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A central challenge in sensory neuroscience is to understand neural computations and circuit mechanisms that underlie the encoding of ethologically relevant, natural stimuli. In multilayered neural circuits, nonlinear processes such as synaptic transmission and spiking dynamics present a significant obstacle to the creation of accurate computational models(More)
Inspired by biophysical principles underlying nonlinear dendritic computation in neural circuits , we develop a scheme to train deep neu-ral networks to make them robust to adversar-ial attacks. Our scheme generates highly nonlin-ear, saturated neural networks that achieve state of the art performance on gradient based adver-sarial examples on MNIST,(More)
Given a random permutation f : [N ] → [N ] as a black box and y ∈ [N ], we want to output x = f −1 (y). Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but not on the input y. Classically, there is a data structure of size˜O(S) and an algorithm that with the(More)
Let R k (n) be the number of representations of an integer n as the sum of a prime and a k-th power for k ≥ 2. Furthermore, set E k (X) = |{n ≤ X, n ∈ I k , n not a sum of a prime and a k-th power}|. In the present paper we use sieve techniques to obtain a strong upper bound on R k (n) for n ≤ X with no exceptions, and we improve upon the results of A.(More)