup. For the local setup we furthermore introduce simple communication protocols. Of all approaches, the mean embedding representation using neural network features enables the richest information exchange between neighboring agents facilitating the development of more complex collective strategies.Comment: 31 pages, 12 figures, version 3 (published in JMLR Volume 20

Convex treemaps with bounded aspect ratio

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree T where the weight of each node is the sum of the weights of its children. A treemap for T is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of T and the area of each region is proportional to the weight of the corresponding node. An important quality criterium for treemaps is the aspect ratio of its regions. Unfortunately, one cannot bound the aspect ratio if the regions are restricted to be rectangles. Hence Onak and Sidiropoulos introduced polygonal partitions, which use convex polygons. We are the first to obtain convex partitions with optimal aspect ratio O(depth(T )). We also consider the important special case that depth(T ) = 1, that is, single-level treemaps. We show how to construct convex single-level treemaps that use only four simple shapes for the regions and have aspect ratio at most 34/7

Angles of Arc-Polygons and Lombardi Drawings of Cacti

We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are at most pi. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.Comment: 12 pages, 8 figures. To be published in Proc. 33rd Canadian Conference on Computational Geometry, 202

visone - Software for the Analysis and Visualization of Social Networks

We present the software tool visone which combines graph-theoretic methods for the analysis of social networks with tailored means of visualization. Our main contribution is the design of novel graph-layout algorithms which accurately reflect computed analyses results in well-arranged drawings of the networks under consideration. Besides this, we give a detailed description of the design of the software tool and the provided analysis methods