New deep learning method preserves orientation in shape matching.
problem Symmetry issues in shape matching.
method Orientation-aware functional maps using complex functional representations and DiffusionNet.
result Stable correspondence predictions with robust orientation preservation.
Method improves deep learning models for datasets with mixed approximate symmetries.
problem Improving deep learning models for datasets with mixed approximate symmetries.
method Regularizer-based approach to build models for datasets with mixed approximate symmetries.
result Our method achieves better accuracy than prior approaches while discovering the approximate symmetry levels correctly.
Computer vision models are unstable due to task symmetries and labelling issues.
problem Instability of computer vision models in classification tasks.
method Analysis of symmetries, categorical nature, and labelling issues.
result Instability is a necessary result of current computer vision formulation.
We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles…
These notes discuss several topics in neoclassical economics and alternatives, with an aim of reviewing fundamental issues in modeling economic markets. I start with a brief, non-rigorous summary of the basic Arrow-Debreu model of general equilibrium, as well as its extensions to include time and contingency. I then ar…
Develops a framework for designing quantum neural networks that respect symmetries.
problem Trainability and generalization issues in quantum neural networks.
method Equivariant quantum neural networks (EQNN) for any symmetry group.
result Efficient construction of equivariant layers for EQNNs, including QCNNs.
We consider clustering problems where the goal is to determine an optimal partition of a given point set in Euclidean space in terms of a collection of affine subspaces. While there is vast literature on heuristics for this kind of problem, such approaches are known to be susceptible to poor initializations and getting…
We investigate the reduction process of a k-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincare field equations. We discuss two issues about reconstructing a solutio…
This foreword discusses the contributions of Bolyai, Gauss, and Lobachevsky to non-Euclidean geometry.
problem The development of non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
method Historical review of the contributions of these mathematicians.
result The foundational work on non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
Theoretical guarantees for permutation-equivariant QNNs avoid barren plateaus.
problem Excessive local minima and barren plateaus in QNNs training landscapes.
method Designing Sn-equivariant QNNs to encode permutation symmetry. result Equivariant QNNs do not suffer from barren plateaus, quickly reach overparametrization, and generalize well.
By means of an analogy with Classical Mechanics and Geometrical Optics, we are able to reduce Lagrangians to a kinetic term only. This form enables us to examine the extended solution set of field theories by finding the geodesics of this kinetic term's metric. This new geometrical standpoint sheds light on some founda…
New framework builds quantum machine learning models respecting data symmetries.
problem Trainability and generalization issues in QML models for large problem sizes.
method Group-invariant models using data symmetries to build QML models.
result Group-invariant models produce outputs invariant under symmetry group actions.
Improved generative models learn structured data better.
problem Training score-based generative models for structured data.
method Nonlinear denoising score matching with neural control variates.
result Enhanced learning of multimodal and symmetric data.
New method for deep learning hierarchies like sequences and graphs.
problem Lack of formalism for dealing with hierarchical structures in deep learning.
method Express equivariant maps for hierarchies using combinations of building block symmetries.
result Demonstrates effectiveness on semantic segmentation of point-cloud data.
We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the boundary, this is the issue of whether any Killing field of the boundary metric extends to a Killing field of any bulk or filling Einstein metric inducing the given data on …
Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding t…
Paper defines conditions for projective links in projective 3-space.
problem Characterizing links in projective 3-space.
method Combinatorial conditions and antipodal symmetry.
result Easy condition to prevent alternating projective links.
Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. "Structure" has long been understood as symmetry which can tak…
Convolutional neural networks handle rotated image symmetries without dimensionality issues.
problem Binary image classification with rotational symmetry.
method Least squares plug-in classifiers based on convolutional neural networks under rotationally symmetric assumptions.
result Convolutional neural networks can circumvent the curse of dimensionality in binary image classification with rotational symmetry.
Study properties of holomorphic p-contact manifolds, including non-Kähler hyperbolicity and deformations.
problem Characterize the geometric and algebraic properties of holomorphic p-contact manifolds. method Explores non-Kähler hyperbolicity, differential calculus, and p-contact deformations, proving unobstructedness theorems. result Proves a Bogomolov-Tian-Todorov-type unobstructedness theorem for p-contact deformations up to order two. RL optimizes resource allocation in MG by balancing experience and exploration.
problem Optimal resource allocation in competitive scenarios.
method Introduced RL to MG, allowing dynamic strategy adjustment based on experience and expected rewards.
result Achieves optimal resource coordination by balancing exploitation and exploration.
Unconstrained MLIPs outperform constrained ones in accuracy and speed.
problem Improving the efficiency and accuracy of machine-learned interatomic potentials.
method Investigated unconstrained models trained on large datasets compared to physically constrained models.
result Unconstrained MLIPs can be superior in accuracy and speed compared to physically constrained models.
The paper develops an asymptotic theory of self-supervised pre-training.
problem Sharpness of current rates in self-supervised pre-training and their accuracy.
method Two-stage M-estimation and tools from Riemannian geometry.
result Characterization of the limiting distribution of the downstream test risk.
Label switching is a phenomenon arising in mixture model posterior inference that prevents one from meaningfully assessing posterior statistics using standard Monte Carlo procedures. This issue arises due to invariance of the posterior under actions of a group; for example, permuting the ordering of mixture components …
This article provides a brief discussion of the functional of super Riemann surfaces from the point of view of classical (i.e. not "super-) differential geometry. The discussion is based on symmetry considerations and aims to clarify the "borderline" between classical and super differential geometry with respect to the…
Researchers resolved ambiguities in gravitational radiation charges.
problem Ambiguities in charges related to gravitational radiation.
method Addressed supertranslation ambiguities in classical and extended BMS algebras.
result Proposed and proved an invariant charge free from supertranslation ambiguity.
Stability of catenoid in hyperbolic space proven without symmetry assumptions.
problem Stability of catenoid in hyperbolic space.
method Profile construction, modulation analysis, integrated local energy decay, vectorfield method.
result Nonlinear asymptotic stability of catenoid for n≥5 without symmetry assumptions. The paper analyzes symmetries of Vaidya-Bonner geodesics.
problem Investigating invariance properties of Vaidya-Bonner geodesics.
method Classification of Lie point symmetries and Noether symmetries, determination of optimal system of subalgebras.
result Determination of optimal system of subalgebras for Vaidya-Bonner geodesics.
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.
Humans take advantage of real world symmetries for various tasks, yet capturing their superb symmetry perception mechanism with a computational model remains elusive. Motivated by a new study demonstrating the extremely high inter-person accuracy of human perceived symmetries in the wild, we have constructed the first …
Classifies symmetries of non-flat 3-webs around a point.
problem Understanding symmetries of non-flat 3-webs.
method Classification and construction methods for symmetries.
result Classification of symmetries for non-flat 3-webs.
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.
Symmetry in loss functions constrains model parameters, leading to specific learning outcomes.
problem Understanding and leveraging symmetries in neural networks to improve learning outcomes.
method Analyzing the impact of loss function symmetries on model parameters and learning behavior.
result Mirror-reflection symmetries in loss functions lead to constraints on model parameters, influencing learning outcomes.
Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…
New symmetry dimensions for higher order ODEs are identified.
problem Determining the maximal and submaximal symmetry dimensions for higher order ODEs.
method Cartan-geometric approach to classify symmetry dimensions.
result Next largest realizable symmetry dimensions for scalar ODEs of order ≥ 4 and vector ODEs of order ≥ 3 are determined.
Generalizes symmetries of curved manifolds.
problem Maximizing symmetries in curved manifolds.
method Replaces torus with abelian group, generalizes results.
result Generalizes symmetry results for positively curved manifolds.
Study of continuous symmetries in Nahm data and BPS monopoles.
problem Solutions to Nahm's equations with continuous symmetries.
method Classification of Ansätze and construction of Nahm data.
result Construction of new BPS monopoles with spherical symmetry.
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.
Researchers discover symmetries in Ricci flows and use them to find invariant solutions.
problem Finding symmetries in Ricci flows on manifolds.
method Developed a method to find Lie point symmetries of Ricci flows and particular metrics.
result Invariant solutions of Ricci flow for specific metric families were obtained.
This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of…
Symmetry in finance is a neglected but potentially valuable concept.
problem The underutilization of symmetry in financial markets.
method Examining symmetry in game theory, technical analysis, and long-term economic growth.
result Symmetry principles can be applied to financial strategies and market dynamics.
Symmetry of neural network densities can be determined from correlation functions.
problem Determining symmetries of neural network densities without knowing the density itself.
method Symmetry-via-duality approach using invariance properties of correlation functions.
result Symmetries of neural network densities can be determined via dual computations of correlation functions.
Symmetries in shrinking Ricci solitons spread outward.
problem Understanding symmetries in shrinking Ricci solitons.
method Propagating approximate symmetries to larger scales.
result Symmetries in shrinking Ricci solitons spread outward.
Clarifies relation between Pfaffian fibrations and relative algebroids.
problem Understanding geometric structures and symmetries in PDEs.
method Introduces and analyzes Pfaffian fibrations and relative algebroids, clarifying their relationship.
result Every Pfaffian fibration induces a relative algebroid, and their prolongations and local solutions coincide.
Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.
problem Classifying Lie symmetry algebras for 2D quasilinear equations.
method Classification based on abelian Lie symmetry algebras of dimension and rank.
result Equations with specific symmetry algebras are linearizable.
This work relaxes GNN symmetries to approximate automorphisms, improving model performance.
problem Improving graph neural network performance on asymmetric graphs.
method Formalizing approximate symmetries via graph coarsening, introducing a bias-variance formula.
result Best generalization performance achieved by choosing a larger symmetry group than automorphisms but smaller than permutations.
Noether's theorem clarifies how symmetries in neural networks influence learning.
problem Understanding how symmetries in neural networks affect learning.
method Systematic study of symmetry interactions with learning algorithms using Noether's theorem.
result Symmetries impose restrictions on the optimization path, leading to conserved quantities.
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.