Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text.

Template fits to observed galaxy fluxes allow calculation of K-corrections and conversions among observations of galaxies at various wavelengths. We present a method for creating model-based template sets given a set of heterogeneous photometric and spectroscopic galaxy data. Our technique, nonnegative matrix factorization, is akin to principal component analysis (PCA), except that it is constrained to produce nonnegative templates, it can use a basis set of models (rather than the delta-function basis of PCA), and it naturally handles uncertainties, missing data, and heterogeneous data (including broadband fluxes at various redshifts). The particular implementation we present here is suitable for ultraviolet, optical, and near-infrared observations in the redshift range 0 < z < 1:5. Since we base our templates on stellar population synthesis models, the results are interpretable in terms of approximate stellar masses and star formation histories. We present templates fitted with this method to data from Galaxy Evolution Explorer, Sloan Digital Sky Survey spectroscopy and photometry, the Two Micron All Sky Survey, the Deep Extragalactic Evolutionary Probe, and the Great Observatories Origins Deep Survey. In addition, we present software for using such data to estimate K-corrections.

Figure 1: Left: An image spoiled by camera shake. Middle: result from Photoshop "unsharp mask". Right: result from our algorithm. AbstractCamera shake during exposure leads to objectionable image blur and ruins many photographs. Conventional blind deconvolution methods typically assume frequency-domain constraints on images, or overly simplified parametric forms for the motion path during camera shake. Real camera motions can follow convoluted paths, and a spatial domain prior can better maintain visually salient image characteristics. We introduce a method to remove the effects of camera shake from seriously blurred images. The method assumes a uniform camera blur over the image and negligible in-plane camera rotation. In order to estimate the blur from the camera shake, the user must specify an image region without saturation effects. We show results for a variety of digital photographs taken from personal photo collections.