Geometric mechanism mimics physics' symmetry breaking.
problem Understanding spontaneous symmetry breaking in geometry.
method Analogous to physics, studying symmetry breaking in differential geometry.
result Symmetry breaking can be used to solve geometric problems.
Researchers create BPS monopoles with any desired symmetry breaking.
problem Creating monopoles with specific symmetry breaking patterns.
method Using a new class of Nahm data to construct finite energy BPS monopoles.
result Arbitrary symmetry breaking monopoles can be constructed.
The paper classifies and proves properties of symmetry breaking operators for specific groups.
problem Classifying and understanding symmetry breaking operators for de Sitter and Lorentz groups.
method Constructing and classifying differential symmetry breaking operators, proving localness, and showing sporadic nature.
result All symmetry breaking operators are differential and sporadic, not obtainable by residue formulas.
We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explic…
Investigates spontaneous symmetry breaking in non-equilibrium systems.
problem Spontaneous symmetry breaking of ergodicity in non-equilibrium systems.
method Mathematical and effective field theory approaches to investigate symmetry breaking.
result Symmetry breaking phenomena observed in stochastic processes.
SymPE breaks symmetries in equivariant networks, improving performance across various tasks.
problem Equivariant networks cannot break symmetries, leading to poor performance in tasks with symmetrical inputs.
method Novel equivariant conditional distributions and randomized canonicalization.
result SymPE significantly improves performance of group-equivariant and graph neural networks.
Noether's framework reveals symmetry-breaking in neural networks.
problem Understanding the role of symmetry breaking in neural networks.
method Developed a theoretical framework using Lagrangian mechanics.
result Identified 'kinetic symmetry breaking' and its effect on learning dynamics.
New method breaks symmetry in neural networks, improving sample efficiency.
problem Symmetry in neural networks limits their ability to learn unique features.
method Introduces 'relaxed equivariance' to overcome symmetry limitations.
result Equivariant multilayer perceptrons (E-MLPs) can now break symmetry at the sample level.
We introduce the concept of spontaneous symmetry breaking to arbitrage modeling. In the model, the arbitrage strategy is considered as being in the symmetry breaking phase and the phase transition between arbitrage mode and no-arbitrage mode is triggered by a control parameter. We estimate the control parameter for mom…
We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, …
Study on spontaneous symmetry breaking in financial markets using quantum mechanics.
problem Analyzing spontaneous symmetry breaking in financial markets.
method Using Hamiltonian form of Black-Scholes and Merton-Garman equations, analyzing symmetry breaking and interpreting Nambu-Goldstone bosons.
result Interpretation of Nambu-Goldstone bosons in financial markets.
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.
Symmetry in inverse problems leads to multiple solutions, but breaking symmetry helps deep learning.
problem Symmetry in physical systems causes multiple solutions in inverse problems, hindering deep learning.
method Careful symmetry breaking on training data helps solve inverse problems and improve deep learning performance.
result Symmetry breaking on training data significantly improves deep learning performance in inverse problems.
Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method…
New theory shows how membranes can break symmetry.
problem Understanding symmetry breaking in membranes with boundaries.
method Applied bifurcation theory and reduced membrane equation.
result Existence of symmetry breaking bifurcation in membrane solutions.
Classifies patterns of symmetry breaking and vacuum degeneracy in scalar and gauge fields.
problem Understanding patterns of symmetry breaking and vacuum degeneracy in complex field systems.
method Uses mathematical classification of singular foliations to encode and classify patterns of spontaneous symmetry breaking and vacuum degeneracy.
result Mathematical classification provides a qualitative understanding of possible patterns of vacuum degeneracy.
We provide some insights in the study of branching problems of reductive groups, and a method of investigations into symmetry breaking operators. First, we give geometric criteria for finiteness property of linearly independent continuous (respectively, differential) operators that intertwine two induced representation…
The paper connects financial vacuum conditions to spontaneous symmetry breaking in quantum finance.
problem Understanding the conditions under which the martingale condition is a non-degenerate vacuum.
method Expressing financial equations in Hamiltonian form and analyzing symmetry breaking.
result Conditions for the martingale condition to be a non-degenerate vacuum are identified.
This paper analyzes how kinetic terms in stock market equations can affect symmetry breaking.
problem Spontaneous symmetry breaking in quantum finance and its impact on stock market dynamics.
method Analyzes the role of kinetic terms in the context of the martingale condition in stock market equations.
result Kinetic terms can shift the effective location of the vacuum state, affecting symmetry breaking patterns.
Study symmetry breaking in quantum mechanics to understand many-body physics.
problem Understanding many-body physics from quantum mechanics.
method Analyzing potentials with unstable critical points and local minima.
result Emergence of many-body physics from spontaneous symmetry breaking.
The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
Symmetric critical points lead to symmetry breaking in neural networks.
problem Understanding symmetry in critical points of invariant functions.
method Analyzing the symmetry of critical points and their neighbors in invariant nonconvex functions.
result Symmetric critical points in invariant nonconvex functions are generically followed by symmetry breaking adjacent points.
We prove an explicit residue formula for a meromorphic continuation of conformally covariant integral operators between differential forms on Rn and on its hyperplane. The results provide a simple and new construction of the conformally covariant differential symmetry breaking operators between differential fo…
Paper proposes a method to break symmetries in Bayesian matrix factorization.
problem Symmetries in posterior distribution reduce MCMC sampling efficiency.
method Modification to Gaussian prior mean and covariance to break symmetries.
result Breaking symmetries leads to lower autocorrelation and reconstruction errors.
Constructs moduli spaces for monopoles with arbitrary symmetry breaking.
problem Finding moduli spaces for monopoles with varying symmetry.
method Defined configuration space with asymptotic conditions, performed quotient construction, used b-calculus and scattering calculus.
result Constructs hyper-Kähler moduli spaces for monopoles with arbitrary symmetry breaking.
We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in …
Rapid progress has been made recently on symmetry breaking operators for real reductive groups. Based on Program A-C for branching problems (T.Kobayashi [Progr.Math.2015]), we illustrate a scheme of the classification of (local and nonlocal) symmetry breaking operators by an example of conformal representations on diff…
Continuous symmetries and their breaking play a prominent role in contemporary physics. Effective low-energy field theories around symmetry breaking states explain diverse phenomena such as superconductivity, magnetism, and the mass of nucleons. We show that such field theories can also be a useful tool in machine lear…
GE-autoencoder identifies spontaneous symmetry breaking in systems.
problem Locating phase boundaries and identifying spontaneously broken symmetries in systems.
method Group-equivariant autoencoder using group theory to constrain parameters and learn invariant order parameters.
result GE-autoencoder accurately determines spontaneous symmetry breaking and estimates critical temperatures more efficiently.
Permutation of any two hidden units yields invariant properties in typical deep generative neural networks. This permutation symmetry plays an important role in understanding the computation performance of a broad class of neural networks with two or more hidden units. However, a theoretical study of the permutation sy…
Binary perceptron's instability linked to replica symmetry breaking.
problem Understanding the relationship between algorithmic instability and replica symmetry breaking in binary perceptron learning.
method Established the connection between algorithmic instability and replica symmetry breaking by comparing the instability condition around the fixed point to the instability for breaking the replica symmetric solution of the free energy function.
result The instability condition around the algorithmic fixed point is identical to the instability for breaking the replica symmetric saddle point solution of the free energy function.
Researchers classify differential operators between 3-sphere and 2-sphere bundles.
problem Classifying differential symmetry breaking operators between 3-sphere and 2-sphere bundles.
method Constructing and classifying all differential symmetry breaking operators D_{λ,ν}^m.
result Necessary and sufficient conditions for the existence of these operators.
Generative diffusion models are analyzed for their information dynamics.
problem Lack of a unified theoretical understanding of generative diffusion models.
method Integrated perspective connecting information-theoretic, dynamical, and thermodynamic aspects.
result Generative bandwidth is directly governed by the divergence of the score function's vector field.
Paper introduces a new symbol map for differential symmetry breaking operators.
problem Generalizing the symbol map to non-abelian settings.
method Introduces and studies the truncated symbol map Symb0(D). result Classified and constructed differential intertwining operators and homomorphisms.
Breaking symmetry in training data is key for generalization in feature learning kernels.
problem Grokking in algebraic tasks, where models perform well on training but fail on unseen data.
method Used Recursive Feature Machine (RFM) with AGOP to learn task-relevant features, breaking symmetry in training data.
result Generalization occurs only when symmetry in the training set is broken, and RFM generalizes by recovering underlying invariance group action.
Improves hypergraph link prediction by breaking symmetry.
problem Limited expressivity of GWL-1 algorithm in hypergraph link prediction.
method Preprocessing algorithm to identify and replace symmetry-inducing subhypergraphs with covering hyperedges.
result Improves expressivity of GWL-1, leading to better link prediction.
We study conformal symmetry breaking differential operators which map differential forms on Rn to differential forms on a codimension one subspace Rn−1. These operators are equivariant with respect to the conformal Lie algebra of the subspace Rn−1. They correspond to homomorphism…
We reveal connections between RBMs and Bosons, explaining symmetry breaking in their energy landscapes.
problem Understanding the relationships among different deep generative models and their learning mechanisms.
method Introducing a reciprocal space formulation to RBMs, revealing connections to diffusion processes and Bosons.
result Symmetry breaking in RBM energy landscapes is characterized by singular values and weight matrix eigenvectors.
Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.
problem Constructing solutions to Bogomolny equations with given boundary and asymptotic conditions.
method Using generalized Nahm pole boundary condition and real symmetry breaking condition.
result Solutions analogous to instanton solutions, satisfying different asymptotic conditions.
A machine learning model with approximate rotational symmetry is tested and found stable.
problem The effects of broken symmetries in machine learning models.
method Testing a model with approximate rotational symmetry in various physical scenarios.
result The model remains stable even with noticeable symmetry artifacts, suggesting potential benefits.
New method shows random, diverse initializations are not essential for deep neural networks.
problem The necessity of random, diverse initializations in deep neural networks.
method Constructed a deep convolutional network with identical features by initializing weights to 0, enabling signal propagation and stable gradients.
result Random, diverse initializations are not necessary for training neural networks.
The goal of this paper is twofold. First we prove a rigidity estimate, which generalises the theorem on geometric rigidity of Friesecke, James and Müller to 1-forms with non-vanishing exterior derivative. Second we use this estimate to prove a kind of spontaneous breaking of rotational symmetry for some models of cryst…
This paper proposes a new approach to Transformers by integrating hierarchical associative memory with MetaFormers.
problem Theoretical framework for Transformers and MLP-Mixers remains underdeveloped.
method Integrating hierarchical associative memory with MetaFormers to create a parallelized MLP-Mixer.
result Symmetry-breaking effects improve the performance of the MLP-Mixer in image recognition tasks.
The paper explains emergent phenomena in deep learning using entropic forces.
problem Understanding the cause of emergent phenomena in deep learning and large language models.
method Proposes a rigorous entropic-force theory for neural networks trained with SGD and variants.
result Shows that representation learning is governed by emergent entropic forces that break continuous symmetries and preserve discrete ones.
We consider the optimization problem associated with fitting two-layer ReLU networks with respect to the squared loss, where labels are assumed to be generated by a target network. Focusing first on standard Gaussian inputs, we show that the structure of spurious local minima detected by stochastic gradient descent (SG…
The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.
problem Nonlinear dynamics of minimal resistance in radial fields.
method Analysis of two non-equilibrium scenarios: scale-invariant free expansion and incompressible source flow.
result Incompressible flow acts as a structural regularizer, admitting unique, smooth, and strictly concave solutions.
In this paper we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addit…
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.