855 research outputs found
Small orders of Hadamard matrices and base sequences
We update the list of odd integers n<10000 for which an Hadamard matrix of
order 4n is known to exist. We also exhibit the first example of base sequences
BS(40,39). Consequently, there exist T-sequences TS(n) of length n=79. The
first undecided case has the length n=97.Comment: 7 page
Scampi: a robust approximate message-passing framework for compressive imaging
Reconstruction of images from noisy linear measurements is a core problem in
image processing, for which convex optimization methods based on total
variation (TV) minimization have been the long-standing state-of-the-art. We
present an alternative probabilistic reconstruction procedure based on
approximate message-passing, Scampi, which operates in the compressive regime,
where the inverse imaging problem is underdetermined. While the proposed method
is related to the recently proposed GrAMPA algorithm of Borgerding, Schniter,
and Rangan, we further develop the probabilistic approach to compressive
imaging by introducing an expectation-maximizaiton learning of model
parameters, making the Scampi robust to model uncertainties. Additionally, our
numerical experiments indicate that Scampi can provide reconstruction
performance superior to both GrAMPA as well as convex approaches to TV
reconstruction. Finally, through exhaustive best-case experiments, we show that
in many cases the maximal performance of both Scampi and convex TV can be quite
close, even though the approaches are a prori distinct. The theoretical reasons
for this correspondence remain an open question. Nevertheless, the proposed
algorithm remains more practical, as it requires far less parameter tuning to
perform optimally.Comment: Presented at the 2015 International Meeting on High-Dimensional Data
Driven Science, Kyoto, Japa
Implementing Hadamard Matrices in SageMath
Hadamard matrices are square matrices with mutually orthogonal
rows. The Hadamard conjecture states that Hadamard matrices of order exist
whenever is , , or a multiple of . However, no construction is
known that works for all values of , and for some orders no Hadamard matrix
has yet been found. Given the many practical applications of these matrices, it
would be useful to have a way to easily check if a construction for a Hadamard
matrix of order exists, and in case to create it. This project aimed to
address this, by implementing constructions of Hadamard and skew Hadamard
matrices to cover all known orders less than or equal to in SageMath, an
open-source mathematical software. Furthermore, we implemented some additional
mathematical objects, such as complementary difference sets and T-sequences,
which were not present in SageMath but are needed to construct Hadamard
matrices.
This also allows to verify the correctness of the results given in the
literature; within the range, just one order, , of a skew
Hadamard matrix claimed to have a known construction, required a fix.Comment: pdflatex+biber, 32 page
Almost-Fisher families
A classic theorem in combinatorial design theory is Fisher's inequality,
which states that a family of subsets of with all pairwise
intersections of size can have at most non-empty sets. One may
weaken the condition by requiring that for every set in , all but
at most of its pairwise intersections have size . We call such
families -almost -Fisher. Vu was the first to study the maximum
size of such families, proving that for the largest family has
sets, and characterising when equality is attained. We substantially refine his
result, showing how the size of the maximum family depends on . In
particular we prove that for small one essentially recovers Fisher's
bound. We also solve the next open case of and obtain the first
non-trivial upper bound for general .Comment: 27 pages (incluiding one appendix
On the ground states of the Bernasconi model
The ground states of the Bernasconi model are binary +1/-1 sequences of
length N with low autocorrelations. We introduce the notion of perfect
sequences, binary sequences with one-valued off-peak correlations of minimum
amount. If they exist, they are ground states. Using results from the
mathematical theory of cyclic difference sets, we specify all values of N for
which perfect sequences do exist and how to construct them. For other values of
N, we investigate almost perfect sequences, i.e. sequences with two-valued
off-peak correlations of minimum amount. Numerical and analytical results
support the conjecture that almost perfect sequences do exist for all values of
N, but that they are not always ground states. We present a construction for
low-energy configurations that works if N is the product of two odd primes.Comment: 12 pages, LaTeX2e; extended content, added references; submitted to
J.Phys.
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