14,139 research outputs found
Approximating Subadditive Hadamard Functions on Implicit Matrices
An important challenge in the streaming model is to maintain small-space
approximations of entrywise functions performed on a matrix that is generated
by the outer product of two vectors given as a stream. In other works, streams
typically define matrices in a standard way via a sequence of updates, as in
the work of Woodruff (2014) and others. We describe the matrix formed by the
outer product, and other matrices that do not fall into this category, as
implicit matrices. As such, we consider the general problem of computing over
such implicit matrices with Hadamard functions, which are functions applied
entrywise on a matrix. In this paper, we apply this generalization to provide
new techniques for identifying independence between two vectors in the
streaming model. The previous state of the art algorithm of Braverman and
Ostrovsky (2010) gave a -approximation for the distance
between the product and joint distributions, using space , where is the length of the stream and denotes the
size of the universe from which stream elements are drawn. Our general
techniques include the distance as a special case, and we give an
improved space bound of
Fast Hadamard transforms for compressive sensing of joint systems: measurement of a 3.2 million-dimensional bi-photon probability distribution
We demonstrate how to efficiently implement extremely high-dimensional
compressive imaging of a bi-photon probability distribution. Our method uses
fast-Hadamard-transform Kronecker-based compressive sensing to acquire the
joint space distribution. We list, in detail, the operations necessary to
enable fast-transform-based matrix-vector operations in the joint space to
reconstruct a 16.8 million-dimensional image in less than 10 minutes. Within a
subspace of that image exists a 3.2 million-dimensional bi-photon probability
distribution. In addition, we demonstrate how the marginal distributions can
aid in the accuracy of joint space distribution reconstructions
Total variation regularization for manifold-valued data
We consider total variation minimization for manifold valued data. We propose
a cyclic proximal point algorithm and a parallel proximal point algorithm to
minimize TV functionals with -type data terms in the manifold case.
These algorithms are based on iterative geodesic averaging which makes them
easily applicable to a large class of data manifolds. As an application, we
consider denoising images which take their values in a manifold. We apply our
algorithms to diffusion tensor images, interferometric SAR images as well as
sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds
(which includes the data space in diffusion tensor imaging) we show the
convergence of the proposed TV minimizing algorithms to a global minimizer
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