Consider the following unequal error protection scenario. One special message, dubbed the “red alert” message, is required to have an extremely small probability of missed detection. The remainder of the messages must keep their average probability of error and probability of false alarm below a certain threshold. The goal then is to design a codebook that maximizes the error exponent of the red alert message while ensuring that the average probability of error and probability of false alarm go to zero as the blocklength goes to infinity. This red alert exponent has previously been characterized for discrete memoryless channels. This paper completely characterizes the optimal red alert exponent for additive white Gaussian noise channels with block power constraints.
The fundamental mechanism through which high red alert exponents are achieved is a binary hypothesis test. By designing the induced distributions at the output of the channel to be far apart as measured by KL divergence, we can distinguish whether the red alert or some standard codeword was sent. The test threshold is biased to minimize the probability of missed detection and is analyzed via an application of Stein's Lemma. This sort of biased hypothesis test occurs in numerous other communication settings with feedback, such as @cite_16 @cite_8 @cite_13 and, as mentioned earlier, these codes are also used as a component in streaming data systems (see, for instance, @cite_3 @cite_19 @cite_15 @cite_2 ). There is also a rich literature on the interplay between hypothesis testing and information theory, which we cannot do justice to here (see, for instance, @cite_0 @cite_4 @cite_18 ).
@cite_18 @cite_4 @cite_8 @cite_3 @cite_0 @cite_19 @cite_2 @cite_15 @cite_16 @cite_13
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The testing of binary hypotheses is developed from an information-theoretic point of view, and the asymptotic performance of optimum hypothesis testers is developed in exact analogy to the asymptotic performance of optimum channel codes. The discrimination, introduced by Kullback, is developed in a role analogous to that of mutual information in channel coding theory. Based on the discrimination, an error-exponent function e(r) is defined. This function is found to describe the behavior of optimum hypothesis testers asymptotically with block length. Next, mutual information is introduced as a minimum of a set of discriminations. This approach has later coding significance. The channel reliability-rate function E(R) is defined in terms of discrimination, and a number of its mathematical properties developed. Sphere-packing-like bounds are developed in a relatively straightforward and intuitive manner by relating e(r) and E (R) . This ties together the aforementioned developments and gives a lower bound in terms of a hypothesis testing model. The result is valid for discrete or continuous probability distributions. The discrimination function is also used to define a source code reliability-rate function. This function allows a simpler proof of the source coding theorem and also bounds the code performance as a function of block length, thereby providing the source coding analog of E (R) . In this paper, we study the detection boundary for minimax hypothesis testing in the context of high-dimensional, sparse binary regression models. Motivated by genetic sequencing association studies for rare variant effects, we investigate the complexity of the hypothesis testing problem when the design matrix is sparse. We observe a new phenomenon in the behavior of detection boundary which does not occur in the case of Gaussian linear regression. We derive the detection boundary as a function of two components: a design matrix sparsity index and signal strength, each of which is a function of the sparsity of the alternative. For any alternative, if the design matrix sparsity index is too high, any test is asymptotically powerless irrespective of the magnitude of signal strength. For binary design matrices with the sparsity index that is not too high, our results are parallel to those in the Gaussian case. In this context, we derive detection boundaries for both dense and sparse regimes. For the dense regime, we show that the generalized likelihood ratio is rate optimal; for the sparse regime, we propose an extended Higher Criticism Test and show it is rate optimal and sharp. We illustrate the finite sample properties of the theoretical results using simulation studies. The problem of binary hypothesis testing is considered in a bandwidth-constrained densely populated low-power wireless sensor network operating over insecure links. Observations of the sensors are quantized and encrypted before transmission. The encryption method maps the output of the quantizer to one of the possible quantizer output levels randomly according to a probability matrix. The intended (ally) fusion center (AFC) is aware of the encryption keys (probabilities) while the unauthorized (third party) fusion center (TPFC) is not. A constrained optimization problem is formulated from the point of view of AFC in order to design its decision rule along with the encryption probabilities. The objective function to be minimized is the error probability of AFC and the constraint is a lower bound on the error probability of TPFC. In the binary case the optimal solution is found and in the nonbinary case a good suboptimal solution is analytically obtained. Numerical results are presented to show that it is possible to degrade the error probability of TPFC significantly and still achieve very low probability of error for AFC. The proposed method which may be considered a PHY-layer security scheme is highly scalable since it does not increase the packet overhead or transmit power of the sensors and has very low computational complexity. A scheme is described to randomize the keys so as to defeat any key space exploration attack. In object detection, an intersection over union (IoU) threshold is required to define positives and negatives. An object detector, trained with low IoU threshold, e.g. 0.5, usually produces noisy detections. However, detection performance tends to degrade with increasing the IoU thresholds. Two main factors are responsible for this: 1) overfitting during training, due to exponentially vanishing positive samples, and 2) inference-time mismatch between the IoUs for which the detector is optimal and those of the input hypotheses. A multi-stage object detection architecture, the Cascade R-CNN, is proposed to address these problems. It consists of a sequence of detectors trained with increasing IoU thresholds, to be sequentially more selective against close false positives. The detectors are trained stage by stage, leveraging the observation that the output of a detector is a good distribution for training the next higher quality detector. The resampling of progressively improved hypotheses guarantees that all detectors have a positive set of examples of equivalent size, reducing the overfitting problem. The same cascade procedure is applied at inference, enabling a closer match between the hypotheses and the detector quality of each stage. A simple implementation of the Cascade R-CNN is shown to surpass all single-model object detectors on the challenging COCO dataset. Experiments also show that the Cascade R-CNN is widely applicable across detector architectures, achieving consistent gains independently of the baseline detector strength. The code will be made available at this https URL.