On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D

Parameterizing and Aggregating Activation Functions in Deep Neural Networks

The nonlinear activation functions applied by each neuron in a neural network are essential for making neural networks powerful representational models. If these are omitted, even deep neural networks reduce to simple linear regression due to the fact that a linear combination of linear combinations is still a linear combination. In much of the existing literature on neural networks, just one or two activation functions are selected for the entire network, even though the use of heterogenous activation functions has been shown to produce superior results in some cases. Even less often employed are activation functions that can adapt their nonlinearities as network parameters along with standard weights and biases. This dissertation presents a collection of papers that advance the state of heterogenous and parameterized activation functions. Contributions of this dissertation include three novel parametric activation functions and applications of each, a study evaluating the utility of the parameters in parametric activation functions, an aggregated activation approach to modeling time-series data as an alternative to recurrent neural networks, and an improvement upon existing work that aggregates neuron inputs using product instead of sum

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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