1,476 research outputs found
Perturbation of matrices and non-negative rank with a view toward statistical models
In this paper we study how perturbing a matrix changes its non-negative rank.
We prove that the non-negative rank is upper-semicontinuos and we describe some
special families of perturbations. We show how our results relate to Statistics
in terms of the study of Maximum Likelihood Estimation for mixture models.Comment: 13 pages, 3 figures. A theorem has been rewritten, and some
improvements in the presentations have been implemente
Fixed points of the EM algorithm and nonnegative rank boundaries
Mixtures of independent distributions for two discrete random variables
can be represented by matrices of nonnegative rank . Likelihood inference
for the model of such joint distributions leads to problems in real algebraic
geometry that are addressed here for the first time. We characterize the set of
fixed points of the Expectation-Maximization algorithm, and we study the
boundary of the space of matrices with nonnegative rank at most . Both of
these sets correspond to algebraic varieties with many irreducible components.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1282 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or
implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k))
floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
Low-rank matrix approximations, such as the truncated singular value
decomposition and the rank-revealing QR decomposition, play a central role in
data analysis and scientific computing. This work surveys and extends recent
research which demonstrates that randomization offers a powerful tool for
performing low-rank matrix approximation. These techniques exploit modern
computational architectures more fully than classical methods and open the
possibility of dealing with truly massive data sets.
This paper presents a modular framework for constructing randomized
algorithms that compute partial matrix decompositions. These methods use random
sampling to identify a subspace that captures most of the action of a matrix.
The input matrix is then compressed---either explicitly or implicitly---to this
subspace, and the reduced matrix is manipulated deterministically to obtain the
desired low-rank factorization. In many cases, this approach beats its
classical competitors in terms of accuracy, speed, and robustness. These claims
are supported by extensive numerical experiments and a detailed error analysis
Toward Guaranteed Illumination Models for Non-Convex Objects
Illumination variation remains a central challenge in object detection and
recognition. Existing analyses of illumination variation typically pertain to
convex, Lambertian objects, and guarantee quality of approximation in an
average case sense. We show that it is possible to build V(vertex)-description
convex cone models with worst-case performance guarantees, for non-convex
Lambertian objects. Namely, a natural verification test based on the angle to
the constructed cone guarantees to accept any image which is sufficiently
well-approximated by an image of the object under some admissible lighting
condition, and guarantees to reject any image that does not have a sufficiently
good approximation. The cone models are generated by sampling point
illuminations with sufficient density, which follows from a new perturbation
bound for point images in the Lambertian model. As the number of point images
required for guaranteed verification may be large, we introduce a new
formulation for cone preserving dimensionality reduction, which leverages tools
from sparse and low-rank decomposition to reduce the complexity, while
controlling the approximation error with respect to the original cone
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