Representing Patterns of autonomous agent dynamics in multi-robot systems

It is proposed that vocabularies for representing complex systems with interacting agents have a natural lattice hierarchical structure. We investigate this for the example of simulated robot soccer, using data taken from the RoboCup simulation competition. Lattice hierarchies provide symbolic representations for reasoning about systems at appropriate levels. We note the difference between relational constructs being human supplied versus systems that abstract their own constructs autonomously. The lattice hierarchical representation underlies both

Non-commutative quantum geometric data in group field theories

We review briefly the motivations for introducing additional group-theoretic data in tensor models, leading to the richer framework of group field theories, themselves a field theory formulation of loop quantum gravity. We discuss how these data give to the GFT amplitudes the structure of lattice gauge theories and simplicial gravity path integrals, and make their quantum geometry manifest. We focus in particular on the non-commutative flux/algebra representation of these models.Comment: 10 pages; to appear in the proceedings of the workshop "Non-commutative field theory and gravity", Corfu', Greece, EU, September 201

Spectra of massive and massless QCD Dirac operators: A novel link

We show that integrable structure of chiral random matrix models incorporating global symmetries of QCD Dirac operators (labeled by the Dyson index beta=1,2, and 4) leads to emergence of a connection relation between the spectral statistics of massive and massless Dirac operators. This novel link established for beta-fold degenerate massive fermions is used to explicitly derive (and prove the random matrix universality of) statistics of low--lying eigenvalues of QCD Dirac operators in the presence of SU(2) massive fermions in the fundamental representation (beta=1) and SU(N_c >= 2) massive adjoint fermions (beta=4). Comparison with available lattice data for SU(2) dynamical staggered fermions reveals a good agreement

The free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model

We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,.. and for the lattice euclidean scalar field theory with quartic self-interaction, on the simple cubic and the body-centered cubic lattices. Our bivariate high-temperature expansions, which extend through K^24, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.Comment: 22 pages, 10 figure

Geometric Representation Learning

Vector embedding models are a cornerstone of modern machine learning methods for knowledge representation and reasoning. These methods aim to turn semantic questions into geometric questions by learning representations of concepts and other domain objects in a lower-dimensional vector space. In that spirit, this work advocates for density- and region-based representation learning. Embedding domain elements as geometric objects beyond a single point enables us to naturally represent breadth and polysemy, make asymmetric comparisons, answer complex queries, and provides a strong inductive bias when labeled data is scarce. We present a model for word representation using Gaussian densities, enabling asymmetric entailment judgments between concepts, and a probabilistic model for weighted transitive relations and multivariate discrete data based on a lattice of axis-aligned hyperrectangle representations (boxes). We explore the suitability of these embedding methods in different regimes of sparsity, edge weight, correlation, and independence structure, as well as extensions of the representation and different optimization strategies. We make a theoretical investigation of the representational power of the box lattice, and propose extensions to address shortcomings in modeling difficult distributions and graphs