3,517 research outputs found
Inferring Rankings Using Constrained Sensing
We consider the problem of recovering a function over the space of
permutations (or, the symmetric group) over elements from given partial
information; the partial information we consider is related to the group
theoretic Fourier Transform of the function. This problem naturally arises in
several settings such as ranked elections, multi-object tracking, ranking
systems, and recommendation systems. Inspired by the work of Donoho and Stark
in the context of discrete-time functions, we focus on non-negative functions
with a sparse support (support size domain size). Our recovery method is
based on finding the sparsest solution (through optimization) that is
consistent with the available information. As the main result, we derive
sufficient conditions for functions that can be recovered exactly from partial
information through optimization. Under a natural random model for the
generation of functions, we quantify the recoverability conditions by deriving
bounds on the sparsity (support size) for which the function satisfies the
sufficient conditions with a high probability as .
optimization is computationally hard. Therefore, the popular compressive
sensing literature considers solving the convex relaxation,
optimization, to find the sparsest solution. However, we show that
optimization fails to recover a function (even with constant sparsity)
generated using the random model with a high probability as . In
order to overcome this problem, we propose a novel iterative algorithm for the
recovery of functions that satisfy the sufficient conditions. Finally, using an
Information Theoretic framework, we study necessary conditions for exact
recovery to be possible.Comment: 19 page
Constructing Linear Encoders with Good Spectra
Linear encoders with good joint spectra are suitable candidates for optimal
lossless joint source-channel coding (JSCC), where the joint spectrum is a
variant of the input-output complete weight distribution and is considered good
if it is close to the average joint spectrum of all linear encoders (of the
same coding rate). In spite of their existence, little is known on how to
construct such encoders in practice. This paper is devoted to their
construction. In particular, two families of linear encoders are presented and
proved to have good joint spectra. The first family is derived from Gabidulin
codes, a class of maximum-rank-distance codes. The second family is constructed
using a serial concatenation of an encoder of a low-density parity-check code
(as outer encoder) with a low-density generator matrix encoder (as inner
encoder). In addition, criteria for good linear encoders are defined for three
coding applications: lossless source coding, channel coding, and lossless JSCC.
In the framework of the code-spectrum approach, these three scenarios
correspond to the problems of constructing linear encoders with good kernel
spectra, good image spectra, and good joint spectra, respectively. Good joint
spectra imply both good kernel spectra and good image spectra, and for every
linear encoder having a good kernel (resp., image) spectrum, it is proved that
there exists a linear encoder not only with the same kernel (resp., image) but
also with a good joint spectrum. Thus a good joint spectrum is the most
important feature of a linear encoder.Comment: v5.5.5, no. 201408271350, 40 pages, 3 figures, extended version of
the paper to be published in IEEE Transactions on Information Theor
Inferring rankings using constrained sensing
We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over n elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size <;<; domain size). Our recovery method is based on finding the sparsest solution (through l[subscript 0] optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through l[subscript 0] optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as n β β. β0 optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, β[subscript 1] optimization, to find the sparsest solution. However, we show that β[subscript 1] optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as n β β. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible
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