Some new two-sided bounds for determinants of diagonally dominant matrices

Li, Wen; Chen, Yanmei

doi:10.1186/1029-242x-2012-61

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…In particular, if S = suppW i.e, the support of W , then as seen from [36, equation (85)], any non zero lower bound on det(A * S A S ) 1/K in the limit is enough. So if the matrix A * S A S possesses strong diagonal dominance then it is possible to have such a non zero lower bound on det(A * S A S ) 1/K for every S [57]. These could be ensured by having codewords that are overwhelmingly close to orthogonal.…”

Section: Conversementioning

confidence: 99%

Fundamental Limits of Many-User MAC With Finite Payloads and Fading

Kowshik

Polyanskiy

2021

IEEE Trans. Inform. Theory

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Consider a (multiple-access) wireless communication system where users are connected to a unique base station over a shared-spectrum radio links. Each user has a fixed number k of bits to send to the base station, and his signal gets attenuated by a random channel gain (quasi-static fading). In this paper we consider the many-user asymptotics of Chen-Chen-Guo'2017, where the number of users grows linearly with the blocklength. Differently, though, we adopt a per-user probability of error (PUPE) criterion (as opposed to classical joint-error probability criterion). Under PUPE the finite energyper-bit communication is possible, and we are able to derive bounds on the tradeoff between energy and spectral efficiencies. We reconfirm the curious behaviour (previously observed for non-fading MAC) of the possibility of almost perfect multi-user interference (MUI) cancellation for user densities below a critical threshold. Further, we demonstrate the suboptimality of standard solutions such as orthogonalization (i.e., TDMA/FDMA) and treating interference as noise (i.e. pseudo-random CDMA without multi-user detection). Notably, the problem treated here can be seen as a variant of support recovery in compressed sensing for the unusual definition of sparsity with one non-zero entry per each contiguous section of 2 k coordinates. This identifies our problem with that of the sparse regression codes (SPARCs) and hence our results can be equivalently understood in the context of SPARCs with sections of length 2 100 . Finally, we discuss the relation of the almost perfect MUI cancellation property and the replica-method predictions.Index Terms-Finite blocklength, many-user MAC, per-user probability of error, approximate message passing, replicamethod I. INTRODUCTIONWe clearly witness two recent trends in the wireless communication technology: the increasing deployment density and miniaturization of radio-equipped sensors. The first trend results in progressively worsening interference environment, while the second trend puts ever more stringent demands on communication energy efficiency. This suggests a bleak picture for the future networks, where a chaos of packet collisions and interference contamination prevents reliable connectivity.This paper is part of a series aimed at elucidating the fundamental tradeoffs in this new "dense-networks" regime

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Section: Conversementioning

confidence: 99%

Fundamental Limits of Many-User MAC With Finite Payloads and Fading

Kowshik

Polyanskiy

2021

IEEE Trans. Inform. Theory

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“…Proof. To prove (13), first we consider a negative definite matrix as a diagonally stable one with a Lyapunov scaling factor D = I and apply Theorem 7. Then we recall that (Sym(A)) −1 and Skew(A) are symmetric and skewsymmetric respectively, thus they are both normal.…”

Section: Lemma 3 Let a B ∈ M N×n Be Symmetric Positive Definite Matri...mentioning

confidence: 99%

Some bounds for determinants of relatively $D$-stable matrices

Kushel¹

2022

Preprint

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In this paper, we study the class of relatively D-stable matrices and provide the conditions, sufficient for relative D-stability. We generalize the well-known Hadamard inequality, to provide upper bounds for the determinants of relatively D-stable and relatively additive D-stable matrices. For some classes of D-stable matrices, we estimate the sector gap between matrix spectra and the imaginary axis. We apply the developed technique to obtain upper bounds for determinants of some classes of D-stable matrices, e.g. diagonally stable, diagonally dominant and matrices with Q 2 -scalings.

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“…Similarly for the highorder coefficients in the other cases ( 12), ( 13), (14), and (18) that apply for large ε. When ε = 1, all three of ( 19)- (21) give the maximal determinant 160 for 6 × 6 {±1}-matrices [26,39].…”

Section: Small Examplesmentioning

confidence: 99%

“…Many bounds on determinants of diagonally dominant matrices A have been given in the literature. See, for example, Muir [24], Ostrowski [32], Price [34], and more recently Bhatia and Jain [2], Elsner [11], Horn and Johnson [18], Ipsen and Rehman [19], Li and Chen [21], and the references given there.…”

Section: Introductionmentioning

confidence: 99%

Bounds on determinants of perturbed diagonal matrices

Brent,

Osborn,

Smith

2014

Preprint

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We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If A = I − E is a real n× n matrix and the elements of E are bounded in absolute value by ε ≤ 1/n, then a lower bound of Ostrowski (1938) is det(A) ≥ 1 − nε. We show that if, in addition, the diagonal elements of E are zero, then a best-possible lower bound is Corresponding upper bounds are respectivelyThe second upper bound generalises Hadamard's inequality, which is the case ε = 1. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order n and all positive ε is the existence of a skew-Hadamard matrix of order n.

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Some new two-sided bounds for determinants of diagonally dominant matrices

Abstract: In this article, we present some new two-sided bounds for the determinant of some diagonally dominant matrices. In particular, the idea of the preconditioning technique is applied to obtain the new bounds.

Cited by 4 publications

References 16 publications

Fundamental Limits of Many-User MAC With Finite Payloads and Fading

Fundamental Limits of Many-User MAC With Finite Payloads and Fading

Some bounds for determinants of relatively $D$-stable matrices

Bounds on determinants of perturbed diagonal matrices

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