On the composition of integral operators acting in tempered Colombeau algebras
We show that the generalised composition of generalised integral operators is well defined on the space Colombeau algebras of tempered generalised functions.
math.FA↗
arXiv subjects
Ian Fuss
Publications and source records attributed to Ian Fuss.
Critical initialisation in continuous approximations of binary neural networks
The training of stochastic neural network models with binary ($\pm1$) weights and activations via continuous surrogate networks is investigated. We derive new surrogates using a novel derivation based on writing the stochastic neural network as a Markov chain. This derivation also encompasses existing variants of the surrogates presented in the literature. Following this, we theoretically study the surrogates at initialisation. We derive, using mean field theory, a set of scalar equations describing how input signals propagate through the randomly initialised networks. The equations reveal whether so-called critical initialisations exist for each surrogate network, where the network can be trained to arbitrary depth. Moreover, we predict theoretically and confirm numerically, that common weight initialisation schemes used in standard continuous networks, when applied to the mean values of the stochastic binary weights, yield poor training performance. This study shows that, contrary to common intuition, the means of the stochastic binary weights should be initialised close to $\pm 1$, for deeper networks to be trainable.
stat.ML↗
An analytic solution for one-dimensional quantum walks
The first general analytic solutions for the one-dimensional walk in position and momentum space are derived. These solutions reveal, among other things, new symmetry features of quantum walk probability densities and further insight into the behaviour of their moments. The analytic expressions for the quantum walk probability distributions provide a means of modelling quantum phenomena that is analogous to that provided by random walks in the classical domain.
quant-ph↗
Momentum Dynamics of One Dimensional Quantum Walks
We derive the momentum space dynamic equations and state functions for one dimensional quantum walks by using linear systems and Lie group theory. The momentum space provides an analytic capability similar to that contributed by the z transform in discrete systems theory. The state functions at each time step are expressed as a simple sum of three Chebyshev polynomials. The functions provide an analytic expression for the development of the walks with time.
quant-ph↗
Quantum Reed-Muller Codes
This paper presents a set of quantum Reed-Muller codes which are typically 100 times more effective than existing quantum Reed-Muller codes.
quant-ph↗