In this paper we study the problem of signal detection in Gaussian noise in a distributed setting where the local machines in the star topology can communicate a single bit of information. We derive a lower bound on the Euclidian norm that the signal needs to have in order to be detectable. Moreover, we exhibit optimal distributed testing strategies that attain the lower bound.
The Rényi information measures are characterized in terms of their Shannon counterparts, and properties of the former are recovered from first principle via the associated properties of the latter. Motivated by this characterization, a two-sensor composite hypothesis testing problem is presented, and the optimal worst case miss-detection exponent is obtained in terms of a Rényi divergence.
This letter proposes a low-computational Bayesian algorithm for noisy sparse recovery in the context of one bit compressed sensing with sensing matrix perturbation. The proposed algorithm which is called BHT-MLE comprises a sparse support detector and an amplitude estimator. The support detector utilizes Bayesian hypothesis test, while the amplitude estimator uses an ML estimator which is obtained by solving a convex optimization problem. Simulation results show that BHT-MLE algorithm offers more reconstruction accuracy than that of an ML estimator (MLE) at a low computational cost.
We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set $Γ$. We would like to design a test to decide which hypothesis is in effect. Under the constraints that the probabilities that the length of the test, a stopping time, exceeds $n$ are bounded by a certain threshold $ε$, we obtain certain fundamental limits on the asymptotic behavior of the sequential test as $n$ tends to infinity. Assuming that $Γ$ is a convex an…
The classical binary hypothesis testing problem is revisited. We notice that when one of the hypotheses is composite, there is an inherent difficulty in defining an optimality criterion that is both informative and well-justified. For testing in the simple normal location problem (that is, testing for the mean of multivariate Gaussians), we overcome the difficulty as follows. In this problem there exists a natural hardness order between parameters as for different parameters the error-probailities curves (when the parameter is known) are either identical, or one dominates the other. We can thu…
We extend quantum Stein's lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states $ρ^{\otimes n}$ against convex combinations of quantum states $σ^{\otimes n}$ can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative ent…
In this paper, an exact bitwise MAP (Maximum A Posteriori) estimation algorithm for group testing problems is presented. We assume a simplest non-adaptive group testing scenario including N-objects with binary status and M-disjunctive tests. If a group contains a positive object, the test result for the group is assumed to be one; otherwise, the test result becomes zero. Our inference problem is to evaluate the posterior probabilities of the objects from the observation of M-test results and from our knowledge on the prior probabilities for objects. The heart of the algorithm is the dual expre…
The minimax robust hypothesis testing problem for the case where the nominal probability distributions are subject to both modeling errors and outliers is studied in twofold. First, a robust hypothesis testing scheme based on a relative entropy distance is designed. This approach provides robustness with respect to modeling errors and is a generalization of a previous work proposed by Levy. Then, it is shown that this scheme can be combined with Huber's robust test through a composite uncertainty class, for which the existence of a saddle value condition is also proven. The composite version o…
We investigate the nonparametric, composite hypothesis testing problem for arbitrary unknown distributions in the asymptotic regime where both the sample size and the number of hypotheses grow exponentially large. Such asymptotic analysis is important in many practical problems, where the number of variations that can exist within a family of distributions can be countably infinite. We introduce the notion of \emph{discrimination capacity}, which captures the largest exponential growth rate of the number of hypotheses relative to the sample size so that there exists a test with asymptotically …
We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold) and the second order asymptotics (at small deviations from the threshold). We apply our results to find operational interpretations of various Renyi information measures. In case the alternative hypothesis is comprised of bipartite product distribu…
This paper considers the sequential composite hypothesis test with multiple sensors. The sensors observe random samples in parallel and communicate with a fusion center, who makes the global decision based on the sensor inputs. On one hand, in the centralized scenario, where local samples are precisely transmitted to the fusion center, the generalized sequential likelihood ratio test (GSPRT) is shown to be asymptotically optimal in terms of the expected sample size as error rates tend to zero. On the other hand, for systems with limited power and bandwidth resources, decentralized solutions th…
This paper studies recursive composite hypothesis testing in a network of sparsely connected agents. The network objective is to test a simple null hypothesis against a composite alternative concerning the state of the field, modeled as a vector of (continuous) unknown parameters determining the parametric family of probability measures induced on the agents' observation spaces under the hypotheses. Specifically, under the alternative hypothesis, each agent sequentially observes an independent and identically distributed time-series consisting of a (nonlinear) function of the true but unknown …
United Technologies Research Center（联合技术研究中心）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
For the universal hypothesis testing problem, where the goal is to decide between the known null hypothesis distribution and some other unknown distribution, Hoeffding proposed a universal test in the nineteen sixties. Hoeffding's universal test statistic can be written in terms of Kullback-Leibler (K-L) divergence between the empirical distribution of the observations and the null hypothesis distribution. In this paper a modification of Hoeffding's test is considered based on a relaxation of the K-L divergence test statistic, referred to as the mismatched divergence. The resulting mismatched …
The group testing problem consists of determining a small set of defective items from a larger set of items based on tests on groups of items, and is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. While rigorous group testing algorithms have long been known with runtime at least linear in the number of items, a recent line of works has sought to reduce the runtime to ${\rm poly}(k \log n)$, where $n$ is the number of items and $k$ is the number of defectives. In this paper, we present such an algorithm for non-adaptive probabilistic …
Two parties observe independent copies of a $d$-dimensional vector and a scalar. They seek to test if their data is correlated or not, namely they seek to test if the norm $\|ρ\|_2$ of the correlation vector $ρ$ between their observations exceeds $τ$ or is it $0$. To that end, they communicate interactively and declare the output of the test. We show that roughly order $d/τ^2$ bits of communication are sufficient and necessary for resolving the distributed correlation testing problem above. Furthermore, we establish a lower bound of roughly $d^2/τ^2$ bits for communication needed for distribut…
A program can be considered as a device that generates discrete time signals, where a signal is an execution. Shannon information rate, or bit rate, of the signals may not be uniformly distributed. When the program is specified by a finite state transition system, algorithms are provided in identifying information-rich components. For a black-box program that has a partial specification or does not even have a specification, a bit rate signal and its spectrum are studied, which make use of data compression and the Fourier transform. The signal provides a bit-rate coverage for testing the black…
Error bounds for composite quantum hypothesis testing and a new characterization of the weighted Kubo-Ando geometric means
Eötvös Loránd University（罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与创新科学院阿尔弗雷德·雷尼数学研究所）；National University of Singapore（新加坡国立大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
Joint radar-communication (JRC) waveform can be used for simultaneous radar detection and communication in the same frequency band. However, radar detection processing requires the prior knowledge of the waveform including the embedded information for matched filtering. To remove this requirement, we propose a unimodular JRC waveform based on composite modulation where the internal modulation embeds information by mapping the bit sequence to different orthogonal signals, and the external modulation performs phase modulation on the internal waveform to satisfy the demand of detection. By adjust…
Thermal jitter (phase noise) from a free-running ring oscillator is a common, easily implementable physical randomness source in True Random Number Generators (TRNGs). We show how to evaluate entropy, autocorrelation, and bit pattern distributions of ring oscillator noise sources, even with low jitter levels or some bias. Entropy justification is required in NIST 800-90B and AIS-31 testing and for applications such as the RISC-V entropy source extension. Our numerical evaluation algorithms outperform Monte Carlo simulations in speed and accuracy. We also propose a new lower bound estimation fo…
