96 research outputs found
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
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
Kernel Spectral Curvature Clustering (KSCC)
Multi-manifold modeling is increasingly used in segmentation and data
representation tasks in computer vision and related fields. While the general
problem, modeling data by mixtures of manifolds, is very challenging, several
approaches exist for modeling data by mixtures of affine subspaces (which is
often referred to as hybrid linear modeling). We translate some important
instances of multi-manifold modeling to hybrid linear modeling in embedded
spaces, without explicitly performing the embedding but applying the kernel
trick. The resulting algorithm, Kernel Spectral Curvature Clustering, uses
kernels at two levels - both as an implicit embedding method to linearize
nonflat manifolds and as a principled method to convert a multiway affinity
problem into a spectral clustering one. We demonstrate the effectiveness of the
method by comparing it with other state-of-the-art methods on both synthetic
data and a real-world problem of segmenting multiple motions from two
perspective camera views.Comment: accepted to 2009 ICCV Workshop on Dynamical Visio
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
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
Subspace Clustering via Optimal Direction Search
This letter presents a new spectral-clustering-based approach to the subspace
clustering problem. Underpinning the proposed method is a convex program for
optimal direction search, which for each data point d finds an optimal
direction in the span of the data that has minimum projection on the other data
points and non-vanishing projection on d. The obtained directions are
subsequently leveraged to identify a neighborhood set for each data point. An
alternating direction method of multipliers framework is provided to
efficiently solve for the optimal directions. The proposed method is shown to
notably outperform the existing subspace clustering methods, particularly for
unwieldy scenarios involving high levels of noise and close subspaces, and
yields the state-of-the-art results for the problem of face clustering using
subspace segmentation
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