Graph Kernels

We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels, and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of Fröhlich et al. (2006) yet provably positive semi-definite

A weighted pair graph representation for reconstructibility of Boolean control networks

A new concept of weighted pair graphs (WPGs) is proposed to represent a new reconstructibility definition for Boolean control networks (BCNs), which is a generalization of the reconstructibility definition given in [Fornasini & Valcher, TAC2013, Def. 4]. Based on the WPG representation, an effective algorithm for determining the new reconstructibility notion for BCNs is designed with the help of the theories of finite automata and formal languages. We prove that a BCN is not reconstructible iff its WPG has a complete subgraph. Besides, we prove that a BCN is reconstructible in the sense of [Fornasini & Valcher, TAC2013, Def. 4] iff its WPG has no cycles, which is simpler to be checked than the condition in [Fornasini & Valcher, TAC2013, Thm. 4].Comment: 20 pages, 10 figures, accepted by SIAM Journal on Control and Optimizatio

Lattice Gauge Tensor Networks

We present a unified framework to describe lattice gauge theories by means of tensor networks: this framework is efficient as it exploits the high amount of local symmetry content native of these systems describing only the gauge invariant subspace. Compared to a standard tensor network description, the gauge invariant one allows to speed-up real and imaginary time evolution of a factor that is up to the square of the dimension of the link variable. The gauge invariant tensor network description is based on the quantum link formulation, a compact and intuitive formulation for gauge theories on the lattice, and it is alternative to and can be combined with the global symmetric tensor network description. We present some paradigmatic examples that show how this architecture might be used to describe the physics of condensed matter and high-energy physics systems. Finally, we present a cellular automata analysis which estimates the gauge invariant Hilbert space dimension as a function of the number of lattice sites and that might guide the search for effective simplified models of complex theories.Comment: 28 pages, 9 figure

M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities

We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing and simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update

Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables

Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a sequence of graphs. In this paper, we formulate Koopman spectral analysis for NLDSs with structures among observables and propose an estimation algorithm for this problem. This method can extract and visualize the underlying low-dimensional global dynamics of NLDSs with structures among observables from data, which can be useful in understanding the underlying dynamics of such NLDSs. To this end, we first formulate the problem of estimating spectra of the Koopman operator defined in vector-valued reproducing kernel Hilbert spaces, and then develop an estimation procedure for this problem by reformulating tensor-based DMD. As a special case of our method, we propose the method named as Graph DMD, which is a numerical algorithm for Koopman spectral analysis of graph dynamical systems, using a sequence of adjacency matrices. We investigate the empirical performance of our method by using synthetic and real-world data.Comment: 34 pages with 4 figures, Published in Neural Networks, 201

Lattice QED photonic wavepackets on ladder geometries

openIn this thesis we explore numerical simulations, including Tensor Networks (TNs) methods, to study Hamiltonian Lattice Gauge Theories (LGTs), a numerical framework for investigating non-perturbative properties of Quantum Field Theories. We develop a model-independent approach for constructing Matrix Product Operators (MPOs) representations of 1-dimensional quasiparticles with definite momenta, and apply it to Hamiltonian Lattice Quantum Electrodynamics (QED) on a ladder geometry. By means of exact diagonalization at intermediate system sizes, we obtain the first excitation band states (the Bloch functions) representing the single-(quasi)particle states (the photons) expressed as entangled states of local lattice gauge fields. We then construct the corresponding maximally-localized Wannier functions through minimization of a spread functional. Once we identify, via a linear algebra problem, the operation that constructs the localized Wannier excitation from the ground state (dressed vacuum), we can express the creation operator, for any wavepacket of such quasiparticles, as a Matrix Product Operator. The aforementioned steps constitute a constructive strategy to prepare an arbitrary input state for a quasiparticle scattering simulation in real time, and the scattering process itself can be carried out with any standard algorithm for time-evolution with Matrix Product States.In this thesis we explore numerical simulations, including Tensor Networks (TNs) methods, to study Hamiltonian Lattice Gauge Theories (LGTs), a numerical framework for investigating non-perturbative properties of Quantum Field Theories. We develop a model-independent approach for constructing Matrix Product Operators (MPOs) representations of 1-dimensional quasiparticles with definite momenta, and apply it to Hamiltonian Lattice Quantum Electrodynamics (QED) on a ladder geometry. By means of exact diagonalization at intermediate system sizes, we obtain the first excitation band states (the Bloch functions) representing the single-(quasi)particle states (the photons) expressed as entangled states of local lattice gauge fields. We then construct the corresponding maximally-localized Wannier functions through minimization of a spread functional. Once we identify, via a linear algebra problem, the operation that constructs the localized Wannier excitation from the ground state (dressed vacuum), we can express the creation operator, for any wavepacket of such quasiparticles, as a Matrix Product Operator. The aforementioned steps constitute a constructive strategy to prepare an arbitrary input state for a quasiparticle scattering simulation in real time, and the scattering process itself can be carried out with any standard algorithm for time-evolution with Matrix Product States

Matrix product symmetries and breakdown of thermalization from hard rod deformations

We construct families of exotic spin-1/2 chains using a procedure called ``hard rod deformation''. We treat both integrable and non-integrable examples. The models possess a large non-commutative symmetry algebra, which is generated by matrix product operators with fixed small bond dimension. The symmetries lead to Hilbert space fragmentation and to the breakdown of thermalization. As an effect, the models support persistent oscillations in non-equilibrium situations. Similar symmetries have been reported earlier in integrable models, but here we show that they also occur in non-integrable cases.Comment: v2: references correcte