Lukasiewicz mu-Calculus

We consider state-based systems modelled as coalgebras whose type incorporates branching, and show that by suitably adapting the definition of coalgebraic bisimulation, one obtains a general and uniform account of the linear-time behaviour of a state in such a coalgebra. By moving away from a boolean universe of truth values, our approach can measure the extent to which a state in a system with branching is able to exhibit a particular linear-time behaviour. This instantiates to measuring the probability of a specific behaviour occurring in a probabilistic system, or measuring the minimal cost of exhibiting a specific behaviour in the case of weighted computations

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n$ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors

Extension of Wirtinger's Calculus to Reproducing Kernel Hilbert Spaces and the Complex Kernel LMS

Over the last decade, kernel methods for nonlinear processing have successfully been used in the machine learning community. The primary mathematical tool employed in these methods is the notion of the Reproducing Kernel Hilbert Space. However, so far, the emphasis has been on batch techniques. It is only recently, that online techniques have been considered in the context of adaptive signal processing tasks. Moreover, these efforts have only been focussed on real valued data sequences. To the best of our knowledge, no adaptive kernel-based strategy has been developed, so far, for complex valued signals. Furthermore, although the real reproducing kernels are used in an increasing number of machine learning problems, complex kernels have not, yet, been used, in spite of their potential interest in applications that deal with complex signals, with Communications being a typical example. In this paper, we present a general framework to attack the problem of adaptive filtering of complex signals, using either real reproducing kernels, taking advantage of a technique called \textit{complexification} of real RKHSs, or complex reproducing kernels, highlighting the use of the complex gaussian kernel. In order to derive gradients of operators that need to be defined on the associated complex RKHSs, we employ the powerful tool of Wirtinger's Calculus, which has recently attracted attention in the signal processing community. To this end, in this paper, the notion of Wirtinger's calculus is extended, for the first time, to include complex RKHSs and use it to derive several realizations of the Complex Kernel Least-Mean-Square (CKLMS) algorithm. Experiments verify that the CKLMS offers significant performance improvements over several linear and nonlinear algorithms, when dealing with nonlinearities.Comment: 15 pages (double column), preprint of article accepted in IEEE Trans. Sig. Pro

Differentiating the multipoint Expected Improvement for optimal batch design

This work deals with parallel optimization of expensive objective functions which are modeled as sample realizations of Gaussian processes. The study is formalized as a Bayesian optimization problem, or continuous multi-armed bandit problem, where a batch of q > 0 arms is pulled in parallel at each iteration. Several algorithms have been developed for choosing batches by trading off exploitation and exploration. As of today, the maximum Expected Improvement (EI) and Upper Confidence Bound (UCB) selection rules appear as the most prominent approaches for batch selection. Here, we build upon recent work on the multipoint Expected Improvement criterion, for which an analytic expansion relying on Tallis' formula was recently established. The computational burden of this selection rule being still an issue in application, we derive a closed-form expression for the gradient of the multipoint Expected Improvement, which aims at facilitating its maximization using gradient-based ascent algorithms. Substantial computational savings are shown in application. In addition, our algorithms are tested numerically and compared to state-of-the-art UCB-based batch-sequential algorithms. Combining starting designs relying on UCB with gradient-based EI local optimization finally appears as a sound option for batch design in distributed Gaussian Process optimization