Tangent space estimation for smooth embeddings of Riemannian manifolds

Numerous dimensionality reduction problems in data analysis involve the recovery of low-dimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the number of samples in the neighborhood (sampling density) affect the performance of learning algorithms. In this work, we propose a theoretical analysis of local sampling conditions for the estimation of the tangent space at a point P lying on a m-dimensional Riemannian manifold S in R^n. Assuming a smooth embedding of S in R^n, we estimate the tangent space T_P S by performing a Principal Component Analysis (PCA) on points sampled from the neighborhood of P on S. Our analysis explicitly takes into account the second order properties of the manifold at P, namely the principal curvatures as well as the higher order terms. We consider a random sampling framework and leverage recent results from random matrix theory to derive conditions on the sampling width and the local sampling density for an accurate estimation of tangent subspaces. We measure the estimation accuracy by the angle between the estimated tangent space and the true tangent space T_P S and we give conditions for this angle to be bounded with high probability. In particular, we observe that the local sampling conditions are highly dependent on the correlation between the components in the second-order local approximation of the manifold. We finally provide numerical simulations to validate our theoretical findings

Sparse Representation Based SAR Vehicle Recognition along with Aspect Angle

As a method of representing the test sample with few training samples from an overcomplete dictionary, sparse representation classification (SRC) has attracted much attention in synthetic aperture radar (SAR) automatic target recognition (ATR) recently. In this paper, we develop a novel SAR vehicle recognition method based on sparse representation classification along with aspect information (SRCA), in which the correlation between the vehicle’s aspect angle and the sparse representation vector is exploited. The detailed procedure presented in this paper can be summarized as follows. Initially, the sparse representation vector of a test sample is solved by sparse representation algorithm with a principle component analysis (PCA) feature-based dictionary. Then, the coefficient vector is projected onto a sparser one within a certain range of the vehicle’s aspect angle. Finally, the vehicle is classified into a certain category that minimizes the reconstruction error with the novel sparse representation vector. Extensive experiments are conducted on the moving and stationary target acquisition and recognition (MSTAR) dataset and the results demonstrate that the proposed method performs robustly under the variations of depression angle and target configurations, as well as incomplete observation

Learning How a Tool Affords by Simulating 3D Models from the Web

Thanks to: UoAs ABVenture Zone, N. Petkov, K. Georgiev, B. Nougier, S. Fichtl, S. Ramamoorthy, M. Beetz, A. Haidu, J. Alexander, M. Schoeler, N. Pugeault, D. Cruickshank, M. Chung and N. Khan. Paulo Abelha is on a PhD studentship supported by the Brazilian agency CAPES through the program Science without Borders. Frank Guerin received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Published in: 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) DOI: 10.1109/IROS.2017.8206372 Date of Conference: 24-28 Sept. 2017 Conference Location: Vancouver, BC, Canada.Postprin