4,562 research outputs found
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 Pic b, which is 9 AU from and embedded in the debris disk of the
young nearby A6V star 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 Pic b at 3.1 m, 3.3 m, , and
, including the first observation in the fundamental CH 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 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(/) . From our
bolometric luminosity and an age of 233 Myr, hot-start evolutionary tracks
give a mass of 12.70.3 , radius of 1.450.02 , and
of 170823 K (model-dependent errors not included). Our
empirically-determined luminosity is in agreement with values from atmospheric
models (typically dex), but brighter than values from the field-dwarf
bolometric correction (typically 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
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