Randomized hybrid linear modeling by local best-fit flats

The hybrid linear modeling problem is to identify a set of d-dimensional affine sets in a D-dimensional Euclidean space. It arises, for example, in object tracking and structure from motion. The hybrid linear model can be considered as the second simplest (behind linear) manifold model of data. In this paper we will present a very simple geometric method for hybrid linear modeling based on selecting a set of local best fit flats that minimize a global l1 error measure. The size of the local neighborhoods is determined automatically by the Jones' l2 beta numbers; it is proven under certain geometric conditions that good local neighborhoods exist and are found by our method. We also demonstrate how to use this algorithm for fast determination of the number of affine subspaces. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the algorithm on synthetic and real hybrid linear data.Comment: To appear in the proceedings of CVPR 201

Median K-flats for hybrid linear modeling with many outliers

We describe the Median K-Flats (MKF) algorithm, a simple online method for hybrid linear modeling, i.e., for approximating data by a mixture of flats. This algorithm simultaneously partitions the data into clusters while finding their corresponding best approximating l1 d-flats, so that the cumulative l1 error is minimized. The current implementation restricts d-flats to be d-dimensional linear subspaces. It requires a negligible amount of storage, and its complexity, when modeling data consisting of N points in D-dimensional Euclidean space with K d-dimensional linear subspaces, is of order O(n K d D+n d^2 D), where n is the number of iterations required for convergence (empirically on the order of 10^4). Since it is an online algorithm, data can be supplied to it incrementally and it can incrementally produce the corresponding output. The performance of the algorithm is carefully evaluated using synthetic and real data

Robust Recovery of Subspace Structures by Low-Rank Representation

In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed Low-Rank Representation (LRR), which seeks the lowest-rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.Comment: IEEE Trans. Pattern Analysis and Machine Intelligenc

Magellan Adaptive Optics first-light observations of the exoplanet beta Pic b. II. 3-5 micron direct imaging with MagAO+Clio, and the empirical bolometric luminosity of a self-luminous giant planet

Young giant exoplanets are a unique laboratory for understanding cool, low-gravity atmospheres. A quintessential example is the massive extrasolar planet $\beta$ Pic b, which is 9 AU from and embedded in the debris disk of the young nearby A6V star $\beta$ Pictoris. We observed the system with first light of the Magellan Adaptive Optics (MagAO) system. In Paper I we presented the first CCD detection of this planet with MagAO+VisAO. Here we present four MagAO+Clio images of $\beta$ Pic b at 3.1 $\mu$m, 3.3 $\mu$m, $L^\prime$, and $M^\prime$, including the first observation in the fundamental CH$_4$ band. To remove systematic errors from the spectral energy distribution (SED), we re-calibrate the literature photometry and combine it with our own data, for a total of 22 independent measurements at 16 passbands from 0.99--4.8 $\mu$m. Atmosphere models demonstrate the planet is cloudy but are degenerate in effective temperature and radius. The measured SED now covers $>$80\% of the planet's energy, so we approach the bolometric luminosity empirically. We calculate the luminosity by extending the measured SED with a blackbody and integrating to find log($L_{bol}$/$L_{Sun}$) $= -3.78\pm0.03$. From our bolometric luminosity and an age of 23$\pm$3 Myr, hot-start evolutionary tracks give a mass of 12.7$\pm$0.3 $M_{Jup}$, radius of 1.45$\pm$0.02 $R_{Jup}$, and $T_{eff}$ of 1708$\pm$23 K (model-dependent errors not included). Our empirically-determined luminosity is in agreement with values from atmospheric models (typically $-3.8$ dex), but brighter than values from the field-dwarf bolometric correction (typically $-3.9$ dex), illustrating the limitations in comparing young exoplanets to old brown dwarfs.Comment: Accepted to ApJ. 27 pages, 22 figures, 19 table

Least squares approximations of measures via geometric condition numbers

For a probability measure on a real separable Hilbert space, we are interested in "volume-based" approximations of the d-dimensional least squares error of it, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of that measure, where the comparison depends on a simple quantitative geometric property of it. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of measure. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo SVD based on volume sampling

Sparse Subspace Clustering: Algorithm, Theory, and Applications

In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we propose and study an algorithm, called Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of subspaces and the distribution of data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm can be solved efficiently and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal with data nuisances, such as noise, sparse outlying entries, and missing entries, directly by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering

SN 2006bp: Probing the Shock Breakout of a Type II-P Supernova

HET optical spectroscopy and unfiltered ROTSE-III photometry spanning the first 11 months since explosion of the Type II-P SN 2006bp are presented. Flux limits from the days before discovery combined with the initial rapid brightening suggest the supernova was first detected just hours after shock breakout. Optical spectra obtained about 2 days after breakout exhibit narrow emission lines corresponding to HeII 4200, HeII 4686, and CIV 5805 in the rest frame, and these features persist in a second observation obtained 5 hours later; however, these emission lines are not detected the following night nor in subsequent observations. We suggest that these lines emanate from material close to the explosion site, possibly in the outer layers of the progenitor that have been ionized by the high energy photons released at shock breakout. A P-Cygni profile is observed around 4450 A in the +2 and +3 day spectra. Previous studies have attributed this feature to high velocity H-beta, but we discuss the possibility that this profile is instead due to HeII 4687. Further HET observations (14 nights in total) covering the spectral evolution across the photometric plateau up to 73 days after breakout and during the nebular phase around day +340 are presented, and expansion velocities are derived for key features. The measured decay slope for the unfiltered light curve is 0.0073 +/- 0.0004 mag/day between days +121 and +335, which is significantly slower than the decay of rate 56Co. We combine our HET measurements with published X-ray, UV, and optical data to obtain a quasi-bolometric light curve through day +60. We see a slow cooling over the first 25 days, but no sign of an early sharp peak; any such feature from the shock breakout must have lasted less than ~1 day.[ABRIDGED]Comment: ApJ accepted, 43 page

lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers

We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the lp-averaged distances of data points from d-dimensional subspaces, where p>0. Unlike other lp minimization problems, this minimization is non-convex for all p>0 and thus requires different methods for its analysis. We show that if 0<p<=1, then for any fraction of outliers the most significant subspace can be recovered by lp minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces the most significant subspace can be nearly recovered by lp minimization for any 0<p<=1 with an error proportional to the noise level. On the other hand, if p>1 and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.Comment: This is a revised version of the part of 1002.1994 that deals with single subspace recovery. V3: Improved estimates (in particular for Lemma 3.1 and for estimates relying on it), asymptotic dependence of probabilities and constants on D and d and further clarifications; for simplicity it assumes uniform distributions on spheres. V4: minor revision for the published versio