99 research outputs found
Capturing a phylogenetic tree when the number of character states varies with the number of leaves
We show that for any two values for which
then there is a value so that for all the
following holds. For any binary phylogenetic tree on leaves there is a
set of characters that capture , and for which
each character takes at most distinct states. Here
`capture' means that is the unique perfect phylogeny for these characters.
Our short proof of this combinatorial result is based on the probabilistic
method.Comment: 3 pages, 0 figure
Large Alphabets and Incompressibility
We briefly survey some concepts related to empirical entropy -- normal
numbers, de Bruijn sequences and Markov processes -- and investigate how well
it approximates Kolmogorov complexity. Our results suggest th-order
empirical entropy stops being a reasonable complexity metric for almost all
strings of length over alphabets of size about when surpasses
Phase retrieval from power spectra of masked signals
In diffraction imaging, one is tasked with reconstructing a signal from its
power spectrum. To resolve the ambiguity in this inverse problem, one might
invoke prior knowledge about the signal, but phase retrieval algorithms in this
vein have found limited success. One alternative is to create redundancy in the
measurement process by illuminating the signal multiple times, distorting the
signal each time with a different mask. Despite several recent advances in
phase retrieval, the community has yet to construct an ensemble of masks which
uniquely determines all signals and admits an efficient reconstruction
algorithm. In this paper, we leverage the recently proposed polarization method
to construct such an ensemble. We also present numerical simulations to
illustrate the stability of the polarization method in this setting. In
comparison to a state-of-the-art phase retrieval algorithm known as PhaseLift,
we find that polarization is much faster with comparable stability.Comment: 18 pages, 3 figure
- …