6,277 research outputs found
Optimal Approximate Minimization of One-Letter Weighted Finite Automata
In this paper, we study the approximate minimization problem of weighted
finite automata (WFAs): to compute the best possible approximation of a WFA
given a bound on the number of states. By reformulating the problem in terms of
Hankel matrices, we leverage classical results on the approximation of Hankel
operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory.
We solve the optimal spectral-norm approximate minimization problem for
irredundant WFAs with real weights, defined over a one-letter alphabet. We
present a theoretical analysis based on AAK theory, and bounds on the quality
of the approximation in the spectral norm and norm. Moreover, we
provide a closed-form solution, and an algorithm, to compute the optimal
approximation of a given size in polynomial time.Comment: 32 pages. arXiv admin note: substantial text overlap with
arXiv:2102.0686
Spectral measure of large random Hankel, Markov and Toeplitz matrices
We study the limiting spectral measure of large symmetric random matrices of
linear algebraic structure. For Hankel and Toeplitz matrices generated by
i.i.d. random variables of unit variance, and for symmetric Markov
matrices generated by i.i.d. random variables of zero mean
and unit variance, scaling the eigenvalues by we prove the almost
sure, weak convergence of the spectral measures to universal, nonrandom,
symmetric distributions , and of unbounded
support. The moments of and are the sum of volumes of
solids related to Eulerian numbers, whereas has a bounded smooth
density given by the free convolution of the semicircle and normal densities.
For symmetric Markov matrices generated by i.i.d. random variables
of mean and finite variance, scaling the eigenvalues by
we prove the almost sure, weak convergence of the spectral measures to
the atomic measure at . If , and the fourth moment is finite, we prove
that the spectral norm of scaled by converges
almost surely to 1.Comment: Published at http://dx.doi.org/10.1214/009117905000000495 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Helson matrix with explicit eigenvalue asymptotics
A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite
matrix with entries for . Here the 'th term
depends on the product . We study a self-adjoint Helson matrix for a
particular sequence , , where , and prove that it is compact and that its eigenvalues
obey the asymptotics as ,
with an explicit constant . We also establish some
intermediate results (of an independent interest) which give a connection
between the spectral properties of a Helson matrix and those of its continuous
analogue, which we call the integral Helson operator
Bulk behaviour of skew-symmetric patterned random matrices
Limiting Spectral Distributions (LSD) of real symmetric patterned matrices
have been well-studied. In this article, we consider
skew-symmetric/anti-symmetric patterned random matrices and establish the LSDs
of several common matrices. For the skew-symmetric Wigner, skew-symmetric
Toeplitz and the skew-symmetric Circulant, the LSDs (on the imaginary axis) are
the same as those in the symmetric cases. For the skew-symmetric Hankel and the
skew-symmetric Reverse Circulant however, we obtain new LSDs. We also show the
existence of the LSDs for the triangular versions of these matrices.
We then introduce a related modification of the symmetric matrices by
changing the sign of the lower triangle part of the matrices. In this case, the
modified Wigner, modified Hankel and the modified Reverse Circulants have the
same LSDs as their usual symmetric counterparts while new LSDs are obtained for
the modified Toeplitz and the modified Symmetric Circulant.Comment: 21 pages, 2 figure
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