We propose a new analysis framework to utilize the full information of brain functional networks for computing the mean of a set of brain functional networks and embedding brain functional networks into a low-dimensional space in which traditional regression and classification analyses can be easily employed. For this, we first represent the brain functional network by a symmetric positive matr...

In this work we present a new generalization of the geometric mean of positive numbers on symmetric positive-definite matrices, called Log-Euclidean. The approach is based on two novel algebraic structures on symmetric positive-definite matrices: first, a lie group structure which is compatible with the usual algebraic properties of this matrix space; second, a new scalar multiplication that sm...

The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace-Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (u...

We study the problem of computing a low-distortion embedding between two metric spaces. More precisely given an input metric space M we are interested in computing in polynomial time an embedding into a host space M ′ with minimum multiplicative distortion. This problem arises naturally in many applications, including geometric optimization, visualization, multi-dimensional scaling, network spa...

In the Gaussian sequence model Y=?+?, we study likelihood ratio test (LRT) for testing H0:?=?0 versus H1:??K, where ?0?K, and K is a closed convex set in Rn. particular, show that under null hypothesis, normal approximation holds log-likelihood statistic general pair (?0,K), high-dimensional regime estimation error of associated least squares estimator diverges an appropriate sense. The further...

Let F be a flat vector bundle over a compact Riemannian manifold M and let f : M → R be a self-indexing Morse function. Let g be a smooth Euclidean metric on F , let g t = e g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g t . Assuming that ∇f satisfies the MorseSmale transversality conditions, we provide an asymptotic expansion for log ρ(t) for t → ∞ of the for...

Unsupervised phoneme segmentation aims at dividing a speech stream into phonemes without using any prior knowledge of linguistic contents and acoustic models. In [1], we formulated this problem into an optimization framework, and developed an objective function, summation of squared error (SSE) based on the Euclidean distance of cepstral features. However, it is unknown whether or not Euclidean...

The isotropic elastic moduli closest to a given anisotropic elasticity tensor are defined using three definitions of elastic distance, the standard Frobenius (Euclidean) norm, the Riemannian distance for tensors, and the log-Euclidean norm. The closest moduli are unique for the Riemannian and the log-Euclidean norms, independent of whether the difference in stiffness or compliance is considered...

The greedy spanner in a low dimensional Euclidean space is fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of good spanner: constant maximum degree and lightness. Recently, Eppstein Khodabandeh [28] showed \(\mathbb {R}^2 \) admits sublinear separator strong sense: any subgraph k vertices size \(O(\sqrt {k}) ....