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Abstruct-Digital communication systemsare frequently operated over nonlinear channels with memory. The analysis of the performance of these systems is difficult and no complete analytical treatment of the problem has been obtained before. Several recent efforts have been directed toward the computation of error probabilities via Monte-Carlo simulation using a complete system model. These simulations require excessively large sample sizes and are not practical for estimating very low values of error probabilities. This paper presents a modified Monte-Carlo simulation technique for estimating error Probabilities in digital. communication systems operating over nonlinear channels.An importance-sampling technique is used to modify the probability density function of the noise process in a way to make simulation possible. Theoretical results as well as realistic examples are presented, showing that the number of samples needed for simulation is reduced considerably.

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In this paper we propose a novel method that provides contrastive explanations justifying the classification of an input by a black box classifier such as a deep neural network. Given an input we find what should be minimally and sufficiently present (viz. important object pixels in an image) to justify its classification and analogously what should be minimally and necessarily absent (viz. certain background pixels). We argue that such explanations are natural for humans and are used commonly in domains such as health care and criminology. What is minimally but critically absent is an important part of an explanation, which to the best of our knowledge, has not been explicitly identified by current explanation methods that explain predictions of neural networks. We validate our approach on three real datasets obtained from diverse domains; namely, a handwritten digits dataset MNIST, a large procurement fraud dataset and a brain activity strength dataset. In all three cases, we witness the power of our approach in generating precise explanations that are also easy for human experts to understand and evaluate.

--A procedure is developed to extract numerical features which characterize the pore structure of reservoir rocks. The procedure is based on a set of descriptors which give a statistical description of porous media. These features are evaluated from digitized photomicrographs of reservoir rocks and they characterize the rock grain structure in term of (1) the linear dependency of grey tones in the photomicrograph image, (2) the degree of "homogeneity" of the image and (3) the angular variations of the image grey tone dependencies. On the basis of these textural features, a simple identification rule using piecewise linear discriminant functions is developed for categorizing the photomicrograph images. The procedure was applied to a set of 243 distinct images comprising 6 distinct rock categories. The coefficients of the discriminant functions were obtained using 143 training samples. The remaining (100) samples were then processed, each sample being assigned to one of 6 possible sandstone categories. Eighty-nine per cent of the test samples were correctly identified.

The standard risk minimization paradigm of machine learning is brittle when operating in environments whose test distributions are different from the training distribution due to spurious correlations. Training on data from many environments and finding invariant predictors reduces the effect of spurious features by concentrating models on features that have a causal relationship with the outcome. In this work, we pose such invariant risk minimization as finding the Nash equilibrium of an ensemble game among several environments. By doing so, we develop a simple training algorithm that uses best response dynamics and, in our experiments, yields similar or better empirical accuracy with much lower variance than the challenging bi-level optimization problem of [1]. One key theoretical contribution is showing that the set of Nash equilibria for the proposed game are equivalent to the set of invariant predictors for any finite number of environments, even with nonlinear classifiers and transformations. As a result, our method also retains the generalization guarantees to a large set of environments shown in [1]. The proposed algorithm adds to the collection of successful game-theoretic machine learning algorithms such as generative adversarial networks.

Edge detection and enhancement are widely used in image processing applications. In this paper we consider the problem of optimizing spatial frequency domain filters for detecting edges in digital pictures. The filter is optimum in that it produces maximum energy within a resolution interval of specified width in the vicinity of the edge.We show that, in the continuous case, the filter transfer function is specified in terms of the prolate spheroidal wave function. In the discrete case, the filter transfer function is specified in terms of the sampled values of the first-order prolate spheroidal wave function or in terms of the sampled values of an asymptotic approximation of the wave function. Both versions can be implemented via the fast Fourier transform (FFT). We show that the optimum filter is very effective for detecting blufred and noisy edges. Finally, we compare the perfor-

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