Edge Kernel Research Articles

On the structure of vertex stabilizers of arc-transitive locally quasiprimitive graphs

Let Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document} be a connected arc-transitive G-graph which is locally finite and locally quasiprimitive. Let {x,y}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x,y\\}$$\\end{document} be an edge of Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}. A relation between Gx[1]/Op(Gx[1])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_x^{[1]}/O_p(G_x^{[1]})$$\\end{document} and the existence of certain normal subgroups of GxΔ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_x^{\\Delta (x)}$$\\end{document} and Gx,yΔ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_{x,y}^{\\Delta (x)}$$\\end{document} is established. This is then used to determine the vertex stabilizers of a class of 2-arc-transitive graphs with trivial edge kernel.

The elliptic Ginibre ensemble: A unifying approach to local and global statistics for higher dimensions

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature β = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions Rd and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions Cd, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For Cd with d = 1, this corresponds to non-interacting fermions in a rotating trap.

Graph metric learning quantifies morphological differences between two genotypes of shoot apical meristem cells inArabidopsis

AbstractWe present a method for learning ‘spectrally descriptive’ edge weights for graphs. We generalize a previously known distance measure on graphs (graph diffusion distance [GDD]), thereby allowing it to be tuned to minimize an arbitrary loss function. Because all steps involved in calculating this modified GDD are differentiable, we demonstrate that it is possible for a small neural network model to learn edge weights which minimize loss. We apply this method to discriminate between graphs constructed from shoot apical meristem images of two genotypes of Arabidopsis thaliana specimens: wild-type and trm678 triple mutants with cell division phenotype. Training edge weights and kernel parameters with contrastive loss produce a learned distance metric with large margins between these graph categories. We demonstrate this by showing improved performance of a simple k-nearest-neighbour classifier on the learned distance matrix. We also demonstrate a further application of this method to biological image analysis. Once trained, we use our model to compute the distance between the biological graphs and a set of graphs output by a cell division simulator. Comparing simulated cell division graphs to biological ones allows us to identify simulation parameter regimes which characterize mutant versus wild-type Arabidopsis cells. We find that trm678 mutant cells are characterized by increased randomness of division planes and decreased ability to avoid previous vertices between cell walls.

Hard edge statistics of products of Pólya ensembles and shifted GUE’s

Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative Pólya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes transform (Green function). Under rather mild conditions for the Pólya ensemble, we prove formulas for the hard edge kernel of the singular value statistics of the Pólya ensemble alone and the product matrix to highlight their very close similarity. Due to these observations, we even propose a conjecture for the hard edge statistics of a multiplicative Pólya ensemble on the complex matrices and a polynomial ensemble on the Hermitian matrices.

Modeling and prediction of tool influence function under complex edge in sub-aperture optical polishing.

Edge mis-figures are regarded as one of the most difficult technical issues in optical fabrication. At present, only the near straight-line edge tool influence function (TIF) can be fitted by a polynomial function, but it is difficult to unify a 2-D analytical model suitable for complex edge workpieces and various tools, due to the lack of the scientific understanding of the edge removal behavior. In this paper, a comprehensive mathematical model is proposed to reveal the mechanism of the edge effect and accurately predict the complex edge TIF. The concept of a nonlinear edge kernel is first proposed and verified that the nonlinear pressure can be characterized by convoluting the kernel with the edge contour, which can be easily adapted to complex edge cases; besides, the edge kernel obtaining algorithm is established. The linear pressure part is verified to be constrained by the moment balance formula, which occurs in universal joint tool. Besides, the basic pressure distribution is presented to compensate the pressure distortion caused by the uneven form of the tool pad. The introducing of these three parts makes the complex edge pressure modeled efficiently and matched perfectly with the FEA results. In addition, a series of TIF experiments were carried out on various complex edge workpieces and different tools, which could be well predicted by the proposed model in 2-D view.

A Non-Hermitian Generalisation of the Marchenko–Pastur Distribution: From the Circular Law to Multi-criticality

We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $N\times (N+\nu)$ are correlated through a non-Hermiticity parameter $\tau\in[0,1]$. For general $\nu=O(N)$ and $\tau$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $\tau=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $\mathcal{D}=\begin{pmatrix} 0 & X_1 \\ X_2^* & 0 \end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $\tau_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.

Complex-Domain Super MDS: A New Framework for Wireless Localization With Hybrid Information

We revisit the super multidimensional scaling (SMDS) wireless localization algorithm first proposed a decade ago, recasting it onto the complex-domain. Under this new formulation, the edge kernel, which carries both angle and distance information simultaneously and plays a central role in the SMDS algorithm, becomes a complex-valued rank-one matrix, resulting in a new complex-domain SMDS framework, which yields several advantages over the original, including the elimination of redundancy, the enhancement of conditions to handle information erasure, and the possibility of designing several algorithmic variations that offer different complexity/performance improvements. To cite some concrete results, it is shown, for instance, that a distance-based localization system with 20 targets employing one of the new algorithms dubbed complex-domain SMDS (CD-SDMDS) outperforms the original SMDS at a tenth of the computational cost, and that the handling of missing information via matrix completion is superior in CD-SMDS compared to SMDS. If the same network collects also anchor-to-target angle information, it is furthermore shown that localization is achieved with nearly $20\times $ lower complexity and still higher accuracy using another new algorithm dubbed Turbo MRC-SMDS, and over $25\times $ faster using yet another method dubbed Iterative MRC-SMDS, with only a slight degradation.

GraphIt: a high-performance graph DSL

The performance bottlenecks of graph applications depend not only on the algorithm and the underlying hardware, but also on the size and structure of the input graph. As a result, programmers must try different combinations of a large set of techniques, which make tradeoffs among locality, work-efficiency, and parallelism, to develop the best implementation for a specific algorithm and type of graph. Existing graph frameworks and domain specific languages (DSLs) lack flexibility, supporting only a limited set of optimizations. This paper introduces GraphIt, a new DSL for graph computations that generates fast implementations for algorithms with different performance characteristics running on graphs with different sizes and structures. GraphIt separates what is computed (algorithm) from how it is computed (schedule). Programmers specify the algorithm using an algorithm language, and performance optimizations are specified using a separate scheduling language. The algorithm language simplifies expressing the algorithms, while exposing opportunities for optimizations. We formulate graph optimizations, including edge traversal direction, data layout, parallelization, cache, NUMA, and kernel fusion optimizations, as tradeoffs among locality, parallelism, and work-efficiency. The scheduling language enables programmers to easily search through this complicated tradeoff space by composing together a large set of edge traversal, vertex data layout, and program structure optimizations. The separation of algorithm and schedule also enables us to build an autotuner on top of GraphIt to automatically find high-performance schedules. The compiler uses a new scheduling representation, the graph iteration space, to model, compose, and ensure the validity of the large number of optimizations. We evaluate GraphIt’s performance with seven algorithms on graphs with different structures and sizes. GraphIt outperforms the next fastest of six state-of-the-art shared-memory frameworks (Ligra, Green-Marl, GraphMat, Galois, Gemini, and Grazelle) on 24 out of 32 experiments by up to 4.8×, and is never more than 43% slower than the fastest framework on the other experiments. GraphIt also reduces the lines of code by up to an order of magnitude compared to the next fastest framework.

A combined Weisfeiler–Lehman graph kernel for structured data

Different graph kernels may correspond to using different notions of similarity or may be using information coming from multiple sources. In this paper, we develop a common method to construct combined graph kernel (CGK) which is based on a family of graph kernels. We define three kinds of CGK. The first one is called the weighted combined graph kernel and is a parametric CGK. The second one is called the accuracy ratio weighted combined graph kernel and is a non-parametric CGK. The third one is called the product combined graph kernel and also belongs to non-parametric CGK. The three kinds of definition of CGK can be applied for constructing CGK based on a family of graph kernels. This family of kernels is demonstrated based on the Weisfeiler–Lehman (WL) sequence of graphs in this paper, including a highly efficient subtree kernel, edge kernel, and shortest path kernel. Experiments demonstrate that our CGK based on WL graph kernels outperforms the corresponding single WL graph kernel on several classification benchmark data sets.

Statistics of fermions in a d-dimensional box near a hard wall

We study N noninteracting fermions in a domain bounded by a hard-wall potential in dimensions. We show that for large N, the correlations at the edge of the Fermi gas (near the wall) at zero temperature are described by a universal kernel, different from the universal edge kernel valid for smooth confining potentials. We compute this d-dimensional hard edge kernel exactly for a spherical domain and argue, using a generalized method of images, that it holds close to any sufficiently smooth boundary. As an application we compute the quantum statistics of the position of the fermion closest to the hard wall. Our results are then extended in several directions, including non-smooth boundaries such as a wedge, and also to finite temperature.

Universal ground-state properties of free fermions in a d-dimensional trap

The ground-state properties of N spinless free fermions in a d-dimensional confining potential are studied. We find that any n-point correlation function has a simple determinantal structure that allows us to compute several properties exactly for large N. We show that the average density has a finite support with an edge, and near this edge the density exhibits a universal (valid for a wide class of potentials) scaling behavior for large N. The associated edge scaling function is computed exactly and generalizes to any d the edge electron gas result of Kohn and Mattsson in d = 3 (Kohn W. and Mattsson A. E., Phys. Rev. Lett., 81 (1998) 3487). In addition, we calculate the kernel (that characterizes any n-point correlation function) for large N and show that, when appropriately scaled, it depends only on dimension d, but has otherwise universal scaling forms, at the edges. The edge kernel, for higher d, generalizes the Airy kernel in one dimension, well known from the random matrix theory.

ON GRAPHS OF PRIME VALENCY ADMITTING A SOLVABLE ARC-TRANSITIVE GROUP

Let $X$ be a simple, connected, $p$-valent, $G$-arc-transitive graph, where the subgroup $G\leq \text{Aut}(X)$ is solvable and $p\geq 3$ is a prime. We prove that $X$ is a regular cover over one of the three possible types of graphs with semi-edges. This enables short proofs of the facts that $G$ is at most 3-arc-transitive on $X$ and that its edge kernel is trivial. For pentavalent graphs, two further applications are given: all $G$-basic pentavalent graphs admitting a solvable arc-transitive group are constructed and an example of a non-Cayley graph of this kind is presented.

A linear edge kernel for two-layer crossing minimization

The two-layer crossing minimization problem (TLCM), given a bipartite graph G with n vertices and a positive integer k, asks whether G has a two-layer drawing with at most k crossings. We consider a simple generalization of TLCM called leaf-edge-weighted TLCM (LEW-TLCM), where we allow positive weights on edges incident to leaves, and show that this problem admits a kernel with O(k) edges provided that the given graph is connected. As a straightforward consequence, LEW-TLCM (and hence TLCM) has a fixed parameter algorithm that runs in 2O(klog⁡k)+nO(1) time which improves on the previously best known algorithm with running time 2O(k3)n.

Universal microscopic correlation functions for products of truncated unitary matrices

We investigate the spectral properties of the product of M complex non-Hermitian random matrices that are obtained by removing L rows and columns of larger unitary random matrices uniformly distributed on the group U(N + L). Such matrices are called truncated unitary matrices or random contractions. We first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed N, L, and M, given by a standard determinantal point process in the complex plane. The weight however is non-standard and can be expressed in terms of the Meijer G-function. The explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large N (and L) limits at fixed M. At strong non-unitarity, with L/N finite, the eigenvalues condense on a domain inside the unit circle. At the edge and in the bulk we find the same universal microscopic kernel as for a single complex non-Hermitian matrix from the Ginibre ensemble. At the origin we find the same new universality classes labeled by M as for the product of M matrices from the Ginibre ensemble. Keeping a fixed size of truncation, L, when N goes to infinity leads to weak non-unitarity, with most eigenvalues on the unit circle as for unitary matrices. Here we find a new microscopic edge kernel that generalizes the known results for M = 1. We briefly comment on the case when each product matrix results from a truncation of different size Lj.

Algebraic Approach for Robust Localization with Heterogeneous Information

We offer a redesigned form of the classical multidimensional scaling (C-MDS) algorithm suitable to handle the localization of multiple sources under line-of-sight (LOS) and non-line-of-sight (NLOS) conditions. To do so we propose to modify the kernel matrix used in the MDS algorithm to allow for both distance and angle information to be processed algebraically (without iteration) and simultaneously. In so doing we also show that the new formulation overcomes two well known limitations of the C-MDS approach, namely the propagation error problem and the possibility to weight the dissimilarities used as measurement information, including, for the case of binary weights, the data erasure problem. Due to the increased size of the proposed edge kernel matrix KE used in the algorithm, the Nystrom approximation is applied to reduce the overall computational complexity to few matrix multiplications. Range only scenarios are also dealt with by approximating the matrix KE. Simulations in range-angle as well as range-only scenarios demonstrate the superiority of our solution under both LOS and NLOS conditions versus semidefinite programming (SDP) formulations of the problem specifically designed to exploit the heterogeneity of the information available.

Locally arc-transitive graphs of valence {3,4} with trivial edge kernel

In this paper, we consider connected locally G-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighbourhood Γ(u)?Γ(v) is trivial. We find 19 finitely presented groups with the property that any such group G is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence {3,4} on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup G with $G_{uv}^{[1]} = 1$ .

More on Moore graphs

Let Δ be a tree such that each vertex has valency at least 3 and let A be a set of regular subgraphs of valency 2. In the early eighties A. Delgado and B. Stellmacher introduced the uniqueness and exchange conditions on the pair (Δ,A) and showed how they relate to generalized polygons. We modify the exchange condition and show how the modified version relates to Moore graphs. This is then used to give the isomorphism type of the amalgam of vertex stabilizers of two adjacent vertices in an s-arc transitive graph with trivial edge kernel and s⩾4.

On locally s -arc transitive graphs with trivial edge kernel

Let Δ be a connected graph, without loops or multiple edges, such that each vertex has valency at least 3. Let G ⩽ Aut(Δ) act locally s-arc transitively on Δ and let {x, y} be an edge. We show that if s ⩾ 4 and |Gz| < ∞ for all z ∈ {x, y}, then G x , y [ 1 ] is non-trivial.

Interactive Point-based Isosurface Exploration and High-quality Rendering

We present an efficient point-based isosurface exploration system with high quality rendering. Our system incorporates two point-based isosurface extraction and visualization methods: edge splatting and the edge kernel method. In a volume, two neighboring voxels define an edge. The intersection points between the active edges and the isosurface are used for exact isosurface representation. The point generation is incorporated in the GPU-based hardware-accelerated rendering, thus avoiding any overhead when changing the isovalue in the exploration. We call this method edge splatting. In order to generate high quality isosurface rendering regardless of the volume resolution and the view, we introduce an edge kernel method. The edge kernel upsamples the isosurface by subdividing every active cell of the volume data. Enough sample points are generated to preserve the exact shape of the isosurface defined by the trilinear interpolation of the volume data. By employing these two methods, we can achieve interactive isosurface exploration with high quality rendering.

SU‐FF‐T‐208: Modeling the Headscatter Off‐Axis in Megavoltage Photon Beams

Purpose: To accurately model off‐axis headscatter from a linear accelerator at off‐axis positions using a parameterized, two‐component model. An improved model is now proposed to model headscatter outside beam collimation. Method and Materials: We define the head‐scatter off‐axis ratio, HOA, as the ratio of the kerma due to head‐scatter photons at the off‐axis position x to the kerma from direct primary photons on the central axis. “Direct primary” are those photons that come from the source without interactions in the intervening structures. We measured HOA along the X jaw direction from the central axis to a distance of 25 cm from the field edge for various collimator settings. We fit the measured HOA using an improved two‐component model. The first component is a Gaussian source projected on a plane below the x‐ray source, representing the headscatter from the flattening filter and the primary collimator. Compared to an earlier model [1], this model models the second component for photons scattered in the secondary (variable) collimators using a linear edge kernel integrated over the four collimator jaws. Headscatter off‐axis outside beam collimation is examined more extensively with this study. Results: The two‐component model can be fit to data obtained from a variety of fields with a mean (maximum) local relative error of 10% (35)%. The region outside the penumbra region is modeled primarily by the collimator component. Conclusion: A two‐component model in which the headscatter from the flattening filter and the collimators are modeled separately can accurately predict HOA values for arbitrary collimator settings.