On Geodesic Completeness for Riemannian Metrics on Smooth Probability Densities

The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding the $L^2$-Wasserstein distance of optimal mass transport, and the Fisher--Rao metric, predominant in the theory of information geometry. On the space of smooth probability densities, none of these Riemannian metrics are geodesically complete---a property desirable for example in imaging applications. That is, the existence interval for solutions to the geodesic flow equations cannot be extended to the whole real line. Here we study a class of Hamilton--Jacobi-like partial differential equations arising as geodesic flow equations for higher-order Sobolev type metrics on the space of smooth probability densities. We give order conditions for global existence and uniqueness, thereby providing geodesic completeness. The system we study is an interesting example of a flow equation with loss of derivatives, which is well-posed in the smooth category, yet non-parabolic and fully non-linear. On a more general note, the paper establishes a link between geometric analysis on the space of probability densities and analysis of Euler-Arnold equations in topological hydrodynamics.Comment: 19 pages, accepted in Calc. Var. Partial Differential Equations (2017

Diffeomorphic density matching by optimal information transport

We address the following problem: given two smooth densities on a manifold, find an optimal diffeomorphism that transforms one density into the other. Our framework builds on connections between the Fisher-Rao information metric on the space of probability densities and right-invariant metrics on the infinite-dimensional manifold of diffeomorphisms. This optimal information transport, and modifications thereof, allows us to construct numerical algorithms for density matching. The algorithms are inherently more efficient than those based on optimal mass transport or diffeomorphic registration. Our methods have applications in medical image registration, texture mapping, image morphing, non-uniform random sampling, and mesh adaptivity. Some of these applications are illustrated in examples.Comment: 35 page

Bridge Simulation and Metric Estimation on Landmark Manifolds

We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood

Weighted Diffeomorphic Density Matching with Applications to Thoracic Image Registration

In this article we study the problem of thoracic image registration, in particular the estimation of complex anatomical deformations associated with the breathing cycle. Using the intimate link between the Riemannian geometry of the space of diffeomorphisms and the space of densities, we develop an image registration framework that incorporates both the fundamental law of conservation of mass as well as spatially varying tissue compressibility properties. By exploiting the geometrical structure, the resulting algorithm is computationally efficient, yet widely general.Comment: Accepted in Proceedings of the 5th MICCAI workshop on Mathematical Foundations of Computational Anatomy, Munich, Germany, 2015 (http://www-sop.inria.fr/asclepios/events/MFCA15/

DEDA: An algorithm for early detection of topology attacks in the internet of things

The internet of things (IoT) is used in domestic, industrial as well as mission-critical systems including homes, transports, power plants, industrial manufacturing and health-care applications. Security of data generated by such systems and IoT systems itself is very critical in such applications. Early detection of any attack targeting IoT system is necessary to minimize the damage. This paper reviews security attack detection methods for IoT Infrastructure presented in the state-of-the-art. One of the major entry points for attacks in IoT system is topology exploitation. This paper proposes a distributed algorithm for early detection of such attacks with the help of predictive descriptor tables. This paper also presents feature selection from topology control packet fields. The performance of the proposed algorithm is evaluated using an extensive simulation carried out in OMNeT++. Performance parameter includes accuracy and time required for detection. Simulation results presented in this paper show that the proposed algorithm is effective in detecting attacks ahead in time

Context Sensitive Search String Composition Algorithm using User Intention to Handle Ambiguous Keywords

Finding the required URL among the first few result pages of a search engine is still a challenging task. This may require number of reformulations of the search string thus adversely affecting user's search time. Query ambiguity and polysemy are major reasons for not obtaining relevant results in the top few result pages. Efficient query composition and data organization are necessary for getting effective results. Context of the information need and the user intent may improve the autocomplete feature of existing search engines. This research proposes a Funnel Mesh-5 algorithm (FM5) to construct a search string taking into account context of information need and user intention with three main steps 1) Predict user intention with user profiles and the past searches via weighted mesh structure 2) Resolve ambiguity and polysemy of search strings with context and user intention 3) Generate a personalized disambiguated search string by query expansion encompassing user intention and predicted query. Experimental results for the proposed approach and a comparison with direct use of search engine are presented. A comparison of FM5 algorithm with K Nearest Neighbor algorithm for user intention identification is also presented. The proposed system provides better precision for search results for ambiguous search strings with improved identification of the user intention. Results are presented for English language dataset as well as Marathi (an Indian language) dataset of ambiguous search strings.

Recommendations in Social Media Applications to Ensure Personification and Safety using Machine Learning

Myriads of social media utilization lead to various issues like personalization hacks, data security problems, and safety. A recommendation is of paramount importance to alleviate this problem when there is a huge amount of data and the number of participants on the platform is increasing exponentially. Unfortunately, modern social media research has enhanced the performance and personalization of recommendations in many fields, yet largely underutilizes the power of artificial intelligence to enable personalized recommendations system for social media platforms like WhatsApp, Facebook, Twitter, etc. With advancement inside the global of technology every hour and every day new features are delivered to the list.  In a manner, social platforms are merging into our actual existence, and to achieve personification and related safety, users can get any one safety factor from all 6 classes with this approach. This factor provides the basis for personification and the implementation of safety precautions. This research proposes recommendations for personification in social media applications. The proposed Modified Inception Resnet V4 Convolutional Neural Network (MInReCNN) outperforms embedded media persona analysis and classification through text, image, and video data. Using these prediction classes better decisions can be made in given social media domain

Matching aggregate posteriors in the variational autoencoder

The variational autoencoder (VAE) is a well-studied, deep, latent-variable model (DLVM) that efficiently optimizes the variational lower bound of the log marginal data likelihood and has a strong theoretical foundation. However, the VAE's known failure to match the aggregate posterior often results in \emph{pockets/holes} in the latent distribution (i.e., a failure to match the prior) and/or \emph{posterior collapse}, which is associated with a loss of information in the latent space. This paper addresses these shortcomings in VAEs by reformulating the objective function associated with VAEs in order to match the aggregate/marginal posterior distribution to the prior. We use kernel density estimate (KDE) to model the aggregate posterior in high dimensions. The proposed method is named the \emph{aggregate variational autoencoder} (AVAE) and is built on the theoretical framework of the VAE. Empirical evaluation of the proposed method on multiple benchmark data sets demonstrates the effectiveness of the AVAE relative to state-of-the-art (SOTA) methods

Matching shapes using the current distance

posterCurrent Distance: It was introduced by Vaillant and Glaunès as a way of comparing shapes (point sets, curves, surfaces). This distance measure is defined by viewing a shape as a linear operator on a k-form field, and constructing a (dual) norm on the space of shapes. Shape Matching: Given two shapes P;Q, a distance measure d on shapes, and a transformation group T , the problem of shape matching is to determine a transformation T that minimizes d(P; T Q). Current Norm: For a point set P, current norm is kPk2 = X i X j K(pi; pj)) (p) (q) Current Distance: Distance between two point sets P and Q is D2(P;Q) = kP + (??1)Qk2 = kPk2 + kQk2 ?? 2 X i X j K(pi; qj)) (p) (q) It takes O(n2) time to compute the current distance between two shapes of size n. Also current distance between 2 surfaces or curves can be reduced to set of distance computations on appropriately weighted point sets