A Comparison of Signalling Alphabets

Gilbert, E. N.

doi:10.1002/j.1538-7305.1952.tb01393.x

Cited by 452 publications

(260 citation statements)

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“…We now compute the first entropy integral on the right-hand side of (25). Since we are now considering the · 2,∞ instead of the · 2 -norm we expect that s does not play a role in this part.…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

Uniform recovery of fusion frame structured sparse signals

Ayaz

Dirksen

Rauhut

2016

Applied and Computational Harmonic Analysis

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We consider the problem of recovering fusion frame sparse signals from incomplete measurements. These signals are composed of a small number of nonzero blocks taken from a family of subspaces. First, we show that, by using a-priori knowledge of a coherence parameter associated with the angles between the subspaces, one can uniformly recover fusion frame sparse signals with a significantly reduced number of vector-valued (sub-)Gaussian measurements via mixed ℓ 1 /ℓ 2 -minimization. We prove this by establishing an appropriate version of the restricted isometry property. Our result complements previous nonuniform recovery results in this context, and provides stronger stability guarantees for noisy measurements and approximately sparse signals. Second, we determine the minimal number of scalar-valued measurements needed to uniformly recover all fusion frame sparse signals via mixed ℓ 1 /ℓ 2 -minimization. This bound is achieved by scalar-valued subgaussian measurements. In particular, our result shows that the number of scalar-valued subgaussian measurements cannot be further reduced using knowledge of the coherence parameter. As a special case it implies that the best known uniform recovery result for block sparse signals using subgaussian measurements is optimal.

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Section: Proof Of Theorem 32mentioning

confidence: 99%

Uniform recovery of fusion frame structured sparse signals

Ayaz

Dirksen

Rauhut

2016

Applied and Computational Harmonic Analysis

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“…As discussed in the introduction, the best lower bound for R(δ) (Varshamov-Gilbert bound [1]) can be obtained by proving the convergence of the virial series [4]. It turns out that the virial series converges for ρV d−1 = ρ < 1/6e, that means ϕ < 0 for n → ∞.…”

Section: Known Bounds On R(δ)mentioning

confidence: 99%

“…Thus we will restrict to δ < 1/2 in the following. This problem is relevant for the theory of error correcting codes [1][2][3][4][5][6]. In "physics language", it is the problem of finding the maximum possible density of a system of hard spheres on the hypercubic lattice.…”

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confidence: 99%

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On the High Density Behavior of Hamming Codes with Fixed Minimum Distance

Parisi

Zamponi

2006

J Stat Phys

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We discuss the high density behavior of a system of hard spheres of diameter d on the hypercubic lattice of dimension n, in the limit n → ∞, d → ∞, d/n = δ. The problem is relevant for coding theory, and the best available bounds state that the maximum density of the system falls in the interval 1 ≤ ρV d ≤ exp(nκ(δ)), being κ(δ) > 0 and V d the volume of a sphere of radius d. We find a solution of the equations describing the liquid up to an exponentially large value of ρ = ρV d , but we show that this solution gives a negative entropy for the liquid phase for ρ > ∼ n. We then conjecture that a phase transition towards a different phase might take place, and we discuss possible scenarios for this transition.A code C is a subset of the binary Hamming space H n 2 = {0, 1} n . The distance between two points s, t ∈ H n 2 is the Hamming distance d(s, t) = n α=1 (s α + t α ) mod2 , i.e. it is given by the number of different bits. We consider the problem of finding the maximal size A(n, d) of a code C such that the minimum distance between two points in C is d, that means, denoting by |C| the number of sequences in C,( 1) In particular we are interested in the quantitywhere the supremum is taken on all possible sequences of codes such that d/n → δ. The problem trivializes for δ > 1/2 as the total number of sequences is finite and R(δ) = 0. An interesting scaling is d/n = 1/2 − εn −α , as for an appropriate choice of α the number A(n, d) might increase polynomially in n, but this will not be investigated here. Thus we will restrict to δ < 1/2 in the following. This problem is relevant for the theory of error correcting codes [1][2][3][4][5][6]. In "physics language", it is the problem of finding the maximum possible density of a system of hard spheres on the hypercubic lattice. This rephrasing of the problem has been shown to be useful as it allows to use well known methods borrowed from the theory of liquids, like the virial expansion [4].In this paper we will discuss the behavior of the system at high density, in order to understand how one can try to compute the maximum density. We will show that, for large n, the problem closely resembles the problem of hard spheres in R n in the limit of large space dimension n. The basic idea is that in this limit the number of neighbors of a sphere is large, much as it happens in the continuum for large space dimension. We will then discuss some recent ideas that have been used in the continuum [7][8][9][10] to make some progress in the direction of deriving bounds on A(n, d).The best known lower bound on A(n, d) (Varshamov-Gilbert bound) states that the density ρ = |C|/2 n ≥ 1/V d−1 , V d being the volume of a sphere of radius d in H n 2 [1]. This bound can be proven from the convergence of the virial series [4] and gives R(δ) ≥ R V G (δ) = 1 − H(δ), H(δ) being the binary entropy function (see below). This means that a "liquid phase", defined by the virial equation of state, exists at least up to ρV d−1 ∼ 1. We will show that the liquid phase can be formally continued up to a ...

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“…i Z i n Here c is a constant in [1000,2000], chosen in such a way that n 0 is a prime number. Now, we apply the following claim, whose proof is essentially a well known probabilistic argument (with an associated greedy algorithm) for existence of codes with good distance (akin to the Gilbert-Varshamov bound (Gilbert, 1952;Varshamov, 1957)). The proof is given in the next subsection.…”

Section: F Y and Proving Lemmamentioning

confidence: 99%

Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

Hegde

2017

Proceedings of the Indian National Science Academy

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An arithmetic circuit (respectively, formula) is a rooted graph (respectively, tree) whose nodes are addition or multiplication gates and input variables/nodes. It computes a polynomial in a natural way. The formal degree of an addition (respectively, multiplication) gate with respect to a variable x is defined as the maximum (respectively, sum) of the formal degrees of its children, with respect to x. The formal degree of an input node with respect to x is 1 if the node is labelled with x, and 0 otherwise. In a multi-r-ic formula, the formal degree of every gate with respect to every variable is at most r. Multi-r-ic formulas make an intermediate model between multilinear formulas (the r = 1 case), for which lower bounds are relatively well-understood, and general formulas (the unbounded-r case), which are conjectured to have superpolynomial size lower bound.On depth four multi-r-ic formulas/circuits computing IMM n,d -the product of d symbolic metrices of size n x n each,

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A Comparison of Signalling Alphabets

Cited by 452 publications

References 0 publications

Uniform recovery of fusion frame structured sparse signals

Uniform recovery of fusion frame structured sparse signals

On the High Density Behavior of Hamming Codes with Fixed Minimum Distance

Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

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