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Register a new userOn the Limits of Sequential Testing in High Dimensions
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Added by
Matt Malloy
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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This paper presents results pertaining to sequential methods for support recovery of sparse signals in noise. Specifically, we show that any sequential measurement procedure fails provided the average number of measurements per dimension grows slower then log s / D(f0||f1) where s is the level of sparsity, and D(f0||f1) the Kullback-Leibler divergence between the underlying distributions. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than log n / D(f1||f0), where n is the total dimension of the problem. Lastly, we show that a simple procedure termed sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than (log s + log log n) / D(f0||f1), a mere additive factor more than the lower bound.
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In this paper study the problem of signal detection in Gaussian noise in a distributed setting. We derive a lower bound on the size that the signal needs to have in order to be detectable. Moreover, we exhibit optimal distributed testing strategies that attain the lower bound.
Nowadays data compressors are applied to many problems of text analysis, but many such applications are developed outside of the framework of mathematical statistics. In this paper we overcome this obstacle and show how several methods of classical mathematical statistics can be developed based on applications of the data compressors.
We consider the problem of decentralized sequential active hypothesis testing (DSAHT), where two transmitting agents, each possessing a private message, are actively helping a third agent--and each other--to learn the message pair over a discrete memoryless multiple access channel (DM-MAC). The third agent (receiver) observes the noisy channel output, which is also available to the transmitting agents via noiseless feedback. We formulate this problem as a decentralized dynamic team, show that optimal transmission policies have a time-invariant domain, and characterize the solution through a dynamic program. Several alternative formulations are discussed involving time-homogenous cost functions and/or variable-length codes, resulting in solutions described through fixed-point, Bellman-type equations. Subsequently, we make connections with the problem of simplifying the multi-letter capacity expressions for the noiseless feedback capacity of the DM-MAC. We show that restricting attention to distributions induced by optimal transmission schemes for the DSAHT problem, without loss of optimality, transforms the capacity expression, so that it can be thought of as the average reward received by an appropriately defined stochastic dynamical system with time-invariant state space.
We revisit the universal outlier hypothesis testing (Li emph{et al.}, TIT 2014) and derive fundamental limits for the optimal test. In outlying hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated according to anomalous distributions. The nominal and anomalous distributions are emph{unknown}. We study the tradeoff among the probabilities of misclassification error, false alarm and false reject for tests that satisfy weak conditions on the rate of decrease of these error probabilities as a function of sequence length. Specifically, we propose a threshold-based universal test that ensures exponential decay of misclassification error and false alarm probabilities. We study two constraints on the false reject probabilities, one is that it be a non-vanishing constant and the other is that it have an exponential decay rate. For both cases, we characterize bounds on the false reject probability, as a function of the threshold, for each pair of nominal and anomalous distributions and demonstrate the optimality of our test in the generalized Neyman-Pearson sense. We first consider the case of at most one outlier and then generalize our results to the case of multiple outliers where the number of outliers is unknown and each outlier can follow a different anomalous distribution.
The problem of designing optimal quantization rules for sequential detectors is investigated. First, it is shown that this task can be solved within the general framework of active sequential detection. Using this approach, the optimal sequential detector and the corresponding quantizer are characterized and their properties are briefly discussed. In particular, it is shown that designing optimal quantization rules requires solving a nonconvex optimization problem, which can lead to issues in terms of computational complexity and numerical stability. Motivated by these difficulties, two performance bounds are proposed that are easier to evaluate than the true performance measures and are potentially tighter than the bounds currently available in the literature. The usefulness of the bounds and the properties of the optimal quantization rules are illustrated with two numerical examples.
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