Graph kernels between point clouds

Point clouds are sets of points in two or three dimensions. Most kernel methods for learning on sets of points have not yet dealt with the specific geometrical invariances and practical constraints associated with point clouds in computer vision and graphics. In this paper, we present extensions of graph kernels for point clouds, which allow to use kernel methods for such ob jects as shapes, line drawings, or any three-dimensional point clouds. In order to design rich and numerically efficient kernels with as few free parameters as possible, we use kernels between covariance matrices and their factorizations on graphical models. We derive polynomial time dynamic programming recursions and present applications to recognition of handwritten digits and Chinese characters from few training examples

Matching Natural Language Sentences with Hierarchical Sentence Factorization

Semantic matching of natural language sentences or identifying the relationship between two sentences is a core research problem underlying many natural language tasks. Depending on whether training data is available, prior research has proposed both unsupervised distance-based schemes and supervised deep learning schemes for sentence matching. However, previous approaches either omit or fail to fully utilize the ordered, hierarchical, and flexible structures of language objects, as well as the interactions between them. In this paper, we propose Hierarchical Sentence Factorization---a technique to factorize a sentence into a hierarchical representation, with the components at each different scale reordered into a "predicate-argument" form. The proposed sentence factorization technique leads to the invention of: 1) a new unsupervised distance metric which calculates the semantic distance between a pair of text snippets by solving a penalized optimal transport problem while preserving the logical relationship of words in the reordered sentences, and 2) new multi-scale deep learning models for supervised semantic training, based on factorized sentence hierarchies. We apply our techniques to text-pair similarity estimation and text-pair relationship classification tasks, based on multiple datasets such as STSbenchmark, the Microsoft Research paraphrase identification (MSRP) dataset, the SICK dataset, etc. Extensive experiments show that the proposed hierarchical sentence factorization can be used to significantly improve the performance of existing unsupervised distance-based metrics as well as multiple supervised deep learning models based on the convolutional neural network (CNN) and long short-term memory (LSTM).Comment: Accepted by WWW 2018, 10 page

Chromatic Ramsey number of acyclic hypergraphs

Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $\chi(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph are colored with $t$ colors in any manner, there is a monochromatic copy of $T$. We observe that $\chi(T,t)$ is well defined and $$\left\lceil {R^r(T,t)-1\over r-1}\right \rceil +1 \le \chi(T,t)\le |E(T)|^t+1$$ where $R^r(T,t)$ is the $t$-color Ramsey number of $H$. We give linear upper bounds for $\chi(T,t)$ when T is a matching or star, proving that for $r\ge 2, k\ge 1, t\ge 1$, $\chi(M_k^r,t)\le (t-1)(k-1)+2k$ and $\chi(S_k^r,t)\le t(k-1)+2$ where $M_k^r$ and $S_k^r$ are, respectively, the $r$-uniform matching and star with $k$ edges. The general bounds are improved for $3$-uniform hypergraphs. We prove that $\chi(M_k^3,2)=2k$, extending a special case of Alon-Frankl-Lov\'asz' theorem. We also prove that $\chi(S_2^3,t)\le t+1$, which is sharp for $t=2,3$. This is a corollary of a more general result. We define $H^{[1]}$ as the 1-intersection graph of $H$, whose vertices represent hyperedges and whose edges represent intersections of hyperedges in exactly one vertex. We prove that $\chi(H)\le \chi(H^{[1]})$ for any $3$-uniform hypergraph $H$ (assuming $\chi(H^{[1]})\ge 2$). The proof uses the list coloring version of Brooks' theorem.Comment: 10 page

Elastic Registration of Geodesic Vascular Graphs

Vascular graphs can embed a number of high-level features, from morphological parameters, to functional biomarkers, and represent an invaluable tool for longitudinal and cross-sectional clinical inference. This, however, is only feasible when graphs are co-registered together, allowing coherent multiple comparisons. The robust registration of vascular topologies stands therefore as key enabling technology for group-wise analyses. In this work, we present an end-to-end vascular graph registration approach, that aligns networks with non-linear geometries and topological deformations, by introducing a novel overconnected geodesic vascular graph formulation, and without enforcing any anatomical prior constraint. The 3D elastic graph registration is then performed with state-of-the-art graph matching methods used in computer vision. Promising results of vascular matching are found using graphs from synthetic and real angiographies. Observations and future designs are discussed towards potential clinical applications

Inferring Sparsity: Compressed Sensing using Generalized Restricted Boltzmann Machines

In this work, we consider compressed sensing reconstruction from $M$ measurements of $K$-sparse structured signals which do not possess a writable correlation model. Assuming that a generative statistical model, such as a Boltzmann machine, can be trained in an unsupervised manner on example signals, we demonstrate how this signal model can be used within a Bayesian framework of signal reconstruction. By deriving a message-passing inference for general distribution restricted Boltzmann machines, we are able to integrate these inferred signal models into approximate message passing for compressed sensing reconstruction. Finally, we show for the MNIST dataset that this approach can be very effective, even for $M < K$.Comment: IEEE Information Theory Workshop, 201