Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $\rho>0$ such that for any $x\in M$, the open $\rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_\rho,x^*)\to (B_\rho(x),x)$, has the maximal volume i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].Comment: The only change from the early version is an improvement on Theorem A: we replace the non-collapsing condition on $M$ by on $\tilde M$ (the Riemannian universal cover), and the corresponding modification is adding "subsection c" in Section

Quadratic Projection Based Feature Extraction with Its Application to Biometric Recognition

This paper presents a novel quadratic projection based feature extraction framework, where a set of quadratic matrices is learned to distinguish each class from all other classes. We formulate quadratic matrix learning (QML) as a standard semidefinite programming (SDP) problem. However, the con- ventional interior-point SDP solvers do not scale well to the problem of QML for high-dimensional data. To solve the scalability of QML, we develop an efficient algorithm, termed DualQML, based on the Lagrange duality theory, to extract nonlinear features. To evaluate the feasibility and effectiveness of the proposed framework, we conduct extensive experiments on biometric recognition. Experimental results on three representative biometric recogni- tion tasks, including face, palmprint, and ear recognition, demonstrate the superiority of the DualQML-based feature extraction algorithm compared to the current state-of-the-art algorithm

A Geometric Approach to the Modified Milnor Problem

The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent to the Nilpotency Conjecture in Riemannian geometry: given $n, d>0$, there exists a constant $\epsilon(n,d)>0$ such that if a compact Riemannian $n$-manifold $M$ satisfies that Ricci curvature \op{Ric}_M\ge -(n-1), diameter d\ge \op{diam}(M) and volume entropy $h(M)<\epsilon(n,d)$, then the fundamental group $\pi_1(M)$ is virtually nilpotent. We will verify the Nilpotency Conjecture in some cases, and we will verify the vanishing gap phenomena for more cases i.e., if $h(M)<\epsilon(n,d)$, then $h(M)=0$.Comment: 25 page

Logistic Regression Based on Statistical Learning Model with Linearized Kernel for Classification

In this paper, we propose a logistic regression classification method based on the integration of a statistical learning model with linearized kernel pre-processing. The single Gaussian kernel and fusion of Gaussian and cosine kernels are adopted for linearized kernel pre-processing respectively. The adopted statistical learning models are the generalized linear model and the generalized additive model. Using a generalized linear model, the elastic net regularization is adopted to explore the grouping effect of the linearized kernel feature space. Using a generalized additive model, an overlap group-lasso penalty is used to fit the sparse generalized additive functions within the linearized kernel feature space. Experiment results on the Extended Yale-B face database and AR face database demonstrate the effectiveness of the proposed method. The improved solution is also efficiently obtained using our method on the classification of spectra data