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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)
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)
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)
We consider the quantum time complexity of the all pairs shortest paths (APSP) problem and some of its variants. The trivial classical algorithm for APSP and most all pairs path problems runs in O(n 3) time, while the trivial algorithm in the quantum setting runs iñ O(n 2.5) time, using Grover search. A major open problem in classical algorithms is to(More)
For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to(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)