Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mode-Seeking on Hypergraphs for Robust Geometric Model Fitting

In this paper, we propose a novel geometric model fitting method, called Mode-Seeking on Hypergraphs (MSH),to deal with multi-structure data even in the presence of severe outliers. The proposed method formulates geometric model fitting as a mode seeking problem on a hypergraph in which vertices represent model hypotheses and hyperedges denote data points. MSH intuitively detects model instances by a simple and effective mode seeking algorithm. In addition to the mode seeking algorithm, MSH includes a similarity measure between vertices on the hypergraph and a weight-aware sampling technique. The proposed method not only alleviates sensitivity to the data distribution, but also is scalable to large scale problems. Experimental results further demonstrate that the proposed method has significant superiority over the state-of-the-art fitting methods on both synthetic data and real images.Comment: Proceedings of the IEEE International Conference on Computer Vision, pp. 2902-2910, 201

Spectral Templates from Multicolor Redshift Surveys

Understanding how the physical properties of galaxies (e.g. their spectral type or age) evolve as a function of redshift relies on having an accurate representation of galaxy spectral energy distributions. While it has been known for some time that galaxy spectra can be reconstructed from a handful of orthogonal basis templates, the underlying basis is poorly constrained. The limiting factor has been the lack of large samples of galaxies (covering a wide range in spectral type) with high signal-to-noise spectrophotometric observations. To alleviate this problem we introduce here a new technique for reconstructing galaxy spectral energy distributions directly from samples of galaxies with broadband photometric data and spectroscopic redshifts. Exploiting the statistical approach of the Karhunen-Loeve expansion, our iterative training procedure increasingly improves the eigenbasis, so that it provides better agreement with the photometry. We demonstrate the utility of this approach by applying these improved spectral energy distributions to the estimation of photometric redshifts for the HDF sample of galaxies. We find that in a small number of iterations the dispersion in the photometric redshifts estimator (a comparison between predicted and measured redshifts) can decrease by up to a factor of 2.Comment: 25 pages, 9 figures, LaTeX AASTeX, accepted for publication in A

Empirical modeling of the stellar spectrum of galaxies

An empirical method of modeling the stellar spectrum of galaxies is proposed, based on two successive applications of Principal Component Analysis (PCA). PCA is first applied to the newly available stellar library STELIB, supplemented by the J, H and K$_{s}$ magnitudes taken mainly from the 2 Micron All Sky Survey (2MASS). Next the resultant eigen-spectra are used to fit the observed spectra of a sample of 1016 galaxies selected from the Sloan Digital Sky Survey Data Release One (SDSS DR1). PCA is again applied, to the fitted spectra to construct the eigen-spectra of galaxies with zero velocity dispersion. The first 9 galactic eigen-spectra so obtained are then used to model the stellar spectrum of the galaxies in SDSS DR1, and synchronously to estimate the stellar velocity dispersion, the spectral type, the near-infrared SED, and the average reddening. Extensive tests show that the spectra of different type galaxies can be modeled quite accurately using these eigen-spectra. The method can yield stellar velocity dispersion with accuracies better than 10%, for the spectra of typical S/N ratios in SDSS DR1.Comment: 34 pages with 18 figures, submitted to A