In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Q: We give a new algorithm for learning a two-layer neural network under a very general class of input distributions. Assuming there is a ground-truth two-layer network
y = A \sigma(Wx) + \xi,
where A, W are weight matrices, \xi represents noise, and the number of neurons in the hidden layer is no larger than the input or output, our algorithm is guaranteed to recover the parameters A, W of the ground-truth network. The only requirement on the input x is that it is symmetric, which still allows highly complicated and structured input.

Our algorithm is based on the method-of-moments framework and extends several results in tensor decompositions. We use spectral algorithms to avoid the complicated non-convex optimization in learning neural networks. Experiments show that our algorithm can robustly learn the ground-truth neural network with a small number of samples for many symmetric input distributions.

A: We give an algorithm for learning a two-layer neural network with symmetric input distribution.
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Q: Temporal logics are useful for describing dynamic system behavior, and have been successfully used as a language for goal definitions during task planning. Prior works on inferring temporal logic specifications have focused on "summarizing" the input dataset -- i.e., finding specifications that are satisfied by all plan traces belonging to the given set. In this paper, we examine the problem of inferring specifications that describe temporal differences between two sets of plan traces. We formalize the concept of providing such contrastive explanations, then present a Bayesian probabilistic model for inferring contrastive explanations as linear temporal logic specifications. We demonstrate the efficacy, scalability, and robustness of our model for inferring correct specifications across various benchmark planning domains and for a simulated air combat mission.

A: We present a Bayesian inference model to infer contrastive explanations (as LTL specifications) describing how two sets of plan traces differ.
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Q: We develop new approximation and statistical learning theories of convolutional neural networks (CNNs) via the ResNet-type structure where the channel size, filter size, and width are fixed. It is shown that a ResNet-type CNN is a universal approximator and its expression ability is no worse than fully-connected neural networks (FNNs) with a \textit{block-sparse} structure even if the size of each layer in the CNN is fixed. Our result is general in the sense that we can automatically translate any approximation rate achieved by block-sparse FNNs into that by CNNs. Thanks to the general theory, it is shown that learning on CNNs satisfies optimality in approximation and estimation of several important function classes.

As applications, we consider two types of function classes to be estimated: the Barron class and H\"older class. We prove the clipped empirical risk minimization (ERM) estimator can achieve the same rate as FNNs even the channel size, filter size, and width of CNNs are constant with respect to the sample size. This is minimax optimal (up to logarithmic factors) for the H\"older class. Our proof is based on sophisticated evaluations of the covering number of CNNs and the non-trivial parameter rescaling technique to control the Lipschitz constant of CNNs to be constructed.

A:
It is shown that ResNet-type CNNs are a universal approximator and its expression ability is not worse than fully connected neural networks (FNNs) with a extit{block-sparse} structure even if the size of each layer in the CNN is fixed.
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