Distributed testing with zero-rate compression

A collaborative distributed binary decision problem is considered. Two statisticians are required to declare the correct probability measure of two jointly distributed memoryless process, denoted by X n = (X 1 , . . . , X n ) and Y n = (Y 1 , . . . , Y n ), out of two possible probability measures on finite alphabets, namely P XY and PXȲ . The marginal samples given by X n and Y n are assumed to be available at different locations. The statisticians are allowed to exchange limited amount of data over multiple rounds of interactions, which differs from previous work that deals mainly with unidirectional communication. A single round of interaction is considered before the result is generalized to any finite number of communication rounds. A feasibility result is shown, guaranteeing the feasibility of an error exponent for general hypotheses, through information-theoretic methods. The special case of testing against independence is revisited as being an instance of this result for which also an unfeasibility result is proven. A second special case is studied where zero-rate communication is imposed (data exchanges grow sub-exponentially with n) for which it is shown that interaction does not improve asymptotic performance. * Primary 94A24,94A15; secondary 94A13,68P30

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“…By (28) and the continuity of the entropy function as well as the KL divergence, we can conclude that…”

Section: 2mentioning

confidence: 90%

“…Interestingly, it was shown that this scenario does not offer any advantage, with relation to complete data compression. This setting, referred to as zero-rate communication, was recently revisited in [28] where both α n and β n are required to decrease exponentially with n.…”

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confidence: 99%

Collaborative distributed hypothesis testing with general hypotheses

Katz

Piantanida

Debbah

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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“…The finite length regime was investigated in [28]. The optimal Type-II error exponents of zerorate hypothesis testing in an interactive setup and in a cascaded-encoders network were presented respectively in [29] and [13], [16], [23], [30]. In particular, in our previous work [16], we proved such a result for the singlesensor two-detectors setup.…”

Section: Introductionmentioning

confidence: 77%

“…The latter scheme was shown to achieve the optimal exponent in the special case of testing against conditional independence by Rahman and Wagner [4]. This line of work has also been extended to networks with multiple sensors [2], [4]- [8], multiple detectors [9], interactive terminals [10]- [12], multi-hop networks [5], [13]- [16], noisy channels [17], [18], and scenarios with privacy constraints [19]- [22]. In this paper, we consider the single-sensor two-detectors system shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

Distributed Hypothesis Testing: Cooperation and Concurrent Detection

Escamilla

Wigger

Zaidi

2020

IEEE Trans. Inform. Theory

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A single-sensor two-detectors system is considered where the sensor communicates with both detectors and Detector 1 communicates with Detector 2, all over noise-free rate-limited links. The sensor and both detectors observe discrete memoryless source sequences whose joint probability mass function depends on a binary hypothesis. The goal at each detector is to guess the binary hypothesis in a way that, for increasing observation lengths, the probability of error under one of the hypotheses decays to zero with largest possible exponential decay, whereas the probability of error under the other hypothesis can decay to zero or to a small positive number arbitrarily slow. For the setting with positive communication rates from the sensor to the detectors and when both detectors are interested in maximizing the error exponent under the same hypothesis, we characterize the set of all possible exponents in a special case of testing against independence. In this case the cooperation link allows Detector 2 to increase its Type-II error exponent by an amount that is equal to the exponent attained at Detector 1. We also provide a general inner bound on the set of achievable error exponents that shows a tradeoff between the exponents at the two detectors in most cases. When the two detectors aim at maximizing the error exponent under different hypotheses and the distribution at the Sensor is different under the two hypotheses, then we show that such a tradeoff does not exist. We propose a general scheme that allows each detector to attain the same exponent as if it was the only detector in the system. For the setting with zerorate communication on both links, we exactly characterize the set of possible exponents and the gain brought up by cooperation, in function of the number of bits that are sent over the two links. Notice that, for this setting, tradeoffs between the exponents achieved at the two detectors arise only in few particular cases. In all other cases, each detector achieves the same performance as if it were the only detector in the system.

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“…This problem, which was first introduced by Berger [1], and then studied further in [2]- [4], arises naturally in many applications. Recent developments on this topic include extensions to networks with multiple sensors [5]- [9], multiple detectors [10], interactive terminals [11], [12], multi-hop networks [6], [13]- [16], noisy channels [17], [18] and scenarios with privacy constraints [19]- [21]. Its theoretical understanding, however, is far from complete, even from seemingly simple instances of it.…”

Section: Introductionmentioning

confidence: 99%

Rate-Exponent Region for a Class of Distributed Hypothesis Testing Against Conditional Independence Problems

Zaidi¹

2021

Preprint

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We study a class of K-encoder hypothesis testing against conditional independence problems. Under the criterion that stipulates minimization of the Type II error exponent subject to a (constant) upper bound on the Type I error rate, we characterize the set of encoding rates and exponent for both discrete memoryless and memoryless vector Gaussian settings. For the DM setting, we provide a converse proof and show that it is achieved using the Quantize-Bin-Test scheme of Rahman and Wagner. For the memoryless vector Gaussian setting, we develop a tight outer bound by means of a technique that relies on the de Bruijn identity and the properties of Fisher information. In particular, the result shows that for memoryless vector Gaussian sources the rate-exponent region is exhausted using the Quantize-Bin-Test scheme with Gaussian test channels; and there is no loss in performance caused by restricting the sensors' encoders not to employ time sharing. Furthermore, we also study a variant of the problem in which the source, not necessarily Gaussian, has finite differential entropy and the sensors' observations noises under the null hypothesis are Gaussian. For this model, our main result is an upper bound on the exponent-rate function. The bound is shown to mirror a corresponding explicit lower bound, except that the lower bound involves the source power (variance) whereas the upper bound has the source entropy power. Part of the utility of the established bound is for investigating asymptotic exponent/rates and losses incurred by distributed detection as function of the number of sensors.

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Distributed testing with zero-rate compression

Cited by 17 publications

References 11 publications

Collaborative distributed hypothesis testing with general hypotheses

Collaborative distributed hypothesis testing with general hypotheses

Distributed Hypothesis Testing: Cooperation and Concurrent Detection

Rate-Exponent Region for a Class of Distributed Hypothesis Testing Against Conditional Independence Problems

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