Statistics on special manifolds

By Y. Chikuse. 2003

416 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/093869a6f8f229a6e2e9ad15cd5e666173e66456

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In Artificial Intelligence

Citations in the corpus

Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow (2007, positive): “Fortunately, the availability of kernel-based methods for estimating probability densities deﬁned over nonlinear manifolds makes it possible to apply our approach in this situation, as well [41].”
MINIMAX SPARSE PRINCIPAL SUBSPACE ESTIMATION IN HIGH DIMENSIONS (2012, background): “[For background on Grassmann and Stiefel manifolds, see Edelman, Arias and Smith (1999), Chikuse (2003).]”
Instrumental Variable Estimation with Heteroskedasticity and Many Instruments (2012, neutral): “17) of Chikuse (2003) , we obtain”
High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation (2010, background): “, Haar) distributed on the orthogonal group [see also Chikuse (2003), page 40, equation (2.”
Fisher Lecture: Dimension Reduction in Regression (2007, methodology): “For background on Grassmann manifolds, see Edelman, Arias and Smith (1998) and Chikuse (2003). Estimation of a sufficient reduction might be based on the method of moments, Bayesian considerations or a concern for robustness, but staying with Fisher we will use the method of maximum likelihood.”
Locally time-invariant models of human activities using trajectories on the grassmannian (2009, methodology): “Using the Procrustes representation and distance metric, the density estimate is given by [2] as”
Likelihood-Based Sufficient Dimension Reduction (2009, background): “The dimension of Gðd;pÞ is pd d2 ¼ d(p d), because a plane is invariant under nonsingular right-side linear transformations of its basis matrix (Chikuse 2003).”
Sure independence screening for ultrahigh dimensional feature space (2006, background): “ove. This invariant distribution is referred to as the uniform distribution on the Grassmann manifold Gpn,n. For more on group action and invariant measures on special manifolds, see Eaton (1989) and Chikuse (2003). The uniform distribution on the Grassmann manifold is not easy to deal with directly”
Model Averaging and Dimension Selection for the Singular Value Decomposition (2006, background): “Because this probability distribution is invariant under left- and right-orthogonal transformations of U (see, e.g., Chikuse 2003), it follows that the rows and columns of U are exchangeable.”
Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization (2009, neutral): “, [11] Section 1.”
Face recognition in unconstrained videos with matched background similarity (2011, neutral): “Several distances are defined based on these notations: The max correlation is defined by the minimal angle θ(1); The projection metricis given by ||U1U 1 − U2U 2 ||F = || sin Θ||; The norm ||U> 1 U2||F seems to be relatively effective; Finally, the Procrustes metric [1] is computed from the vector-”
A Blur-Robust Descriptor with Applications to Face Recognition (2012, background): “…study 1) the effect of q on (rank-1) recognition when blur is the only source of variation between the gallery and probe (i.e., f ¼ 0), and 2) when there are other facial variations f between the gallery and probe, where we analyze the role of learning from data representative of such conditions.”
Statistical analysis on high-dimensional spheres and shape spaces (2005, background): “a uniformly distributed matrix X on the Stiefel manifold Vm,p tend to an hm-dimensional zero-mean Gaussian distribution with identity covariance matrix as p → ∞. Chikuse [6, 7 ] provides many extensions.”
A statistical approach to persistent homology (2006, methodology): “diﬀerent distributions for a given manifold, as well as sample spaces that are diﬀerent manifolds. References that discuss these topics can be found in the books by Mardia and Jupp [MJ00] and Chikuse [Chi03]. Furthermore, although nonparametric statistical procedures on compact Riemannian manifolds ”
High dimensional ordinary least squares projection for screening variables (2015, background): “(Chikuse, 2003, Page 41-44) Supposed that a p × n random matrix Z has the density function of the form which is invariant under the right-orthogonal transformation of Z , where Σ is a p × p positive deﬁnite matrix.”
Simulation of the Matrix Bingham–von Mises–Fisher Distribution, With Applications to Multivariate and Relational Data (2007, methodology): “Following chapter 3 of Chikuse (2003), the conditional distribution of z given X[,−1] is given by”