Statistical Mechanics of MAP Estimation: General Replica Ansatz

Bereyhi, Ali; Müller, Ralf R.; Schulz-Baldes, Hermann

doi:10.48550/arxiv.1612.01980

Cited by 14 publications

(40 citation statements)

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“…Thereby, it is easy to see from the expressions in (36) and (39) that σ x (t) → 1 and C θ (t, t) → 1/v. In other words, the field statistics asymptotically obey the equations of the static solutions of the replica ansatz (10) [8], [14].…”

Section: A Asymptotic Consistency With the Replica Ansatzmentioning

confidence: 97%

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Dynamical functional theory for compressed sensing

Çakmak

Opper

Winther

et al. 2017

2017 IEEE International Symposium on Information Theory (ISIT)

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We introduce a theoretical approach for designing generalizations of the approximate message passing (AMP) algorithm for compressed sensing which are valid for large observation matrices that are drawn from an invariant random matrix ensemble. By design, the fixed points of the algorithm obey the Thouless-Anderson-Palmer (TAP) equations corresponding to the ensemble. Using a dynamical functional approach we are able to derive an effective stochastic process for the marginal statistics of a single component of the dynamics. This allows us to design memory terms in the algorithm in such a way that the resulting fields become Gaussian random variables allowing for an explicit analysis. The asymptotic statistics of these fields are consistent with the replica ansatz of the compressed sensing problem.

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Section: A Asymptotic Consistency With the Replica Ansatzmentioning

confidence: 97%

“…it has the same distribution as O † J O for any orthogonal matrix O independent of J . 2 These ensembles are now favoured in performance analyses of compressed sensing [8]- [10].…”

Section: Introductionmentioning

confidence: 99%

Dynamical functional theory for compressed sensing

Çakmak

Opper

Winther

et al. 2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…For sake of compactness, we state the one-step RSB ansatz, i.e., b = 1, in this paper. The result, however, is extended to an arbitrary number of breaking steps by taking the approach in Appendix D of [12].…”

Section: B Rsb Ansätzementioning

confidence: 99%

“…The one-step RSB decoupled precoder is similar to RS; however, the "decoupled input" s dec , which in RS is s rs , is replaced by s rs + s rsb 1 . Taking the same approach as in [12], it is shown that under b-steps of RSB, the decoupled precoder has a same form, and s dec = s rs + b κ=1 s rsb κ . In this case, s rsb κ is obtained from s rs and s rsb ς b ς=κ+1 through p rsb κ (u κ |u κ+1 , .…”

Section: Rsb Marginal Decoupling Propertymentioning

confidence: 99%

Asymptotics of nonlinear LSE precoders with applications to transmit antenna selection

Bereyhi

Sedaghat

Müller

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Abstract-This paper studies the large-system performance of Least Square Error (LSE) precoders which minimize the inputoutput distortion over an arbitrary support subject to a general penalty function. The asymptotics are determined via the replica method in a general form which encloses the Replica Symmetric (RS) and Replica Symmetry Breaking (RSB) ansätze. As a result, the "marginal decoupling property" of LSE precoders for b-steps of RSB is derived. The generality of the studied setup enables us to address special cases in which the number of active transmit antennas are constrained. Our numerical investigations depict that the computationally efficient forms of LSE precoders based on "ℓ1-norm" minimization perform close to the cases with "zeronorm" penalty function which have a considerable improvements compared to the random antenna selection. For the case with BPSK signals and restricted number of active antennas, the results show that RS fails to predict the performance while the RSB ansatz is consistent with theoretical bounds.

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“…For various problems, the RS structure results in solutions which are consistent with the available rigorous bounds [12], [13], [18], [20]. There are, however, several examples in which the RS ansatz does not give valid solutions [6], [14], [21], [22]. For these cases, it is believed that the problem is caused by the assumed structure, i.e., RS, and therefore, more general structures are supposed to result in valid solutions.…”

Section: E Rs Assumptionmentioning

confidence: 69%

Nonlinear Precoders for Massive MIMO Systems with General Constraints

Bereyhi¹,

Sedaghat²,

Asaad³

et al. 2017

Preprint

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We introduce a class of nonlinear least square error precoders with a general penalty function for multiuser massive MIMO systems. The generality of the penalty function allows us to consider several hardware limitations including transmitters with a predefined constellation and restricted number of active antennas. The large-system performance is then investigated via the replica method under the assumption of replica symmetry. It is shown that the least square precoders exhibit the "marginal decoupling property" meaning that the marginal distributions of all precoded symbols converge to a deterministic distribution. As a result, the asymptotic performance of the precoders is described by an equivalent single-user system. To address some applications of the results, we further study the asymptotic performance of the precoders when both the peak-to-average power ratio and number of active transmit antennas are constrained. Our numerical investigations show that for a desired distortion at the receiver side, proposed forms of the least square precoders need to employ around 35% fewer number of active antennas compared to cases with random transmit antenna selection.

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Statistical Mechanics of MAP Estimation: General Replica Ansatz

Cited by 14 publications

References 0 publications

Dynamical functional theory for compressed sensing

Dynamical functional theory for compressed sensing

Asymptotics of nonlinear LSE precoders with applications to transmit antenna selection

Nonlinear Precoders for Massive MIMO Systems with General Constraints

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