Local Invariance Research Articles

Local invariants of divergence-free webs

The objects of our study are webs in the geometry of volume-preserving diffeomorphisms. We introduce two local invariants of divergence-free webs: a differential one, directly related to the curvature of the natural connection of a divergence-free 2-web introduced by Tabachnikov (Diff Geom Appl 3:265-284, 1993), and a geometric one, inspired by the classical notion of planar 3-web holonomy defined by Blaschke and Bol (Geometrie der Gewebe. Grundlehren der mathematischen Wissenschaften, vol. 49. Springer, Berlin, 1938). We show that triviality of either of these invariants characterizes trivial divergence-free web-germs up to equivalence. We also establish some preliminary results regarding the full classification problem, which jointly generalize the theorem of Tabachnikov on normal forms of divergence-free 2-webs. They are used to provide a canonical form and a complete set of invariants of a generic divergence-free web in the planar case. Lastly, the relevance of local triviality conditions and their potential applications in numerical relativity are discussed.

The effective theory of gravity and dynamical vacuum energy

Gravity and general relativity are considered as an Effective Field Theory (EFT) at low energies and macroscopic distances. The effective action of the conformal anomaly of light or massless quantum fields has significant effects on macroscopic scales, due to associated light cone singularities that are not captured by an expansion in local curvature invariants. A compact local form for the Wess-Zumino effective action of the conformal anomaly and stress tensor is given, requiring the introduction of a new light scalar field, which it is argued should be included in the low energy effective action for gravity. This scalar conformalon couples to the conformal part of the spacetime metric and allows the effective value of the vacuum energy, described as a condensate of an exact 4-form abelian gauge field strength F = dA, to change in space and time. This is achieved by the identification of the torsion dependent part of the Chern-Simons 3-form of the Euler class with the gauge potential A, which enters the effective action of the conformal anomaly as a J · A interaction analogous to electromagnetism. The conserved 3-current J describes the worldtube of 2-surfaces that separate regions of differing vacuum energy. The resulting EFT thus replaces the fixed constant Λ of classical gravity, and its apparently unnaturally large sensitivity to UV physics, with a dynamical condensate whose ground state value in empty flat space is Λeff = 0 identically. By allowing Λeff to vary rapidly near the 2-surface of a black hole horizon, the proposed EFT of dynamical vacuum energy provides an effective Lagrangian framework for gravitational condensate stars, as the final state of complete gravitational collapse consistent with quantum theory. The possible consequences of dynamical vacuum dark energy for cosmology, the cosmic coincidence problem, and the role of conformal invariance for other fine tuning issues in the Standard Model are discussed.

Ants and bracket generating distributions in dimensions 5 and 6

We consider a mechanical system of three ants on the floor, in two situations. In the first situation ants move according to Rule A, which forces the velocity of any given ant to always point at a neighboring ant; in the second situation ants move according to Rule B, which forces the velocity of every ant to be parallel to the line defined by the two other ants. We observe that Rule A equips the 6-dimensional configuration space of the ants with a structure of a homogeneous (3,6) distribution, and that Rule B foliates this 6-dimensional configuration space onto 5-dimensional leaves, each of which is equipped with a homogeneous (2,3,5) distribution. The symmetry properties and the local invariants of these distributions are determined.In the case of Rule B we study and determine the singular trajectories (abnormal extremals) of the corresponding distributions. We show that these satisfy an interesting system of two ODEs of Fuchsian type.

Coastline matching via a graph-based approach

This paper studies the problem of unsupervised detection of geometrically similar fragments (segments) in curves, in the context of boundary matching. The goal is to determine all pairs of sub-curves that are geometrically similar, under local scale invariance. In particular, we aim to locate the existence of a similar section (independent of length and/or orientation) in the second curve, to a section of the first curve, as indicated by the user. The proposed approach is based on a suitable distance matrix of the two given curves. Additionally, a suitable objective function is proposed to capture the trade-off between the similarity of the common sub-sequences and their lengths. The goal of the algorithm is to minimize this objective function via an efficient graph-based approach that capitalizes on Dynamic Time Warping to compare the two subcurves. We apply the proposed technique in the context of geometric matching of coastline pairs. This application is crucial for investigating the forcing factors related to the coastline evolution. The proposed method was successfully applied to global coastline data, yielding a bipartite graph with analytical point-to-point correspondences.

Adaptive Propagation Graph Convolutional Networks Based on Attention Mechanism

The main steps in a graph neural network are message propagation and aggregation between nodes. Message propagation allows messages from distant nodes in the graph to be transmitted to the central node, while feature aggregation allows the central node to obtain messages regarding its neighbors and update itself, so that it can express deep-layer features. Because the graph structure data have no local translation invariance, the number of neighbors of each central node is different, and there is no order, there are two difficulties: (1) how to design a reliable message propagation method to better express all network topologies; (2) how to design a feature aggregation function so that it can weigh local features and global features. In this paper, a new adaptive propagation graph convolutional network model based on the attention mechanism (APAT-GCN) is proposed, which enables GNNs to adaptively complete the process of message propagation and feature aggregation, according to the neighbors of the central node, and set the influence degree of local and global messages on the aggregation of the central node. Compared with other classical models, this method is superior to the baseline model and can improve the accuracy of node- and graph-level classification tasks in downstream tasks.

Automatic Registration for Panoramic Images and Mobile LiDAR Data Based on Phase Hybrid Geometry Index Features

The registration of panoramic images and mobile light detection and ranging (LiDAR) data is quite challenging because different imaging mechanisms and viewing angle differences generate significant geometric and radiation distortions between the two multimodal data sources. To address this problem, we propose a registration method for panoramic images and mobile LiDAR data based on the hybrid geometric structure index feature of phase. We use the initial GPS/IMU to transform the mobile LiDAR data into an intensity map and align the two images to complete registration. Firstly, a novel feature descriptor called a hybrid geometric structure index of phase (HGIFP) is built to capture the structural information of the images. Then, a set of corresponding feature points is obtained from the two images using the constructed feature descriptor combined with a robust false-match elimination algorithm. The average pixel distance of the corresponding feature points is used as the error function. Finally, in order to complete the accurate registration of the mobile LiDAR data and panoramic images and improve computational efficiency, we propose the assumption of local motion invariance of 3D–2D corresponding feature points and minimize the error function through multiple reprojections to achieve the best registration parameters. The experimental results show that the method in this paper can complete the registration of panoramic images and the mobile LiDAR data under a rotation error within 12° and a translation error within 2 m. After registration, the average error of rotation is about 0.15°, and the average error of translation is about 1.27 cm. Moreover, it achieves a registration accuracy of less than 3 pixels in all cases, which outperforms the current five state-of-the-art methods, demonstrating its superior registration performance.

SIFT-CNN: When Convolutional Neural Networks Meet Dense SIFT Descriptors for Image and Sequence Classification.

Despite the success of hand-crafted features in computer visioning for many years, nowadays, this has been replaced by end-to-end learnable features that are extracted from deep convolutional neural networks (CNNs). Whilst CNNs can learn robust features directly from image pixels, they require large amounts of samples and extreme augmentations. On the contrary, hand-crafted features, like SIFT, exhibit several interesting properties as they can provide local rotation invariance. In this work, a novel scheme combining the strengths of SIFT descriptors with CNNs, namely SIFT-CNN, is presented. Given a single-channel image, one SIFT descriptor is computed for every pixel, and thus, every pixel is represented as an M-dimensional histogram, which ultimately results in an M-channel image. Thus, the SIFT image is generated from the SIFT descriptors for all the pixels in a single-channel image, while at the same time, the original spatial size is preserved. Next, a CNN is trained to utilize these M-channel images as inputs by operating directly on the multiscale SIFT images with the regular convolution processes. Since these images incorporate spatial relations between the histograms of the SIFT descriptors, the CNN is guided to learn features from local gradient information of images that otherwise can be neglected. In this manner, the SIFT-CNN implicitly acquires a local rotation invariance property, which is desired for problems where local areas within the image can be rotated without affecting the overall classification result of the respective image. Some of these problems refer to indirect immunofluorescence (IIF) cell image classification, ground-based all-sky image-cloud classification and human lip-reading classification. The results for the popular datasets related to the three different aforementioned problems indicate that the proposed SIFT-CNN can improve the performance and surpasses the corresponding CNNs trained directly on pixel values in various challenging tasks due to its robustness in local rotations. Our findings highlight the importance of the input image representation in the overall efficiency of a data-driven system.

Relativistic fluids, hydrodynamic frames and their Galilean versus Carrollian avatars

We comprehensively study Galilean and Carrollian hydrodynamics on arbitrary backgrounds, in the presence of a matter/charge conserved current. For this purpose, we follow two distinct and complementary paths. The first is based on local invariance, be it Galilean or Carrollian diffeomorphism invariance, possibly accompanied by Weyl invariance. The second consists in analyzing the relativistic fluid equations at large or small speed of light, after choosing an adapted gauge, Arnowitt-Deser-Misner-Zermelo for the former and Papapetrou-Randers for the latter. Unsurprisingly, the results agree, but the second approach is superior as it effortlessly captures more elaborate situations with multiple degrees of freedom. It furthermore allows to investigate the fate of hydrodynamic-frame invariance in the two limits at hand, and conclude that its breaking (in the Galilean) or its preservation (in the Carrollian) are fragile consequences of the behaviour of transport attributes at large or small c. Both methods do also agree on the doom of Nœtherian currents generated in the relativistic theory by isometries: conserved currents are not always guaranteed in Newton-Cartan or Carroll spacetimes as a consequence of Galilean or Carrollian isometries. Comparison of Galilean and Carrollian fluid equations exhibits a striking but often superficial resemblance, which we comment in relation to black-hole horizon dynamics, awkwardly akin to Navier-Stokes equations. This congruity is authentic in one instance though and turns out then to describe Aristotelian dynamics, which is the last item in our agenda.

Effectiveness of Remote PPG Construction Methods: A Preliminary Analysis.

The contactless recording of a photoplethysmography (PPG) signal with a Red-Green-Blue (RGB) camera is known as remote photoplethysmography (rPPG). Studies have reported on the positive impact of using this technique, particularly in heart rate estimation, which has led to increased research on this topic among scientists. Therefore, converting from RGB signals to constructing an rPPG signal is an important step. Eight rPPG methods (plant-orthogonal-to-skin (POS), local group invariance (LGI), the chrominance-based method (CHROM), orthogonal matrix image transformation (OMIT), GREEN, independent component analysis (ICA), principal component analysis (PCA), and blood volume pulse (PBV) methods) were assessed using dynamic time warping, power spectrum analysis, and Pearson’s correlation coefficient, with different activities (at rest, during exercising in the gym, during talking, and while head rotating) and four regions of interest (ROI): the forehead, the left cheek, the right cheek, and a combination of all three ROIs. The best performing rPPG methods in all categories were the POS, LGI, and OMI methods; each performed well in all activities. Recommendations for future work are provided.

Generalized multifractality at metal-insulator transitions and in metallic phases of two-dimensional disordered systems

We study generalized multifractality characterizing fluctuations and correlations of eigenstates in disordered systems of symmetry classes AII, D, and DIII. Both metallic phases and Anderson-localization transitions are considered. By using the nonlinear sigma-model approach, we construct pure-scaling eigenfunction observables. The construction is verified by numerical simulations of appropriate microscopic models, which also yield numerical values of the corresponding exponents. In the metallic phases, the numerically obtained exponents satisfy Weyl symmetry relations as well as generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator). At the same time, the generalized parabolicity is strongly violated at critical points of metal-insulator transitions, signaling violation of local conformal invariance. Moreover, in classes D and DIII, even the Weyl symmetry breaks down at critical points of metal-insulator transitions. This last feature is related to a peculiarity of the sigma-model manifolds in these symmetry classes: they consist of two disjoint components. Domain walls associated with these additional degrees of freedom are crucial for ensuring Anderson localization and, at the same time, lead to the violation of the Weyl symmetry.

Generalized parton distributions of sea quarks in the proton from nonlocal chiral effective theory

We calculate the spin-averaged generalized parton distributions (GPDs) of sea quarks in the proton at zero skewness from nonlocal covariant chiral effective theory, including one-loop contributions from intermediate states with pseudoscalar mesons and octet and decuplet baryons. A relativistic regulator is generated from the nonlocal Lagrangian where a gauge link is introduced to guarantee local gauge invariance, with additional diagrams from the expansion of the gauge link ensuring conservation of electric charge and strangeness. Flavor asymmetries for sea quarks at zero and finite momentum transfer, as well as strange form factors, are obtained from the calculated GPDs, and results compared with phenomenological extractions and lattice QCD.

Multiple similarity drug-target interaction prediction with random walks and matrix factorization.

The discovery of drug-target interactions (DTIs) is a very promising area of research with great potential. The accurate identification of reliable interactions among drugs and proteins via computational methods, which typically leverage heterogeneous information retrieved from diverse data sources, can boost the development of effective pharmaceuticals. Although random walk and matrix factorization techniques are widely used in DTI prediction, they have several limitations. Random walk-based embedding generation is usually conducted in an unsupervised manner, while the linear similarity combination in matrix factorization distorts individual insights offered by different views. To tackle these issues, we take a multi-layered network approach to handle diverse drug and target similarities, and propose a novel optimization framework, called Multiple similarity DeepWalk-based Matrix Factorization (MDMF), for DTI prediction. The framework unifies embedding generation and interaction prediction, learning vector representations of drugs and targets that not only retain higher order proximity across all hyper-layers and layer-specific local invariance, but also approximate the interactions with their inner product. Furthermore, we develop an ensemble method (MDMF2A) that integrates two instantiations of the MDMF model, optimizing the area under the precision-recall curve (AUPR) and the area under the receiver operating characteristic curve (AUC), respectively. The empirical study on real-world DTI datasets shows that our method achieves statistically significant improvement over current state-of-the-art approaches in four different settings. Moreover, the validation of highly ranked non-interacting pairs also demonstrates the potential of MDMF2A to discover novel DTIs.

Discrete gravity with local Lorentz invariance

A novel structure-preserving algorithm for general relativity in vacuum is derived from a lattice gauge theoretic discretization of the tetradic Palatini action. The resulting model of discrete gravity is demonstrated to preserve local Lorentz invariance and symplectic structure.

Ultra-short-term prediction method of PV power output based on the CNN–LSTM hybrid learning model driven by EWT

This study proposes a new method for ultra-short-term prediction of photovoltaic (PV) power output using a convolutional neural network (CNN) and long short-term memory (LSTM) hybrid model driven by empirical wavelet transform (EWT) to address the intermittent and stochastic nature of PV power generation. Given the differences in the spatial and temporal distribution of features between PV sample data and meteorological conditions, a hybrid learning model for multibranch feature extraction was designed. First, the frequency band of PV output data was adaptively selected using EWT and decomposed into the amplitude modulation–frequency modulation single components with frequencies ranging from low to high. Second, data reconstruction was performed on the obtained power components to exploit the extraction ability of the two-dimensional CNN model for short-term local invariance and periodic features. Third, the combined one-dimensional CNN–LSTM model was used for the sample daily meteorological conditions to extract their spatiotemporal features, and the LSTM model was used to learn the correlation between the power data features and the predicted daily weather conditions and to obtain the corresponding component prediction results. Finally, the prediction results of each component were reconstructed to achieve the ultra-short-term prediction. Using Hangzhou Dianzi University's PV microgrid system as an example, the training and testing sets were randomly selected based on different seasons and weather. The results show that this method outperforms traditional learning models in terms of overall prediction performance. The proposed method of a hybrid deep learning model will provide a novel approach for ultra-short-term prediction of PV output.

Constraint-Induced Symmetric Nonnegative Matrix Factorization for Accurate Community Detection

As a fundamental characteristic of an undirected network, community reveals its networking organization and functional mechanisms, making community detection be a highly-interesting issue in network representation learning. With great interpretability, a symmetric and nonnegative matrix factorization (SNMF)-based approach is frequently adopted to tackle this issue. However, it only adopts a unique feature matrix for describing the symmetry of an undirected network, which unfortunately results in a reduced feature space that evidently impairs its representation learning ability. Motivated by this discovery, this paper proposes a novel Constraint-induced Symmetric Nonnegative Matrix Factorization (C-SNMF) model that adopts three-fold ideas: a) Representing a target undirected network with multiple latent feature matrices, thus preserving its representation learning capacity; b) Incorporating a symmetry-regularizer into its objective function, which preserves the symmetry of the learnt low-rank approximation to the adjacency matrix, thereby making the resultant detector precisely illustrate the target network's symmetry; and c) Introducing a graph-regularizer that preserves local invariance of the network's intrinsic geometry into its learning objective, thus making the achieved detector well-aware of community structure within the target network. Note that the regularization coefficients are selected according to the modularity of the learnt community structure on the training data only, thereby greatly improving the achieved model's practical significance for real applications. Experimental results on six real-world networks demonstrate that the proposed C-SNMF model significantly outperforms the benchmarks and state-of-the-art models in achieving highly-accurate community detection results.

Local invariants identify topology in metals and gapless systems

Although topological band theory has been used to discover and classify a wide array of novel topological phases in insulating and semimetal systems, it is not well suited to identifying topological phenomena in metallic or gapless systems. Here, we develop a theory of topological metals based on the system's spectral localizer and associated Clifford pseudospectrum, which can both determine whether a system exhibits boundary-localized states despite the presence of degenerate bulk bands and provide a measure of these states' topological protection even in the absence of a bulk band gap. We demonstrate the generality of this method across symmetry classes in two lattice systems, a Chern metal and a higher-order topological metal, and prove the topology of these systems is robust to relatively strong perturbations. The ability to define invariants for metallic and gapless systems allows for the possibility of finding topological phenomena in a broad range of natural, photonic, and other artificial materials that could not be previously explored.

Supersymmetric action for 6D (4, 0) supergravity

We give a linearized but otherwise complete supersymmetric action for mathcal{N} = (4, 0) supergravity in six dimensions, using a Kaluza-Klein-type 5 + 1 split of coordinates and fields. We provide in particular a significantly simplified version of the bosonic action derived by us recently. This formulation employs fields that are no longer irreducible, subject to a local Lorentz invariance, which in turn simplifies the supersymmetry transfor mations including the exotic gravitino.

The origin of Weyl gauging in metric-affine theories

In the first part, we discuss the interplay between local scale invariance and metric-affine degrees of freedom from few distinct points of view. We argue, rather generally, that the gauging of Weyl symmetry is a natural byproduct of requiring that scale invariance is a symmetry of a gravitational theory that is based on a metric and on an independent affine structure degrees of freedom. In the second part, we compute the Nöther identities associated with all the gauge symmetries, including Weyl, Lorentz and diffeomorphisms invariances, for general actions with matter degrees of freedom, exploiting a gauge covariant generalization of the Lie derivative. We find two equivalent ways to approach the problem, based on how we regard the spin-connection degrees of freedom, either as an independent object or as the sum of two Weyl invariant terms. The latter approach, which rests upon the use of a Weyl-covariant connection with desirable properties, denoted , is particularly convenient and constitutes one of our main results.

Development of Novel Therapeutics for Schizophrenia Treatment Based on a Selective Positive Allosteric Modulation of α1-Containing GABAARs-In Silico Approach.

For the development of atypical antipsychotics, the selective positive allosteric modulation of the ionotropic GABAA receptor (GABAAR) has emerged as a promising approach. In the presented research, two unrelated methods were used for the development of QSAR models for selective positive allosteric modulation of 1-containing GABAARs with derivatives of imidazo [1,2-a]-pyridine. The development of conformation-independent QSAR models, based on descriptors derived from local molecular graph invariants and SMILES notation, was achieved with the Monte Carlo optimization method. From the vast pool of 0D, 1D, and 2D molecule descriptors, the GA-MLR method developed additional QSAR models. Various statistical methods were utilised for the determination of the developed models’ robustness, predictability, and overall quality, and according to the obtained results, all QSAR models are considered good. The molecular fragments that have a positive or negative impact on the studied activity were obtained from the studied molecules’ SMILES notations, and according to the obtained results, nine novel compounds were designed. The binding affinities to GABAAR of designed compounds were assessed with the application of molecular docking studies and the obtained results showed a high correlation with results obtained from QSAR modeling. To assess all designed molecules’ “drug-likeness”, their physicochemical descriptors were computed and utilised for the prediction of medicinal chemistry friendliness, pharmacokinetic properties, ADME parameters, and druglike nature.

Symmetry evolution for the imperfect fluid under perturbations

Ever since a new symmetry was found for the imperfect fluid with vorticity, the question of the effect of perturbations on the symmetry itself has been raised. This new symmetry arose when realizing that local four-velocity gauge-like transformations would render the left-hand side of the Einstein equations invariant because the metric tensor would be invariant under this new kind of local transformations. Then the point was raised about the invariance of such a kind of transformations of the stress–energy tensor on the right-hand side of the Einstein equations in curved four-dimensional Lorentz spacetimes. It was verified that these invariances do not work with plain perfect fluid but they do work for imperfect fluids. The imperfect fluid stress–energy tensor will be invariant under local four-velocity gauge-like transformations when additional transformations are introduced for several variables included in the stress–energy tensor itself. This local invariance was also the criteria introduced in order to present a new stress–energy tensor for vorticity as well. New tetrads are at the core of the realization of the existence of this new symmetry because it is through these new tetrads that this new symmetry is realized. It is through the local transformation of these new tetrad vectors that we can prove that the metric tensor is invariant. This new kind of symmetry has its origins in a similar tetrad formulation as to the Einstein–Maxwell spacetimes formalism presented in previous manuscripts. In this paper, we will introduce local perturbations by external agents to the relevant objects in the imperfect fluid geometry. We will demonstrate a theorem that proves that the symmetries under four-velocity gauge-like transformations will be instantaneously broken but at the same time transformed into new symmetries. Because the local orthogonal planes determined by these new tetrads, which happen to be the local planes of symmetry will tilt under local perturbations. There will be a symmetry evolution under perturbations.