The geometry of the double gyroid wire network: quantum and classical

Quantum wire networks have recently become of great interest. Here we deal with a novel nano material structure of a Double Gyroid wire network. We use methods of commutative and non-commutative geometry to describe this wire network. Its non--commutative geometry is closely related to non-commutative 3-tori as we discuss in detail.Comment: pdflatex 9 Figures. Minor changes, some typos and formulation

High-Resolution Shape Completion Using Deep Neural Networks for Global Structure and Local Geometry Inference

We propose a data-driven method for recovering miss-ing parts of 3D shapes. Our method is based on a new deep learning architecture consisting of two sub-networks: a global structure inference network and a local geometry refinement network. The global structure inference network incorporates a long short-term memorized context fusion module (LSTM-CF) that infers the global structure of the shape based on multi-view depth information provided as part of the input. It also includes a 3D fully convolutional (3DFCN) module that further enriches the global structure representation according to volumetric information in the input. Under the guidance of the global structure network, the local geometry refinement network takes as input lo-cal 3D patches around missing regions, and progressively produces a high-resolution, complete surface through a volumetric encoder-decoder architecture. Our method jointly trains the global structure inference and local geometry refinement networks in an end-to-end manner. We perform qualitative and quantitative evaluations on six object categories, demonstrating that our method outperforms existing state-of-the-art work on shape completion.Comment: 8 pages paper, 11 pages supplementary material, ICCV spotlight pape

Tensor network states and geometry

Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D=1 dimensions, as well as projected entangled pair states (PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D=1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law -- that is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D>1 dimensions.Comment: 18 pages, 18 figure

Ising Spin Network States for Loop Quantum Gravity: a Toy Model for Phase Transitions

Non-perturbative approaches to quantum gravity call for a deep understanding of the emergence of geometry and locality from the quantum state of the gravitational field. Without background geometry, the notion of distance should entirely emerge from the correlations between the gravity fluctuations. In the context of loop quantum gravity, quantum states of geometry are defined as spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. Here, we develop the condensed matter point of view on extracting the physical and geometrical information out of spin network states: we introduce new Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. We construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints which entirely characterize our states. We discuss their phase diagram and show how the distance can be reconstructed from the correlations in the various phases. Finally, we propose generalizations of these Ising states, which open the perspective to study the coarse graining and dynamics of spin network states using well-known condensed matter techniques and results.Comment: 17 page

The Network Nullspace Property for Compressed Sensing of Big Data over Networks

We present a novel condition, which we term the net- work nullspace property, which ensures accurate recovery of graph signals representing massive network-structured datasets from few signal values. The network nullspace property couples the cluster structure of the underlying network-structure with the geometry of the sampling set. Our results can be used to design efficient sampling strategies based on the network topology

Hyperbolic Geometry of Complex Networks

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as non-interacting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure