14 research outputs found
Estimation of pure qubits on circles
Gisin and Popescu [PRL, 83, 432 (1999)] have shown that more information
about their direction can be obtained from a pair of anti-parallel spins
compared to a pair of parallel spins, where the first member of the pair (which
we call the pointer member) can point equally along any direction in the Bloch
sphere. They argued that this was due to the difference in dimensionality
spanned by these two alphabets of states. Here we consider similar alphabets,
but with the first spin restricted to a fixed small circle of the Bloch sphere.
In this case, the dimensionality spanned by the anti-parallel versus parallel
alphabet is now equal. However, the anti-parallel alphabet is found to still
contain more information in general. We generalize this to having N parallel
spins and M anti-parallel spins. When the pointer member is restricted to a
small circle these alphabets again span spaces of equal dimension, yet in
general, more directional information can be found for sets with smaller |N-M|
for any fixed total number of spins. We find that the optimal POVMs for
extracting directional information in these cases can always be expressed in
terms of the Fourier basis. Our results show that dimensionality alone cannot
explain the greater information content in anti-parallel combinations of spins
compared to parallel combinations. In addition, we describe an LOCC protocol
which extract optimal directional information when the pointer member is
restricted to a small circle and a pair of parallel spins are supplied.Comment: 23 pages, 8 figure
Slepian functions and their use in signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden,
Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla
Scalar and vector Slepian functions, spherical signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and, particularly for applications in the
geosciences, for scalar and vectorial signals defined on the surface of a unit
sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics,
edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be
published by Springer Verlag. This is a slightly modified but expanded
version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the
Handbook, when it was called: Slepian functions and their use in signal
estimation and spectral analysi
Structured Sparsity: Discrete and Convex approaches
Compressive sensing (CS) exploits sparsity to recover sparse or compressible
signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity
is also used to enhance interpretability in machine learning and statistics
applications: While the ambient dimension is vast in modern data analysis
problems, the relevant information therein typically resides in a much lower
dimensional space. However, many solutions proposed nowadays do not leverage
the true underlying structure. Recent results in CS extend the simple sparsity
idea to more sophisticated {\em structured} sparsity models, which describe the
interdependency between the nonzero components of a signal, allowing to
increase the interpretability of the results and lead to better recovery
performance. In order to better understand the impact of structured sparsity,
in this chapter we analyze the connections between the discrete models and
their convex relaxations, highlighting their relative advantages. We start with
the general group sparse model and then elaborate on two important special
cases: the dispersive and the hierarchical models. For each, we present the
models in their discrete nature, discuss how to solve the ensuing discrete
problems and then describe convex relaxations. We also consider more general
structures as defined by set functions and present their convex proxies.
Further, we discuss efficient optimization solutions for structured sparsity
problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure
Clifford-Jacobi Polynomials and the Associated Continuous Wavelet Transform in Euclidean Space
Specific wavelet kernel functions for a continuous wavelet transform in Euclidean space are presented within the framework of Clifford analysis. These multi-dimensional wavelets are constructed by taking the Clifford-monogenic extension to Rm+1 of specific functions in R-m generalizing the traditional Jacobi weights. The notion of Clifford-monogenic function is a direct higher dimensional generalization of that of holomorphic function in the complex plane. Moreover, crucial to this construction is the orthogonal decomposition of the space of square integrable functions into the Hardy space H-2(R-m) and its orthogonal complement. In this way a nice relationship is established between the theory of the Clifford Continuous Wavelet Transform on the one hand, and the theory of Hardy spaces on the other hand. Furthermore, also new multi-dimensional polynomials, the so-called Clifford-Jacobi polynomials, are obtained