773 research outputs found
On perfect hashing of numbers with sparse digit representation via multiplication by a constant
Consider the set of vectors over a field having non-zero coefficients only in
a fixed sparse set and multiplication defined by convolution, or the set of
integers having non-zero digits (in some base ) in a fixed sparse set. We
show the existence of an optimal (resp. almost-optimal in the latter case)
`magic' multiplier constant that provides a perfect hash function which
transfers the information from the given sparse coefficients into consecutive
digits. Studying the convolution case we also obtain a result of non-degeneracy
for Schur functions as polynomials in the elementary symmetric functions in
positive characteristic.Comment: 5 page
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
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