New method detects symmetries beyond affine transformations.
problem Current methods limit symmetry detection to affine transformations.
method Framework for discovering continuous symmetry beyond affine transformations.
result Method is competitive for large sample sizes and superior for small sample sizes.
Study on symmetries of differential equations using gauge transformations and coverings.
problem Understanding the geometry of symmetries in differential equations.
method Interpreting symmetries as gauge transformations and coverings.
result Geometrical insights into λ and μ-symmetries. New minimal hypersurfaces found via transformations.
problem Finding new axially symmetric minimal hypersurfaces in 4D Minkowski space.
method Combining scaling symmetries and a non-obvious symmetry (analogous to Bianchi's transformation) to generate new hypersurfaces.
result Infinitely many axially symmetric minimal hypersurfaces can be generated from any given one.
Transformers reduce redundancy by focusing on invariant relational quantities.
problem Substantial internal redundancy in Transformer models due to coordinate-dependent representations and continuous symmetries.
method Reformulate representations, attention mechanisms, and optimization dynamics in terms of invariant relational quantities, eliminating redundant degrees of freedom by construction.
result Architectures that operate directly on relational structures, providing a principled geometric framework for reducing parameter redundancy and analyzing optimization.
New model learns symmetry transformations from complex data.
problem Learning symmetry transformations in complex domains like chemical space.
method Two latent subspaces, deep information bottleneck, continuous mutual information regularizer.
result Model outperforms state-of-the-art methods on artificial and molecular datasets.
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.
We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.
In this paper we study a symmetry group of vector space. Basis manifold is a homogeneous space of a symmetry group. This concept leads us to the definition of active and passive transformations on basis manifold. Active transformation can be expressed as a transformation of vector space. Passive transformation gives ab…
New mesh network preserves symmetries in deep learning.
problem No existing mesh processing architecture is equivariant to all symmetries.
method Equivariant attention-based mesh network using relative tangential features.
result The network achieves improved performance and is equivariant to various transformations.
Proves upper bounds for Bäcklund transformations in hyperbolic systems.
problem Bounding the generality of Bäcklund transformations in hyperbolic systems.
method Cartan's Method of Equivalence, classification results for specific symmetry groups.
result Obtains classification results and new examples of auto-Bäcklund transformations.
We find a remarkable subalgebra of higher symmetries of the elliptic Euler-Darboux equation. To this aim we map such equation into its hyperbolic analogue already studied by Shemarulin. Taking into consideration how symmetries and recursion operators transform by this complex contact transformation, we explicitly give …
We approach the construction of Backlund transformations for Darboux integrable hyperbolic partial differential equations in the plane through the reduction of exterior differential systems. For example it is shown that all the Backlund transformations in arXiv:0707.4408v2 can be constructed using symmetry reduction.
New CMC surfaces with dihedral symmetry constructed from Darboux transforms.
problem Constructing closed CMC surfaces with specific symmetries.
method Using Darboux transforms on multiple covers to create new CMC surfaces.
result Explicit parametrisations of new closed CMC surfaces with dihedral symmetry.
The paper tackles learning symmetries in data without expert knowledge.
problem Learning symmetries in data from raw data without prior knowledge.
method Develops methods to select eigenvectors for orthogonal symmetries and compares their effectiveness.
result The problem of learning symmetries is as hard as the graph automorphism problem in the worst case, but can be simplified with certain restrictions.
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of geometric symmetry of a linearly singular differential equation is introduced as a trans…
New method detects projective equivalences and symmetries in rational 3D curves.
problem Detecting projective equivalences and symmetries in rational 3D curves.
method Using differential invariants and Möbius transformations to avoid solving large polynomial systems.
result Efficient algorithm for detecting projective equivalences and symmetries without solving large polynomial systems.
Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.
problem Existence and properties of very stable Higgs bundles.
method Bialynicki-Birula theory, C∗-actions, Hecke transformations, Fourier-Mukai transforms. result Precise formula for multiplicity of very stable components of global nilpotent cone.
New symmetries found for scalar and vector ODEs of arbitrary dimensions.
problem Identifying symmetries for scalar and vector ODEs of arbitrary dimensions.
method Explicit expressions and abelian Lie algebra for non-Cartan symmetries in arbitrary dimensions.
result Non-Cartan symmetries characterize linearizable systems of ODEs but not nonlinear ones.
Symmetry in neural networks affects generalization, as shown by CLT and RG transformations.
problem Improving generalization in neural networks by incorporating physical symmetries.
method Evaluation of symmetry constraints and expressivity in MLPs and GNNs using the CLT as a test case.
result Overly complex or overconstrained models generalize poorly, revealing a competition between symmetry constraints and expressivity.
Method discovers symmetries in data with neural networks.
problem Discovering symmetries in high-dimensional data.
method Fully connected neural networks trained with a loss function for symmetry.
result Generators for symmetries in latent space.
New framework discovers non-affine continuous symmetries in neural networks.
problem Lack of efficient methods for detecting non-affine continuous symmetries in neural networks.
method Computational framework for discovering infinitesimal generators of multi-parameter group actions.
result Framework can discover non-affine continuous symmetries in neural networks.
Based on the recent work \cite{PII} we put forward a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid which contain…
We define a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid. Some of their properties are investigated and we give…
Functions with constant geodesic X-ray transform are restricted to manifolds with specific geometrical properties.
problem Existence of functions with constant geodesic X-ray transform on manifolds.
method Analyzing the geometrical properties of manifolds based on the existence of such functions.
result Functions with constant geodesic X-ray transform impose specific geometrical restrictions on the manifold.
We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them for both equations. For certain simple choices of partner symmetries the resulting differential cons…
The paper develops methods to generate invariant quantities in Metric-Affine Geometry.
problem Developing methods to generate invariant quantities in Metric-Affine Geometry.
method The paper introduces a theorem to generate invariant quantities under transformations of the affine connection, proving invariance conditions.
result Theorem establishing conditions for invariance of functionals under transformations of the affine connection.
In this paper, we study generalized symmetric Finsler spaces. We first study symmetry preserving diffeomorphisms, then we show that the group of symmetry preserving diffeomorphisms is a transitive Lie transformation group. Finally we give some existence theorems.
L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.
problem Extracting scientific understanding from particle-physics experiments with high precision and efficiency.
method L-GATr, a geometric algebra Transformer, representing data in 4D space-time and being equivariant under Lorentz transformations.
result L-GATr achieves performance comparable to or better than domain-specific baselines on regression, classification, and generative tasks.
Unified framework connects different neural network models.
problem Understanding the geometry of neural network loss landscapes.
method Unified framework capturing four symmetry classes.
result First discovery of low- and zero-barrier linear interpolation paths.
New definition of Bäcklund transformation for surface isometric deformation.
problem Defining Bäcklund transformation in surface isometric deformation.
method Proving generic 4D integrable rolling distribution splits into 1D family of 3D distributions.
result Introducing new definition of Bäcklund transformation.
This paper aims to incorporate passive symmetries in machine learning for better generalization.
problem Machine learning's reliance on arbitrary choices leads to passive symmetries that can limit generalization.
method Translation among physics, mathematics, and machine learning to understand and implement passive symmetries.
result Respecting passive symmetries can improve machine learning's ability to generalize.
Method learns equivariances from data without custom architecture design.
problem Learning equivariances for tasks without manually designed architectures.
method Reparameterization to learn equivariant parameter sharing.
result Can learn equivariances for any finite group of transformations.
In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are …
Symmetry principles help in creating better AI representations.
problem Creating efficient and generalizable AI representations.
method Using symmetry transformations to guide representation learning.
result Symmetry principles improve data efficiency and generalizability in AI.
New definition of disentangled representations using symmetry transformations.
problem Lack of a generally agreed-upon definition of disentangled representations.
method Focus on transformation properties of the world using symmetry transformations and group representation theory.
result First formal definition of disentangled representations.
We give geometric explanations and proofs of various mirror symmetry conjectures for Tn-invariant Calabi-Yau manifolds when instanton corrections are absent. This uses fiberwise Fourier transformation together with base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex stru…
For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies are related by a Legendre type transformation. We also consider relationships b…
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, w…
Classifies vector equations with higher symmetries.
problem Classifying integrable vector equations with higher symmetries.
method Complete classification of isotropic vector equations of geometric type with higher symmetries.
result New examples of integrable multi-component systems and auto-Backlund transformations found.
Guichard's transformations generate Voss surfaces from sine-Gordon solutions.
problem Generating Voss surfaces from sine-Gordon solutions.
method Using Guichard transformations and recursion operators for sine-Gordon symmetries.
result Explicit derivation of Voss nets and length of Guichard sequences.
For two solutions of the WDVV equations that are related by two types of symmetries of the equations given by Dubrovin, we show that the associated principal hierarchies of integrable systems are related by certain reciprocal transformation, and the tau functions of the hierarchies are either identical or related by a …
This work is dedicated to the study of the Moebius invariant class of constrained Willmore surfaces and its symmetries. We define a spectral deformation by the action of a loop of flat metric connections; Baecklund transformations, by applying a dressing action; and, in 4-space, Darboux transformations, based on the so…
Metric evaluates symmetry-breaking in datasets, revealing severe biases.
problem Symmetry-breaking in datasets can hinder the performance of symmetry-aware methods.
method Developed a metric to quantify symmetry-breaking using a two-sample classifier test.
result Symmetry-breaking can prevent optimal performance of invariant methods, even when labels are invariant.
Transformer struggles with arithmetic length but improves with explicit structure encoding.
problem Transformers fail to generalize length in arithmetic tasks.
method Explicitly encoding structural symmetries via modified number formatting and custom positional encodings.
result Transformer can generalize up to 50-digit numbers without additional data.
We describe mirror symmetry on higher dimensional tori, paying special attention to the behaviour of D-branes under mirror symmetry. To find the mirror D-branes the description of mirror symmetry on D-branes due to Ooguri, Oz en Yin is used. This method allows us to deal with the coisotropic D-branes recently introduce…
Researchers develop Orlov-Schulman symmetries for self-dual conformal structures.
problem Developing symmetries for self-dual conformal structures.
method Explicit proof of compatibility with Lax-Sato flows, dressing scheme based on Riemann-Hilbert problem.
result Construction and proof of compatibility of Orlov-Schulman symmetries.
Study generalizes Yang-Mills equations for special complex surfaces.
problem Deriving equations for self-dual Yang-Mills fields on complex surfaces.
method Generalization of flat space Yang's and Newman's equations to conformally Kahler Riemannian 4-manifolds.
result Continuous group of hidden symmetries found only for conformally half-flat geometry.
Bayesian quadrature improves integration efficiency with invariant priors.
problem Efficient numerical integration with known structure.
method Invariance priors for bijective transformations in input domain.
result Superior performance in synthetic and real-world applications.