A simple class of asymptotically optimal quantizers

Cambanis, Stamatis; Gerr, Neil L.

doi:10.1109/tit.1983.1056733

Cited by 35 publications

(18 citation statements)

References 18 publications

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“…Both studies assume that in representing a cardinality, the neural system maps an input number n to a quantized form n̂ , and that the “right” thing to do is minimize the value of the relative error of the quantization (Sun & Goyal, 2011), given by

{double-struckE}_{n} [{∣ n - \hat{n} ∣}^{2} ∕ n^{2}]

(for history and related results, see Gray & Neuhoff, 1998 ; Cambanis & Gerr, 1983). This is not the same as assuming Weber's law, but rather assumes an objective function (over representations) that is based in relative error.…”

Section: Previous Derivations Of the Logarithmic Mappingmentioning

confidence: 99%

A rational analysis of the approximate number system

Piantadosi

2016

Psychon Bull Rev

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It is well-known in numerical cognition that higher numbers are represented with less absolute fidelity than lower numbers, often formalized as a logarithmic mapping. Previous derivations of this psychological law have worked by assuming that relative change in physical magnitude is the key psychologically-relevant measure (Fechner, 1860 ; Sun, Wang, Goyal, & Varshney, 2012 ; Portugal & Svaiter, 2011). Ideally, however, this property of psychological scales would be derived from more general, independent principles. This paper shows that a logarithmic number line is the one which minimizes the error between input and representation, relative to the probability that subjects would need to represent each number. This need probability is measured here through natural language and matches the form of need probabilities found in other literatures. The derivation does not presuppose anything like Weber's law and makes minimal assumptions about both the nature of internal representations and the form of the mapping. More generally, the results prove in a general setting that the optimal psychological scale will change with the square root of the probability of each input. For stimuli that follow a power-law need distribution this approach recovers either a logarithmic or power-law psychophysical mapping (Stevens, 1957, 1961, 1975).

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{double-struckE}_{n} [{∣ n - \hat{n} ∣}^{2} ∕ n^{2}]

Section: Previous Derivations Of the Logarithmic Mappingmentioning

confidence: 99%

A rational analysis of the approximate number system

Piantadosi

2016

Psychon Bull Rev

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“…Using Hölder's inequality, the optimal point density for fixed-rate quantization for each source (communicated with rate ) is asymptotically (24) over the support of , with fMSE (25) Similarly, the best point density for the entropy-constrained case is asymptotically (26) over the support of , leading to a fMSE of (27) We present performance while leaving the fractional allocation as a parameter. Given a total communication rate constraint , we can also optimize .…”

Section: A Asymptotically Optimal Quantizer Sequencesmentioning

confidence: 99%

Distributed Functional Scalar Quantization Simplified

Sun

Misra

Goyal

2013

IEEE Trans. Signal Process.

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Abstract-Distributed functional scalar quantization (DFSQ) theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired. We address two limitations of previous works: prohibitively expensive decoder design and a restriction to source distributions with bounded support. We show that a much simpler decoder has equivalent asymptotic performance to the conditional expectation estimator studied previously, thus reducing decoder design complexity. The simpler decoder features decoupled communication and computation blocks. Moreover, we extend the DFSQ framework with the simpler decoder to source distributions with unbounded support. Finally, through simulation results, we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis, and we give new insight on the rate of convergence.

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“…We recall the fidelity criterion of interest is fMSE as defined in (1) and the performance of nonuniform scalar quantizers is given in (4)- (6). In this section, we will understand the achievable rate region with respect to fMSE using Theorem 1 and compare the sum rate to the DFSQ results; this determines the rate loss from using scalar quantizers.…”

Section: Rate Loss Of Dfsqmentioning

confidence: 99%

Rate loss in distributed functional source coding

Sun

Goyal

2013

2013 IEEE International Symposium on Information Theory

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Abstract-For point-to-point and distributed communication of continuous sources, high-resolution quantization theory provides an achievable rate-distortion trade-off that is simple to compute and motivates practical compression architectures. Moreover, high-resolution analysis gives good inner bounds for the Shannon rate-distortion region when a more general characterization is difficult. In this paper, we analyze the sum-rate gap between coded nonuniform scalar quantization and the Shannon ratedistortion region for a system that requires fidelity in a computation applied to the source variables. We find that the loss can be as low as 0.255 bits/sample, which has previously been observed in the point-to-point setting, and it is achieved using a simple architecture of nonuniform quantization followed by Slepian-Wolf coding.

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A simple class of asymptotically optimal quantizers

Cited by 35 publications

References 18 publications

A rational analysis of the approximate number system

A rational analysis of the approximate number system

Distributed Functional Scalar Quantization Simplified

Rate loss in distributed functional source coding

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