Incremental Clustering: The Case for Extra Clusters

The explosion in the amount of data available for analysis often necessitates a transition from batch to incremental clustering methods, which process one element at a time and typically store only a small subset of the data. In this paper, we initiate the formal analysis of incremental clustering methods focusing on the types of cluster structure that they are able to detect. We find that the incremental setting is strictly weaker than the batch model, proving that a fundamental class of cluster structures that can readily be detected in the batch setting is impossible to identify using any incremental method. Furthermore, we show how the limitations of incremental clustering can be overcome by allowing additional clusters

Formal study of plane Delaunay triangulation

This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe triangulations, we rely on a combinatorial hypermap specification framework we have been developing for years. We embed hypermaps in the plane by attaching coordinates to elements in a consistent way. We then describe what are legal and illegal Delaunay edges and a flipping operation which we show preserves hypermap, triangulation, and embedding invariants. To prove the termination of the algorithm, we use a generic approach expressing that any non-cyclic relation is well-founded when working on a finite set

High posterior density ellipsoids of quantum states

Regions of quantum states generalize the classical notion of error bars. High posterior density (HPD) credible regions are the most powerful of region estimators. However, they are intractably hard to construct in general. This paper reports on a numerical approximation to HPD regions for the purpose of testing a much more computationally and conceptually convenient class of regions: posterior covariance ellipsoids (PCEs). The PCEs are defined via the covariance matrix of the posterior probability distribution of states. Here it is shown that PCEs are near optimal for the example of Pauli measurements on multiple qubits. Moreover, the algorithm is capable of producing accurate PCE regions even when there is uncertainty in the model.Comment: TL;DR version: computationally feasible region estimator

Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs

This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex notions are those of boundary and separating edges. When performing proofs about this algorithm, questions of symmetry appear and this exposition attempts to give an account of how these symmetries can be handled. All this work relies on formal developments made with Coq and the mathematical components library

Push sum with transmission failures

The push-sum algorithm allows distributed computing of the average on a directed graph, and is particularly relevant when one is restricted to one-way and/or asynchronous communications. We investigate its behavior in the presence of unreliable communication channels where messages can be lost. We show that exponential convergence still holds and deduce fundamental properties that implicitly describe the distribution of the final value obtained. We analyze the error of the final common value we get for the essential case of two nodes, both theoretically and numerically. We provide performance comparison with a standard consensus algorithm