Distributed Hypothesis Testing with Privacy Constraints

We revisit the distributed hypothesis testing (or hypothesis testing with communication constraints) problem from the viewpoint of privacy. Instead of observing the raw data directly, the transmitter observes a sanitized or randomized version of it. We impose an upper bound on the mutual information between the raw and randomized data. Under this scenario, the receiver, which is also provided with side information, is required to make a decision on whether the null or alternative hypothesis is in effect. We first provide a general lower bound on the type-II exponent for an arbitrary pair of hypotheses. Next, we show that if the distribution under the alternative hypothesis is the product of the marginals of the distribution under the null (i.e., testing against independence), then the exponent is known exactly. Moreover, we show that the strong converse property holds. Using ideas from Euclidean information theory, we also provide an approximate expression for the exponent when the communication rate is low and the privacy level is high. Finally, we illustrate our results with a binary and a Gaussian example

Near-Optimal Noisy Group Testing via Separate Decoding of Items

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. In this paper, we revisit an efficient algorithm for noisy group testing in which each item is decoded separately (Malyutov and Mateev, 1980), and develop novel performance guarantees via an information-theoretic framework for general noise models. For the special cases of no noise and symmetric noise, we find that the asymptotic number of tests required for vanishing error probability is within a factor $\log 2 \approx 0.7$ of the information-theoretic optimum at low sparsity levels, and that with a small fraction of allowed incorrectly decoded items, this guarantee extends to all sublinear sparsity levels. In addition, we provide a converse bound showing that if one tries to move slightly beyond our low-sparsity achievability threshold using separate decoding of items and i.i.d. randomized testing, the average number of items decoded incorrectly approaches that of a trivial decoder.Comment: Submitted to IEEE Journal of Selected Topics in Signal Processin

Finite-Block-Length Analysis in Classical and Quantum Information Theory

Coding technology is used in several information processing tasks. In particular, when noise during transmission disturbs communications, coding technology is employed to protect the information. However, there are two types of coding technology: coding in classical information theory and coding in quantum information theory. Although the physical media used to transmit information ultimately obey quantum mechanics, we need to choose the type of coding depending on the kind of information device, classical or quantum, that is being used. In both branches of information theory, there are many elegant theoretical results under the ideal assumption that an infinitely large system is available. In a realistic situation, we need to account for finite size effects. The present paper reviews finite size effects in classical and quantum information theory with respect to various topics, including applied aspects

Distributed Hypothesis Testing over a Noisy Channel: Error-exponents Trade-off

A two-terminal distributed binary hypothesis testing (HT) problem over a noisy channel is studied. The two terminals, called the observer and the decision maker, each has access to $n$ independent and identically distributed samples, denoted by $\mathbf{U}$ and $\mathbf{V}$, respectively. The observer communicates to the decision maker over a discrete memoryless channel (DMC), and the decision maker performs a binary hypothesis test on the joint probability distribution of $(\mathbf{U},\mathbf{V})$ based on $\mathbf{V}$ and the noisy information received from the observer. The trade-off between the exponents of the type I and type II error probabilities in HT is investigated. Two inner bounds are obtained, one using a separation-based scheme that involves type-based compression and unequal error-protection channel coding, and the other using a joint scheme that incorporates type-based hybrid coding. The separation-based scheme is shown to recover the inner bound obtained by Han and Kobayashi for the special case of a rate-limited noiseless channel, and also the one obtained by the authors previously for a corner point of the trade-off. Exact single-letter characterization of the optimal trade-off is established for the special case of testing for the marginal distribution of $\mathbf{U}$, when $\mathbf{V}$ is unavailable. Our results imply that a separation holds in this case, in the sense that the optimal trade-off is achieved by a scheme that performs independent HT and channel coding. Finally, we show via an example that the joint scheme achieves a strictly tighter bound than the separation-based scheme for some points of the error-exponent trade-off

Distributed hypothesis testing over discrete memoryless channels

A distributed binary hypothesis testing (HT) problem involving two parties, one referred to as the observer and the other as the detector is studied. The observer observes a discrete memoryless source (DMS) and communicates its observations to the detector over a discrete memoryless channel (DMC). The detector observes another DMS correlated with that at the observer, and performs a binary HT on the joint distribution of the two DMS’s using its own observed data and the information received from the observer. The trade-off between the type I error probability and the type II error-exponent of the HT is explored. Single-letter lower bounds on the optimal type II errorexponent are obtained by using two different coding schemes, a separate HT and channel coding scheme and a joint HT and channel coding scheme based on hybrid coding for the matched bandwidth case. Exact single-letter characterization of the same is established for the special case of testing against conditional independence, and it is shown to be achieved by the separate HT and channel coding scheme. An example is provided where the joint scheme achieves a strictly better performance than the separation based scheme