Explicit Decomposition Research Articles

Modelling network motifs as higher order interactions: a statistical inference based approach

The prevalent approach to motif analysis seeks to describe the local connectivity structure of networks by identifying subgraph patterns that appear significantly more often in a network then expected under a null model that conserves certain features of the original network. In this article we advocate for an alternative approach based on statistical inference of generative models where nodes are connected not only by edges but also copies of higher order subgraphs. These models naturally lead to the consideration of latent states that correspond to decompositions of networks into higher order interactions in the form of subgraphs that can have the topology of any simply connected motif. Being based on principles of parsimony the method can infer concise sets of motifs from within thousands of candidates allowing for consistent detection of larger motifs. The inferential approach yields not only a set of statistically significant higher order motifs but also an explicit decomposition of the network into these motifs, which opens new possibilities for the systematic study of the topological and dynamical implications of higher order connectivity structures in networks. After briefly reviewing core concepts and methods, we provide example applications to empirical data sets and discuss how the inferential approach addresses current problems in motif analysis and explore how concepts and methods common to motif analysis translate to the inferential framework.

Implications of climate and litter quality for simulations of litterbag decomposition at high latitudes

Abstract. Litter decomposition is a vital part of the carbon cycle and is thoroughly studied both in the field and with models. Although temporally and spatially limited, litterbag decomposition experiments are often used to calibrate and evaluate soil models, coupled to land models, that are intended for use on large scales. We used the microbially explicit soil decomposition model MIMICS+ to replicate two high-latitude litterbag decomposition experiments of different spatial and temporal scales. We investigated how well the model represented observed mass loss in terms of the controlling factors of climate and litter quality and their relative importance with time. In addition to default model forcing, we used measured and site-specific model-derived microclimatic variables (soil moisture and temperature), hypothesizing that this would improve model results. We found that MIMICS+ represented mass loss after 1, 3, and 6 years well across a climatic gradient of Canadian sites but had more variable results for 1-year mass loss across a climate grid in southern Norway. In terms of litter quality, the litter metabolic fraction had more influence on modeled mass loss than the carbon-to-nitrogen ratio of the litter. Using alternative microclimate sources led to up to 23 % more mass remaining and down to 22 % less mass remaining compared to the simulations using default model inputs. None of the input alternatives significantly improved results compared to using the default model setup. We discuss possible causes for our findings and suggest measures to better utilize short-term field experiments to inform microbially explicit decomposition models.

Irreducible Cartesian tensor decomposition: A computational approach

Cartesian tensors are widely used in physics and chemistry, especially for the formulation of linear and nonlinear spectroscopies as well as for molecular response properties. In this work, we review the problem of irreducible Cartesian tensor (ICT) decomposition of a generic Cartesian tensor of rank n into its irreducible parts, each characterized by a specific symmetry. The matrix formulation of the ICT decomposition is structurally similar to the problem of rotational averaging using isotropic Cartesian tensors. Analogously to the latter, the ICT decomposition can be considered as a problem of selecting a set of permutations of n indices that provides a linearly independent set of mappings between Cartesian tensor subspaces. This selection can be performed using a simple computational approach based on the reduced row echelon form (rref) algorithm. This protocol has been implemented in a computer code used to re-derive the already known ICT decomposition for 2 ≤ n ≤ 4. Finally, for the first time, we performed the explicit ICT decomposition of a Cartesian tensor of rank n = 5.

Ansatz optimization of the variational quantum eigensolver tested on the atomic Anderson model

We present a detailed analysis and optimization of the variational quantum algorithms required to find the ground state of a correlated electron model, using several types of variational ansatz. Specifically, we apply our approach to the atomic limit of the Anderson model, which is widely studied in condensed matter physics since it can simulate fundamental physical phenomena, ranging from magnetism to superconductivity. The method is developed by presenting efficient state preparation circuits that exhibit total spin, spin projection, particle number and time-reversal symmetries. These states contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace allowing to avoid irrelevant sectors of Hilbert space. Then, we show how to construct quantum circuits, providing explicit decomposition and gate count in terms of standard gate sets. We test these quantum algorithms looking at ideal quantum computer simulations as well as implementing quantum noisy simulations. We finally perform an accurate comparative analysis among the approaches implemented, highlighting their merits and shortcomings.

Hyperbolic one-relator groups

Abstract We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $2$ -free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.

Mutually unbiased bases: polynomial optimization and symmetry

A set of k orthonormal bases of Cd is called mutually unbiased if |⟨e,f⟩|2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d. The (well-known) upper bound k≤d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk.We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk-module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12.

Generative adversarial network-enhanced directional seismic wavefield decomposition and its application in reverse time migration

Reverse time migration (RTM) is an accurate method for imaging complex geologic structures without imposing any dip limitations. However, a large amount of high-amplitude low-frequency noise, which is mainly generated by the crosscorrelation of source and receiver wavefields propagating in the same directions, seriously contaminates the image quality. The causal imaging condition with separated up- and downgoing wavefields is an effective approach to reduce these low-frequency artifacts. Explicit up- and downgoing wavefield decomposition based on the Hilbert transform is computationally expensive due to additional wavefield extrapolation and storage for the imaginary parts. Directionally propagating wavefields have distinctive kinematic patterns such as traveltime and wavefront curvature, which provide us an opportunity to implement the wavefield decomposition using the statistical neural network method. Using extrapolated wavefields as the input and the decomposed up-, down-, left-, and rightgoing wavefields as the labeled data, we train a pair of generative adversarial networks to predict the directional wavefields. The training data sets are generated using seismic full-waveform modeling and explicit wavefield decomposition based on the Hilbert transform. Then, the decomposed directional wavefields are incorporated into a novel imaging condition that depends on the subsurface dip angles to compute the reflectivity perpendicular to the reflectors. Numerical experiments demonstrate that our method can produce accurate directional wavefield decomposition results and high-quality reflectivity images without low-wavenumber artifacts.

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

This paper presents a method for parameterizing new Lorentz spacetime coordinates based on coupled parameters. The role of symmetry in rapidity in special relativity is explored, and invariance is obtained for new spacetime intervals with respect to the Lorentz transformation. Using the Euler–Hamilton equations, an additional angular rapidity and perpendicular rapidity are obtained, and the Hamiltonian and Lagrangian of a relativistic particle are expanded into rapidity spectra. A so-called passage to the limit is introduced that makes it possible to decompose physical quantities into spectra in terms of elementary functions when explicit decomposition is difficult. New rapidity-dependent Lorentz spacetime coordinates are obtained. The descriptions of particle motion using the old and new Lorentz spacetime coordinates as applied to plane laser pulses are compared in terms of the particle kinetic energy. Based on a classical model of particle motion in the field of a plane monochromatic electromagnetic wave and that of a plane laser pulse, rapidity-dependent spectral decompositions into elementary functions are presented, and the Euler–Hamilton equations are derived as rapidity functions in 3+1 dimensions. The new and old Lorentz spacetime coordinates are compared with the Fermi spacetime coordinates. The proper Lorentz groups SO(1,3) with coupled parameters using the old and new Lorentz spacetime coordinates are also compared. As a special case, the application of Lorentz spacetime coordinates to a relativistic hydrodynamic system with coupled parameters in 1+1 dimensions is demonstrated.

GBP: Graph convolutional network embedded in bilinear pooling for fine-grained encoding

In fine-grained recognition, classical high-order coding has inherent contradiction between visual burstiness and feature redundancy, the core of which is the inherent instability of high-order features. Existing methods mainly use EIG and SVD decomposition to maintain feature stability, but this process increases feature redundancy. To address this problem, this paper proposes a Graph Bilinear Pooling (GBP) model to obtain stable fine-grained features through the effective aggregation ability of graph networks. GBP avoids explicit feature decomposition and reconciles the contradiction between visual burstiness and feature redundancy. First, GBP transforms images into a graph spectrum through feature correlation measurement. Then, an improved multi-head graph convolution structure is proposed by using Graph Isomorphism Networks (GIN) to realize feature aggregation. Finally, bilinear pooling operations are performed between graph convolution feature maps and original feature maps to obtain more compact and stable fine-grained feature representations. Experiments on CUB, Cars, and Aircrafts datasets demonstrate that the accuracy of the proposed method is 87.8 %, 93.5 %, and 89.6 % respectively, with a feature representation of 2048 dimensions. Compared to the baseline, the feature number is only 25 % of the baseline model, and the accuracy is increased by 2.6 %, 1.7 %, and 1.3 % respectively. These results demonstrate the effectiveness of graph neural network embedding in improving feature stability.

Nodal Decompositions of a Symmetric Matrix

Abstract Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of a signed Erdős-Rényi graph is $\Omega (n/\log n)$ and $o(n)$.

Qudit Dicke state preparation

Qudit Dicke states are higher-dimensional analogues of an important class of highly-entangled completely symmetric quantum states known as (qubit) Dicke states. A circuit for preparing arbitrary qudit Dicke states deterministically is formulated. An explicit decomposition of the circuit in terms of elementary gates is presented, and is implemented in cirq for the qubit and qutrit cases.

DC-programming for neural network optimizations

AbstractWe discuss two key problems related to learning and optimization of neural networks: the computation of the adversarial attack for adversarial robustness and approximate optimization of complex functions. We show that both problems can be cast as instances of DC-programming. We give an explicit decomposition of the corresponding functions as differences of convex functions (DC) and report the results of experiments demonstrating the effectiveness of the DCA algorithm applied to these problems.

On the structure of repeated-root polycyclic codes over local rings

This paper provides the Generalized Mattson Solomon polynomial for repeated-root polycyclic codes over local rings that gives an explicit decomposition of them in terms of idempotents. It also states some structural properties of repeated-root polycyclic codes over finite fields in terms of matrix product codes. Both approaches provide a description of the ⊥0-dual code for a given polycyclic code.

New Approaches for Studying Conformal Embeddings and Collapsing Levels for W–Algebras

Abstract In this paper, we prove a general result saying that under certain hypothesis an embedding of an affine vertex algebra into an affine $W$–algebra is conformal if and only if their central charges coincide. This result extends our previous result obtained in the case of minimal affine $W$-algebras [ 3]. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to find some levels $k$ where $W_{k}(sl(N), x, f )$ collapses to its affine part when $f$ is of hook or rectangular type. Our methods can be applied to non-admissible levels. In particular, we prove Creutzig’s conjecture [ 18] on the conformal embedding in the hook type $W$-algebra $W_{k}(sl(n+m), x, f_{m,n})$ of its affine vertex subalgebra. Quite surprisingly, the problem of showing that certain conformal levels are not collapsing turns out to be very difficult. In the cases when $k$ is admissible and conformal, we prove that $W_{k}(sl(n+m), x, f_{m,n})$ is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from [ 2], [ 5], we find many cases in which $W_{k}(sl(n+m), x, f_{m,n})$ is semi-simple as a module for its affine subalgebra at conformal level and we provide explicit decompositions.

An efficient implementation of a conservative finite volume scheme with constant and linear reconstructions for solving the coagulation equation

Population balances provide an economic means of describing dense particulate phases with particle-particle interactions. Here, we focus on particle coagulation and address the evaluation of the corresponding integral source terms within the scope of a conservative finite volume formulation. Based on one kernel evaluation for every pair of parent cells, efficient analytical formulas are derived that obviate the explicit decomposition of the integration domains into elementary geometrical shapes and are valid for arbitrary volume-grids. Considering the volume-weighted cell averages as degrees of freedom, the formulas are presented for both cell-wise constant and linear reconstructions of the particle volume distribution. We find the latter to be effective at mitigating the unphysically heavy tails that are characteristic of solutions obtained with piecewise constant reconstructions. For Brownian coagulation, the computational expense is mainly caused by the kernel evaluations, while the cost of the parent-daughter pairing and integration algorithm is almost negligibly small.

Decomposition and reproducing property of local polynomial equivalent kernels in varying coefficient models

ABSTRACT We consider local polynomial estimation for varying coefficient models and derive corresponding equivalent kernels that provide insights into the role of smoothing on the data and fill a gap in the literature. We show that the asymptotic equivalent kernels have an explicit decomposition with three parts: the inverse of the conditional moment matrix of covariates given the smoothing variable, the covariate vector, and the equivalent kernels of univariable local polynomials. We discuss finite-sample reproducing property which leads to zero bias in linear models with interactions between covariates and polynomials of the smoothing variable. By expressing the model in a centered form, equivalent kernels of estimating the intercept function are asymptotically identical to those of univariable local polynomials and estimators of slope functions are local analogues of slope estimators in linear models with weights assigned by equivalent kernels. Two examples are given to illustrate the weighting schemes and reproducing property.

Computational framework for complex flow and transport in heterogeneous porous media

We present a flexible scalable open-source computational framework, named SECUReFoam, based on the finite-volume library OpenFOAM®, for flow and transport problems in highly heterogeneous geological media and other porous materials. The framework combines geostatistical pre- and post-processing tools with specialised partial differential equations solvers. Random fields, for permeability and other physical properties, are generated by means of continuous or thresholded Gaussian random fields with various covariance/variogram functions. The generation process is based on an explicit spectral Fourier decomposition of the field which, although more computationally intensive than Fast Fourier Transform methods, allows a more flexible choice of statistical parameters and can be used for general geometries and grids. Flow and transport equations are solved for single-phase and variable density problems, with and without the Boussinesq approximation, and for a wide range of density, viscosity, and dispersion models, including dual-continuum (dual permeability or dual porosity) formulations. The mathematical models are here presented in details and the numerical strategies to deal with heterogeneities, equation coupling, and boundary conditions are discussed and benchmarked for the heterogeneous Henry and Horton–Rogers–Lapwood problems, and other test cases. We show that our framework is capable of dealing with large permeability variances, viscous instabilities, and large-scale three-dimensional transport problems.

Duality and bicrystals on infinite binary matrices

The set of finite binary matrices of a given size is known to carry a finite type A bicrystal structure. We first review this classical construction, explain how it yields a short proof of the equality between Kostka polynomials and one-dimensional sums together with a natural generalisation of the 2M-X Pitman transform. Next, we show that, once the relevant formalism on families of infinite binary matrices is introduced, this is a particular case of a much more general phenomenon. Each such family of matrices is proved to be endowed with Kac–Moody bicrystal and tricrystal structures defined from the classical root systems. Moreover, we give an explicit decomposition of these multicrystals, reminiscent of the decomposition of characters yielding the Cauchy identities.

Differential polynomial identities of upper triangular matrices of size three

We determine the differential polynomial identities of 3×3 upper triangular matrices over a base field of characteristic zero, under the action of its full Lie algebra of derivations. We compute the exact differential codimension sequence of the multilinear ones and describe their Sn-structure by means of an explicit decomposition of the Sn-cocharacter of their proper part.

The Kudla-Rapoport conjecture at a ramified prime for 𝑈(1,1)

In this paper, we proved a local arithmetic Siegel-Weil formula for a U ( 1 , 1 ) U(1, 1) -Shimura variety at a ramified prime, a.k.a. a Kudla-Rapoport conjecture at a ramified case. The formula needs to be modified from the original Kudla-Rapoport conjecture. In the process, we also give an explicit decomposition of the special divisors of the Rapoport-Zink space of unitary type ( 1 , 1 ) (1, 1) (Krämer model). A key ingredient is to relate the Rapoport-Zink space to the Drinfeld upper plane.