We provide new bounds for rates of convergence of the multivariate Central Limit Theorem in Wasserstein distances of order p ≥ 2. In particular, we obtain an asymptotic bound for measures with a continuous component which we conjecture to be optimal.

In this paper, we propose a novel pooling approach for shape classification and recognition using the bag-of-words pipeline, based on topological persistence, a recent tool from Topological Data Analysis. Our technique extends the standard max-pooling, which summarizes the distribution of a visual feature with a single number, thereby losing any notion of spatiality. Instead, we propose to use topological persistence, and the derived persistence diagrams, to provide significantly more informative and spatially sensitive characterizations of the feature functions, which can lead to better recognition performance. Unfortunately, despite their conceptual appeal, persistence diagrams are difficult to handle, since they are not naturally represented as vectors in Euclidean space and even the standard metric, the bottleneck distance is not easy to compute. Furthermore, classical distances between diagrams, such as the bottleneck and Wasserstein distances, do not allow to build positive definite kernels that can be used for learning. To handle this issue, we provide a novel way to transform persistence diagrams into vectors, in which comparisons are trivial. Finally, we demonstrate the performance of our construction on the Non-Rigid 3D Human Models SHREC 2014 dataset, where we show that topological pooling can provide significant improvements over the standard pooling methods for the shape pose recognition within the bag-of-words pipeline.

In this paper, we propose a new fuzzy clustering algorithm based on the modeseeking framework. Given a dataset in R d , we define regions of high density that we call cluster cores. We then consider a random walk on a neighborhood graph built on top of our data points which is designed to be attracted by high density regions. The strength of this attraction is controlled by a temperature parameter β > 0. The membership of a point to a given cluster is then the probability for the random walk to hit the corresponding cluster core before any other. While many properties of random walks (such as hitting times, commute distances, etc. . . ) have been shown to enventually encode purely local information when the number of data points grows, we show that the regularization introduced by the use of cluster cores solves this issue. Empirically, we show how the choice of β influences the behavior of our algorithm: for small values of β the result is close to hard modeseeking whereas when β is close to 1 the result is similar to the output of a (fuzzy) spectral clustering. Finally, we demonstrate the scalability of our approach by providing the fuzzy clustering of a protein configuration dataset containing a million data points in 30 dimensions.