Strong regularity of matrices in a discrete bounded bottleneck algebra

AbstractThe results concerning strong regularity of matrices over bottleneck algebras are reviewed. We extend the known conditions to the discrete bounded case and modify the known algorithms for testing strong regularity

Persistence Modules on Commutative Ladders of Finite Type

We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems. A new algebraic framework deals with persistence modules as representations on associative algebras and the Auslander-Reiten theory is applied to develop the theoretical and algorithmic foundations. In particular, we prove that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander-Reiten quivers. Furthermore, a generalization of persistence diagrams is introduced by using Auslander-Reiten quivers. We provide an algorithm for computing persistence diagrams for the commutative ladders of length 3 by using the structure of Auslander-Reiten quivers.Comment: 48 page

The general trapezoidal algorithm for strongly regular max–min matrices

AbstractThe problem of the strong regularity for square matrices over a general max–min algebra is considered. An O(n2logn) algorithm for recognition of the strong regularity of a given n×n matrix is proposed. The algorithm works without any restrictions on the underlying max–min algebra, concerning the density, or the boundedness

Efficient Instantiation of Parameterised Boolean Equation Systems to Parity Games

Parameterised Boolean Equation Systems (PBESs) are sequences of Boolean fixed point equations with data variables, used for, e.g., verification of modal μ-calculus formulae for process algebraic specifications with data. Solving a PBES is usually done by instantiation to a Parity Game and then solving the game. Practical game solvers exist, but the instantiation step is the bottleneck. We enhance the instantiation in two steps. First, we transform the PBES to a Parameterised Parity Game (PPG), a PBES with each equation either conjunctive or disjunctive. Then we use LTSmin, that offers transition caching, efficient storage of states and both distributed and symbolic state space generation, for generating the game graph. To that end we define a language module for LTSmin, consisting of an encoding of variables with parameters into state vectors, a grouped transition relation and a dependency matrix to indicate the dependencies between parts of the state vector and transition groups. Benchmarks on some large case studies, show that the method speeds up the instantiation significantly and decreases memory usage drastically

The Ongoing Impact of Modular Localization on Particle Theory

Modular localization is the concise conceptual formulation of causal localization in the setting of local quantum physics. Unlike QM it does not refer to individual operators but rather to ensembles of observables which share the same localization region, as a result it explains the probabilistic aspects of QFT in terms of the impure KMS nature arising from the local restriction of the pure vacuum. Whereas it played no important role in the perturbation theory of low spin particles, it becomes indispensible for interactions which involve higher spin $s\geq1$ fields, where is leads to the replacement of the operator (BRST) gauge theory setting in Krein space by a new formulation in terms of stringlocal fields in Hilbert space. The main purpose of this paper is to present new results which lead to a rethinking of important issues of the Standard Model concerning massive gauge theories and the Higgs mechanism. We place these new findings into the broader context of ongoing conceptual changes within QFT which already led to new nonperturbative constructions of models of integrable QFTs. It is also pointed out that modular localization does not support ideas coming from string theory, as extra dimensions and Kaluza-Klein dimensional reductions outside quasiclassical approximations. Apart from hologarphic projections on null-surfaces, holograhic relations between QFT in different spacetime dimensions violate the causal completeness property, this includes in particular the Maldacena conjecture. Last not least, modular localization sheds light onto unsolved problems from QFT's distant past since it reveals that the Einstein-Jordan conundrum is really an early harbinger of the Unruh effect.Comment: a small text overlap with unpublished arXiv:1201.632

On the dimension of max-min convex sets

We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvability of linear systems and trapezoidal property.Comment: 19 pages, v2: many corrections in the proof

Copula models in machine learning

The introduction of copulas, which allow separating the dependence structure of a multivariate distribution from its marginal behaviour, was a major advance in dependence modelling. Copulas brought new theoretical insights to the concept of dependence and enabled the construction of a variety of new multivariate distributions. Despite their popularity in statistics and financial modelling, copulas have remained largely unknown in the machine learning community until recently. This thesis investigates the use of copula models, in particular Gaussian copulas, for solving various machine learning problems and makes contributions in the domains of dependence detection between datasets, compression based on side information, and variable selection. Our first contribution is the introduction of a copula mixture model to perform dependency-seeking clustering for co-occurring samples from different data sources. The model takes advantage of the great flexibility offered by the copula framework to extend mixtures of Canonical Correlation Analyzers to multivariate data with arbitrary continuous marginal densities. We formulate our model as a non-parametric Bayesian mixture and provide an efficient Markov Chain Monte Carlo inference algorithm for it. Experiments on real and synthetic data demonstrate that the increased flexibility of the copula mixture significantly improves the quality of the clustering and the interpretability of the results. The second contribution is a reformulation of the information bottleneck (IB) problem in terms of a copula, using the equivalence between mutual information and negative copula entropy. Focusing on the Gaussian copula, we extend the analytical IB solution available for the multivariate Gaussian case to meta-Gaussian distributions which retain a Gaussian dependence structure but allow arbitrary marginal densities. The resulting approach extends the range of applicability of IB to non-Gaussian continuous data and is less sensitive to outliers than the original IB formulation. Our third and final contribution is the development of a novel sparse compression technique based on the information bottleneck (IB) principle, which takes into account side information. We achieve this by introducing a sparse variant of IB that compresses the data by preserving the information in only a few selected input dimensions. By assuming a Gaussian copula we can capture arbitrary non-Gaussian marginals, continuous or discrete. We use our model to select a subset of biomarkers relevant to the evolution of malignant melanoma and show that our sparse selection provides reliable predictors

Expectation Propagation for Approximate Inference: Free Probability Framework

We study asymptotic properties of expectation propagation (EP) -- a method for approximate inference originally developed in the field of machine learning. Applied to generalized linear models, EP iteratively computes a multivariate Gaussian approximation to the exact posterior distribution. The computational complexity of the repeated update of covariance matrices severely limits the application of EP to large problem sizes. In this study, we present a rigorous analysis by means of free probability theory that allows us to overcome this computational bottleneck if specific data matrices in the problem fulfill certain properties of asymptotic freeness. We demonstrate the relevance of our approach on the gene selection problem of a microarray dataset.Comment: Both authors are co-first authors. The main body of this paper is accepted for publication in the proceedings of the 2018 IEEE International Symposium on Information Theory (ISIT