Unified treatment of gauge theories and Yang-Mills theory duality.
problem Unified treatment of gauge theories and Yang-Mills theory duality.
method Cohomological localization techniques and Atiyah-Singer index theorem.
result Unified framework and simplified derivations of localization formulas.
The squashed 7-sphere S7 is a 7-sphere with an Einstein metric given by the canonical variation and its cone R8−{0} has full holonomy Spin(7). There is a canonical calibrating 4-form Φ on R8−{0}. A minimal 3-submanifold in S7 is called associative if its cone …
Study calculates deformations of instantons on a specific Spin(7)-manifold.
problem Analyzing instantons on a specific Spin(7)-manifold. method Deformation theory developed by the author.
result Virtual dimensions of moduli spaces of instantons calculated.
We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed $Λ^3_{27…
MPNNs over-squash distant node information, study shows.
problem Over-squashing in MPNNs where node features ignore distant nodes.
method Theoretical analysis of MPNNs' over-squashing, focusing on width, depth, and graph topology.
result Width mitigates over-squashing but makes network more sensitive, depth doesn't help, graph topology is key.
Graph rewiring method alleviates over-squashing in GNNs.
problem Over-squashing and bottlenecks in graph neural networks.
method Curvature-based graph rewiring method.
result Negatively curved edges cause over-squashing in GNNs.
New Riemannian GNNs reduce over-squashing in graphs with negative curvature.
problem Over-squashing in Riemannian graph neural networks.
method Generalization of Hyperbolic GNNs to Riemannian manifolds of variable curvature.
result Bounds on sensitivity of node features in Riemannian GNNs as layers increase.
New solutions found for G2 system on squashed manifolds.
problem Finding solutions to the heterotic G2 system on squashed manifolds.
method Constructing families of solutions using squashed metrics on 7-sphere or Aloff-Wallach space.
result Obtained AdS3 solutions for most squashing parameters, excluding a specific case.
Graphs can be smoothed or squashed too, study finds.
problem Graph Neural Networks struggle with over-smoothing and over-squashing issues.
method Unified framework using Ollivier-Ricci curvature to address both issues.
result Over-smoothing and over-squashing linked to positive and negative graph curvature respectively.
gLSTM improves graph neural networks by increasing storage capacity to prevent over-squashing.
problem Over-squashing in GNNs collapses information from a large receptive field into a single vector, creating an information bottleneck.
method Introduced a new synthetic task to measure over-squashing and adapted ideas from sequence modeling to develop gLSTM, a novel GNN architecture with improved capacity.
result gLSTM architecture demonstrates strong performance on synthetic and real-world graph benchmarks, mitigating over-squashing.
Graph pruning improves neural network performance by addressing squashing and smoothing issues.
problem Over-squashing and over-smoothing in Graph Neural Networks.
method Proposes edge deletions to simultaneously address over-squashing and over-smoothing, optimizing spectral gap.
result Edge deletions improve generalization and distinguishability of nodes of different classes.
Graph neural networks struggle to propagate long-range information, causing over-squashing.
problem Graph neural networks struggle to propagate long-range information.
method Identified over-squashing as the bottleneck in GNNs, demonstrated on various models.
result Breaking the bottleneck improves GNNs' performance on long-range problems.
The study constructs associative 3-folds in squashed 3-Sasakian manifolds.
problem Understanding associative submanifolds in squashed 3-Sasakian manifolds.
method Analyzes foliations and geodesic ruled associative 3-folds correspondence.
result Infinitely many topological types of non-trivial associative 3-folds constructed.
Advocates against over-smoothing and over-squashing in GNNs, suggesting they are less critical than previously thought.
problem Over-smoothing and over-squashing in Graph Neural Networks (GNNs).
method Challenged the prevailing focus on these phenomena, proposing that performance decreases are due to uninformative receptive fields and localised information distribution.
result Performance decreases are mostly uncorrelated with over-smoothing and over-squashing, and optimal model depths remain small.
A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface Σ so that the surfa…
New method uses curvature to improve graph neural networks.
problem Graph Neural Networks struggle with over-smoothing and over-squashing.
method Augmented Forman-Ricci curvature (AFRC) for scalable rewiring.
result AFRC effectively mitigates over-smoothing and over-squashing.
New analysis shows over-squashing limits GNNs' power.
problem Understanding the expressive power of GNNs, especially under over-squashing.
method Quantitative measure of pairwise interactions in MPNNs, proving capacity requirements.
result Over-squashing hinders GNNs' ability to learn complex node feature functions.
Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class W1,p, 1<p<∞. If f almost minimizes its p Dirichlet energy then f is Hölder continuous. If p=2 and f is squeeze and squash stationary then f is in VMO.
We simplify word embeddings by removing sigmoid in SGNS, revealing connections to hyperbolic spaces.
problem Improving word embeddings quality and understanding their relationship with hyperbolic spaces.
method Analyzing squashed shifted PMI matrix and its relation to graph properties and hyperbolic geometry.
result Word embeddings can be connected to hyperbolic spaces through squashed shifted PMI matrix.
A new method boosts graph neural networks by preventing over-smoothing and over-squashing.
problem Graph Neural Networks struggle with long-range signals and over-smoothing/over-squashing.
method Proposes PowerEmbed, a layer-wise normalization technique inspired by spectral graph embedding.
result PowerEmbed prevents over-smoothing and avoids over-squashing, improving performance on heterophilous graphs.
Several Einstein-Sasaki 7-metrics appearing in the physical literature are fibered over four dimensional Kahler-Einstein metrics. Instead we consider here the natural Kahler-Einstein metrics defined over the twistor space Z of any quaternion Kahler 4-space, together with the corresponding Einstein-Sasaki metrics. We wo…
In this paper, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza--Klein black hole spacetime in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space-time metric using Lie point symmetries are presented and then it the n-dimensional optima…
We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing G-equivariance on the homogeneous space G/H=SU(4)/SU(3) endowed with its Sasaki-Einstein structure, and G/H=Sp(2)/Sp(1) as a 3-Sasakian manifold. In both cases …
DRew dynamically rewires message passing to improve long-range tasks.
problem Over-squashing in MPNNs due to local message passing.
method Layer-dependent rewiring and delay mechanism for gradual densification.
result DRew outperforms graph Transformers and multi-hop MPNNs on long-range tasks.
A new graph neural network (NBA-GNN) avoids revisiting nodes to improve accuracy.
problem Redundancy in graph neural network updates causes over-squashing and inaccurate recognition.
method Proposes non-backtracking graph neural networks (NBA-GNN) that update messages without revisiting nodes.
result The NBA-GNN alleviates over-squashing and improves performance on graph benchmarks.
New formalism solves kinematical constraints in curved backgrounds and non-trivial states.
problem Solving kinematical constraints due to Weyl invariance in curved backgrounds and non-trivial states.
method Constructing Weyl covariant geometric objects and identifying them as building blocks of correlation functions.
result Exact agreement with thermal OPEs and holographic computations for thermal 2-point functions.
In this paper we study Spin(7)-instantons on asymptotically conical Spin(7)-orbifolds (and manifolds) obtained by filling in certain squashed 3-Sasakian 7-manifolds. We construct a 1-parameter family of explicit Spin(7)-instantons. Taking the parameter to infinity, the family (a) bubbles o…
While Recurrent Neural Networks (RNNs) are famously known to be Turing complete, this relies on infinite precision in the states and unbounded computation time. We consider the case of RNNs with finite precision whose computation time is linear in the input length. Under these limitations, we show that different RNN va…
Machine learning finds Z/2 eigenfunctions on a sphere.
problem Finding Z/2 eigenfunctions on the sphere.
method Created a multivalued neural network and used JAX to implement it. Fixed branch points at tetrahedron and cube vertices, and allowed AI to move them in the third case.
result Found Z/2 eigenfunctions for three cases.
Latent Dirichlet Allocation models discrete data as a mixture of discrete distributions, using Dirichlet beliefs over the mixture weights. We study a variation of this concept, in which the documents' mixture weight beliefs are replaced with squashed Gaussian distributions. This allows documents to be associated with e…
We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhof…
In 1981, covariantly constant spinors were introduced into Kaluza-Klein theory as a way of counting the number of supersymmetries surviving compactification. These are related to the holonomy group of the compactifying manifold. The first non-trivial example was provided in 1982 by D=11 supergravity on the squashed S7,…
We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d N=2 "Lens space theory" T[L(p,1)] and the partition function of complex Chern-Simons theory on L(p,1). In particular, for p=1, we show how the familiar S3 partition func…
Graph neural networks (GNNs) are a class of neural networks that allow to efficiently perform inference on data that is associated to a graph structure, such as, e.g., citation networks or knowledge graphs. While several variants of GNNs have been proposed, they only consider simple nonlinear activation functions in th…
Rewiring networks using discrete geometry improves GNN training accuracy and reduces runtime.
problem Inefficient information propagation between distant nodes in graph neural networks.
method Discrete analogues of classical geometric curvature to model and rewire networks.
result Classical geometric notions achieve state-of-the-art GNN training accuracy and significantly reduce runtime.
This study develops an unsupervised learning algorithm for products of expert capsules with dynamic routing. Analogous to binary-valued neurons in Restricted Boltzmann Machines, the magnitude of a squashed capsule firing takes values between zero and one, representing the probability of the capsule being on. This analo…
Combines PCA and message passing for better graph node embeddings.
problem Challenges of learning long-range dependencies in graph data.
method PCAPass: PCA + message passing for unsupervised node embeddings.
result Empirically shows competitive performance on node classification benchmarks.
Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.
problem Analyzing geometric flows of G2-structures on 3-Sasakian manifolds.
method Study of Laplacian flow and Laplacian coflow of G2-structures on 3-Sasakian manifolds.
result Distinct behavior of flows, notably regarding stability of nearly parallel G2-structures.
A mathematical model describes deforming manifolds with precise vectors and fields.
problem Modeling and describing the deformation of complex manifolds in practical applications.
method Proposes a modified differential dynamic model with constraints on spatial and temporal continuity, presenting deforming vector and field.
result Demonstrates the effectiveness of an autonomous deforming field in data dimension reduction tasks.
Study YB operators and their deformations, finding integrable and nontrivial cases.
problem Understanding deformations of Yang-Baxter operators and their integrability.
method Relating deformations to Lie algebra deformations, analyzing cohomology groups.
result Existence of integrable YB deformations and nontrivial cases not arising from SD deformations.
Smooth deformation of Moishezon manifolds preserves their Moishezon property.
problem Preserving Moishezon property under smooth deformation.
method Smooth deformation over a unit disk in C.
result Deformation limit of Moishezon manifolds is Moishezon.
In this paper, we study deformations of holomorphic Poisson maps which extend Horikawa's series of papers on deformations of holomorphic maps in the context of holomorphic Poisson deformations. In appendices, we present deformations of Poisson morphisms in the language of functors of Artin rings which is the algebraic …
A method for accurate pricing of multidimensional derivatives under uncertain volatility.
problem High-dimensional stochastic control problem in uncertain volatility model.
method Backward actor-critic stochastic policy gradient scheme combining DP, PPO, and neural networks.
result Accurate and efficient pricing of multidimensional derivatives compared to benchmarks.
In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian…
Study on deformations of Lie groupoid morphisms and their properties.
problem Understanding the deformation theory of Lie groupoid morphisms.
method Established deformation theory, cohomology, and properties of morphisms.
result Invariance and stability properties of morphisms, Morita invariance of cohomology, and simultaneous deformations.
Study canonical deformations of complex forms and their cohomology properties.
problem Understanding canonical deformations and cohomology of complex manifolds.
method Analyzes canonical Aeppli deformations and their relations to deformed cohomology.
result Proves the jumping formula for deformed Aeppli cohomology and constant dimension conditions.
In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Kodaira's series of papers on semi-regularity (deformations of compact complex submanifolds of codimension 1), deformations of compact complex submanifolds of arbitrary codimensions, and stability of compact complex submanifol…
The L∞-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one L∞-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We a…