3,555 research outputs found
The Simplest Solution to an Underdetermined System of Linear Equations
Consider a d*n matrix A, with d<n. The problem of solving for x in y=Ax is
underdetermined, and has infinitely many solutions (if there are any). Given y,
the minimum Kolmogorov complexity solution (MKCS) of the input x is defined to
be an input z (out of many) with minimum Kolmogorov-complexity that satisfies
y=Az. One expects that if the actual input is simple enough, then MKCS will
recover the input exactly. This paper presents a preliminary study of the
existence and value of the complexity level up to which such a complexity-based
recovery is possible. It is shown that for the set of all d*n binary matrices
(with entries 0 or 1 and d<n), MKCS exactly recovers the input for an
overwhelming fraction of the matrices provided the Kolmogorov complexity of the
input is O(d). A weak converse that is loose by a log n factor is also
established for this case. Finally, we investigate the difficulty of finding a
matrix that has the property of recovering inputs with complexity of O(d) using
MKCS.Comment: Proceedings of the IEEE International Symposium on Information Theory
Seattle, Washington, July 9-14, 200
Cauchy problem for integrable discrete equations on quad-graphs
Initial value problems for the integrable discrete equations on quad-graphs
are investigated. A geometric criterion of the well-posedness of such a problem
is found. The effects of the interaction of the solutions with the localized
defects in the regular square lattice are discussed for the discrete potential
KdV and linear wave equations. The examples of kinks and solitons on various
quad-graphs, including quasiperiodic tilings, are presented.Comment: Corrected version with the assumption of nonsingularity of solutions
explicitly state
Truncated Moment Problem for Dirac Mixture Densities with Entropy Regularization
We assume that a finite set of moments of a random vector is given. Its
underlying density is unknown. An algorithm is proposed for efficiently
calculating Dirac mixture densities maintaining these moments while providing a
homogeneous coverage of the state space.Comment: 18 pages, 6 figure
Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples
This paper presents a novel power spectral density estimation technique for
band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The
technique employs multi-coset sampling and incorporates the advantages of
compressed sensing (CS) when the power spectrum is sparse, but applies to
sparse and nonsparse power spectra alike. The estimates are consistent
piecewise constant approximations whose resolutions (width of the piecewise
constant segments) are controlled by the periodicity of the multi-coset
sampling. We show that compressive estimates exhibit better tradeoffs among the
estimator's resolution, system complexity, and average sampling rate compared
to their noncompressive counterparts. For suitable sampling patterns,
noncompressive estimates are obtained as least squares solutions. Because of
the non-negativity of power spectra, compressive estimates can be computed by
seeking non-negative least squares solutions (provided appropriate sampling
patterns exist) instead of using standard CS recovery algorithms. This
flexibility suggests a reduction in computational overhead for systems
estimating both sparse and nonsparse power spectra because one algorithm can be
used to compute both compressive and noncompressive estimates.Comment: 26 pages, single spaced, 9 figure
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