65,960 research outputs found
A Polyhedral Method to Compute All Affine Solution Sets of Sparse Polynomial Systems
To compute solutions of sparse polynomial systems efficiently we have to
exploit the structure of their Newton polytopes. While the application of
polyhedral methods naturally excludes solutions with zero components, an
irreducible decomposition of a variety is typically understood in affine space,
including also those components with zero coordinates. We present a polyhedral
method to compute all affine solution sets of a polynomial system. The method
enumerates all factors contributing to a generalized permanent. Toric solution
sets are recovered as a special case of this enumeration. For sparse systems as
adjacent 2-by-2 minors our methods scale much better than the techniques from
numerical algebraic geometry
Semi-Supervised First-Person Activity Recognition in Body-Worn Video
Body-worn cameras are now commonly used for logging daily life, sports, and
law enforcement activities, creating a large volume of archived footage. This
paper studies the problem of classifying frames of footage according to the
activity of the camera-wearer with an emphasis on application to real-world
police body-worn video. Real-world datasets pose a different set of challenges
from existing egocentric vision datasets: the amount of footage of different
activities is unbalanced, the data contains personally identifiable
information, and in practice it is difficult to provide substantial training
footage for a supervised approach. We address these challenges by extracting
features based exclusively on motion information then segmenting the video
footage using a semi-supervised classification algorithm. On publicly available
datasets, our method achieves results comparable to, if not better than,
supervised and/or deep learning methods using a fraction of the training data.
It also shows promising results on real-world police body-worn video
Symmetry groups, semidefinite programs, and sums of squares
We investigate the representation of symmetric polynomials as a sum of
squares. Since this task is solved using semidefinite programming tools we
explore the geometric, algebraic, and computational implications of the
presence of discrete symmetries in semidefinite programs. It is shown that
symmetry exploitation allows a significant reduction in both matrix size and
number of decision variables. This result is applied to semidefinite programs
arising from the computation of sum of squares decompositions for multivariate
polynomials. The results, reinterpreted from an invariant-theoretic viewpoint,
provide a novel representation of a class of nonnegative symmetric polynomials.
The main theorem states that an invariant sum of squares polynomial is a sum of
inner products of pairs of matrices, whose entries are invariant polynomials.
In these pairs, one of the matrices is computed based on the real irreducible
representations of the group, and the other is a sum of squares matrix. The
reduction techniques enable the numerical solution of large-scale instances,
otherwise computationally infeasible to solve.Comment: 38 pages, submitte
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