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

Algorithmic counting of nonequivalent compact Huffman codes

It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees (level-greedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Efficient Higher Order Derivatives of Objective Functions Composed of Matrix Operations

This paper is concerned with the efficient evaluation of higher-order derivatives of functions $f$ that are composed of matrix operations. I.e., we want to compute the $D$-th derivative tensor $\nabla^D f(X) \in \mathbb R^{N^D}$, where $f:\mathbb R^{N} \to \mathbb R$ is given as an algorithm that consists of many matrix operations. We propose a method that is a combination of two well-known techniques from Algorithmic Differentiation (AD): univariate Taylor propagation on scalars (UTPS) and first-order forward and reverse on matrices. The combination leads to a technique that we would like to call univariate Taylor propagation on matrices (UTPM). The method inherits many desirable properties: It is easy to implement, it is very efficient and it returns not only $\nabla^D f$ but yields in the process also the derivatives $\nabla^d f$ for $d \leq D$. As performance test we compute the gradient $\nabla f(X)$ % and the Hessian $\nabla_A^2 f(A)$ by a combination of forward and reverse mode of f(X) = \trace (X^{-1}) in the reverse mode of AD for $X \in \mathbb R^{n \times n}$. We observe a speedup of about 100 compared to UTPS. Due to the nature of the method, the memory footprint is also small and therefore can be used to differentiate functions that are not accessible by standard methods due to limited physical memory