Expanding the Family of Grassmannian Kernels: An Embedding Perspective

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g, support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as we will show- is universal, which limits their ability to approximate a target function arbitrarily well. Here, we introduce several positive definite Grassmannian kernels, including universal ones, and demonstrate their superiority over previously-known kernels in various tasks, such as classification, clustering, sparse coding and hashing

Observations of the post shock break-out emission of SN 2011dh with XMM-Newton

After the occurrence of the type cIIb SN 2011dh in the nearby spiral galaxy M 51 numerous observations were performed with different telescopes in various bands ranging from radio to gamma-rays. We analysed the XMM-Newton and Swift observations taken 3 to 30 days after the SN explosion to study the X-ray spectrum of SN 2011dh. We extracted spectra from the XMM-Newton observations, which took place ~7 and 11 days after the SN. In addition, we created integrated Swift/XRT spectra of 3 to 10 days and 11 to 30 days. The spectra are well fitted with a power-law spectrum absorbed with Galactic foreground absorption. In addition, we find a harder spectral component in the first XMM-Newton spectrum taken at t ~ 7 d. This component is also detected in the first Swift spectrum of t = 3 - 10 d. While the persistent power-law component can be explained as inverse Compton emission from radio synchrotron emitting electrons, the harder component is most likely bremsstrahlung emission from the shocked stellar wind. Therefore, the harder X-ray emission that fades away after t ~ 10 d can be interpreted as emission from the shocked circumstellar wind of SN 2011dh.Comment: Accepted for publication as a Research Note in Astronomy and Astrophysic

Interferometric Space Missions for the Search for Terrestrial Exoplanets: Requirements on the Rejection Ratio

The requirements on space missions designed to study Terrestrial exoplanets are discussed. We then investigate whether the design of such a mission, specifically the Darwin nulling interferometer, can be carried out in a simplified scenario. The key element here is accepting somewhat higher levels of stellar leakage. We establish detailed requirements resulting from the scientific rationale for the mission, and calculate detailed parameters for the stellar suppression required to achieve those requirements. We do this utilizing the Darwin input catalogue. The dominating noise source for most targets in this sample is essentially constant for all targets, while the leakage diminishes with the square of the distance. This means that the stellar leakage has an effect on the integration time only for the nearby stars, while for the more distant targets its influence decreases significantly. We assess the impact of different array configurations and nulling profiles and identify the stars for which the detection efficiency can be maximized.Comment: 21 pages, 8 figures; TBP in Astrophysics and Space Science 200

Interpolation and Regression of Rotation Matrices

The problem of fitting smooth curves to data on the group of rotations is considered. This problem arises when resampling or denoising data points that consist in rotation matrices measured at different times. The rotation matrices typically correspond to the orientation of some physical object, such as a camera or a flying or submarine device. We propose to compute sequences of rotations (discretized curves) that strike a tunable balance between data fidelity and smoothness, where smoothness is assessed by means of a proposed notion of velocity and acceleration along discrete curves on the group of rotations. The best such curve is obtained via optimization on a manifold. Leveraging the simplicity of the cost, we present an efficient algorithm based on second-order Riemannian trust-region methods, implemented using the Manopt toolbox

Soft dimension reduction for ICA by joint diagonalization on the Stiefel manifold

Joint diagonalization for ICA is often performed on the orthogonal group after a pre-whitening step. Here we assume that we only want to extract a few sources after pre-whitening, and hence work on the Stiefel manifold of $p$-frames in $R^n$. The resulting method does not only use second-order statistics to estimate the dimension reduction and is therefore denoted as soft dimension reduction. We employ a trust-region method for minimizing the cost function on the Stiefel manifold. Applications to a toy example and functional MRI data show a higher numerical efficiency, especially when $p$ is much smaller than $n$, and more robust performance in the presence of strong noise than methods based on pre-whitening

A Modified Particle Swarm Optimization Algorithm for the Best Low Multilinear Rank Approximation of Higher-Order Tensors

Abstract. The multilinear rank of a tensor is one of the possible gener-alizations for the concept of matrix rank. In this paper, we are interested in finding the best low multilinear rank approximation of a given ten-sor. This problem has been formulated as an optimization problem over the Grassmann manifold [14] and it has been shown that the objec-tive function presents multiple minima [15]. In order to investigate the landscape of this cost function, we propose an adaptation of the Parti-cle Swarm Optimization algorithm (PSO). The Guaranteed Convergence PSO, proposed by van den Bergh in [23], is modified, including a gradi-ent component, so as to search for optimal solutions over the Grassmann manifold. The operations involved in the PSO algorithm are redefined using concepts of differential geometry. We present some preliminary nu-merical experiments and we discuss the ability of the proposed method to address the multimodal aspects of the studied problem

Convergence of Linesearch and Trust-Region Methods Using the Kurdyka-Łojasiewicz Inequality

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