In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-man…
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48 results for “invariant classification”
Classifies invariant generalised Killing spinors on Lie groups.
problem Classifying invariant generalised Killing spinors on Lie groups.
method Complete classification using invariant properties and computational methods.
result Existence of non-trivial invariant generalised Killing spinors implies all invariant spinors are generalised Killing with the same endomorphism.
Improved CNNs for image classification tasks with rotational invariance.
problem Improving CNN performance for tasks with rotation invariance.
method Rotation-equivariant and rotation-invariant encoding using 2D-DFT magnitude responses and efficient convolutional schemes.
result Significant improvement in classification accuracy and robustness to hyperparameters.
Classifies and computes cohomologies of complex structures on Lie groups.
problem Classifying and computing cohomologies of complex structures on Lie groups.
method Complete classification and computation of invariant cohomologies for left invariant structures.
result Computed invariant cohomologies for various generalized complex and Kähler structures.
New method classifies symplectic structures on Lie groups.
problem Classifying left-invariant symplectic structures on Lie groups.
method Using moduli space of left-invariant nondegenerate 2-forms.
result Classified left-invariant symplectic structures on specific Lie groups.
Reduces gender classification bias by learning race-invariant face representations.
problem Societal bias in gender recognition systems.
method Adversarially trained autoencoder model to learn race-invariant face representations.
result Achieved a significant drop of over 40% in racial bias surrogate metric with race invariant representations.
New method classifies spin 4-manifolds using Kervaire-Milnor invariant.
problem Classifying spin 4-manifolds up to stabilisation.
method Using Kervaire-Milnor invariant to compute Arf invariants and classify.
result New stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
New quantum invariants for planar knotoids improve knot classification.
problem Classifying and distinguishing planar knotoids with up to five crossings.
method Define biframed planar knotoids and construct new invariants.
result Improved classification of planar knotoids with up to five crossings.
New proof for higher rank subvarieties in genus three.
problem Classifying higher rank invariant subvarieties in genus three.
method Uses recent techniques developed by Apisa and Wright.
result Short and simplified proof of classification.
The paper classifies left-invariant pseudo-Riemannian metrics on specific Lie groups.
problem Classifying left-invariant pseudo-Riemannian metrics on Lie groups.
method Analyzing left-invariant metrics on specific Lie groups with .
result A complete classification of left-invariant pseudo-Riemannian metrics for Lie groups of dimension .
This research classifies invariant complex structures and Kähler metrics on principal bundles.
problem Classifying invariant complex structures and Kähler metrics on principal bundles.
method Using Wang's theory of invariant connections and the Levi-Civita connection, the study provides direct geometric proofs and extends the classification from Hermitian to general symmetric spaces.
result The invariant integrable complex structures are unique in the reduced frame bundles of the upper half-plane and complex projective spaces.
Following a purely algebraic procedure, we provide an exhaustive classification of local Weyl-invariant scalar densities in dimension D=8.
New framework improves neural network robustness without sacrificing accuracy.
problem Adversarial inputs compromise neural network robustness.
method Extract and model invariances of objects to enhance classification robustness.
result Invariances improve both robustness and accuracy in classification tasks.
The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.
New approach improves classification guarantees by focusing on direction rather than regression risk.
problem Improving classification guarantees in binary classification problems.
method Establishing a geometric distinction between classification and regression, leveraging scale invariance.
result Improved guarantees for classification risk compared to regression risk.
Graph homomorphism numbers embed graphs for classification.
problem Graph classification using graph homomorphisms.
method Embed graphs into vectors using homomorphism numbers.
result Homomorphism vectors are universal for approximating graph invariants.
A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds. From this classification we then prove the Poincaré conjecture.
PiNet learns graph representations invariant to node permutations.
problem Graph classification and representation learning invariant to node permutations.
method Differentiable node attention pooling, permutation invariant graph neural network.
result Significant accuracy improvement in isomorphic graph classification with limited training data.
We add prior knowledge to deep networks to make them invariant to transformations.
problem Creating deep networks invariant to transformations like rotation.
method A novel layer based on invariant integration to enforce feature space invariances.
result State-of-the-art performance on the Rotated-MNIST dataset.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
New proof of link classification using map invariants.
problem Classifying two-component links in and string links.
method Invariants of link maps and their variations.
result New proof of Nakanishi-Ohyama classification for two-component links.
In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the eq…
The paper extends the classification of bi-invariant 2-forms to infinite-dimensional Lie groups.
problem Classifying bi-invariant 2-forms on infinite-dimensional Lie groups.
method Generalized the classification from compact Lie groups to arbitrary finite-dimensional Lie groups and then to Milnor regular infinite-dimensional Lie groups.
result The classification of bi-invariant 2-forms extends to all Milnor regular infinite-dimensional Lie groups.
Vacuum Kundt Wavesgr-qc
We discuss the invariant classification of vacuum Kundt waves using the Cartan-Karlhede algorithm, and the upper bound on the number of iterations of the Karlhede algorithm to classify the vacuum Kundt waves. By choosing a particular coordinate system we partially construct the canonical coframe used in the classificat…
We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the literature, in particular the one obtained in [Falbel-Gorodski, 1996] in terms of curvatu…
Machine learning classifies braids and discovers new invariants.
problem Classifying and discovering invariants of braids and flat braids.
method Supervised learning with neural networks to classify braids as trivial or non-trivial.
result Found new convenient invariants of braids, including a complete invariant of flat braids.
Simplified classification of special measures in CAT(-1) spaces.
problem Classifying horospherical invariant measures in higher rank.
method Expository approach, simplified argument for special case.
result Simplified classification of measures in CAT(-1) spaces.
A new rotation invariant method for 3D medical imaging classification.
problem Computational expense and lack of rotation invariance in 3D medical image processing.
method Proposes a rotation invariant convolution operator using hypersphere topology.
result Demonstrates improved classification accuracy and rotation invariance.
Classifies homogeneous Riemannian structures on 3D Lie groups.
problem Classifying homogeneous Riemannian structures on 3D Lie groups.
method Classification based on left invariant metrics and previous classifications.
result Complete classification of homogeneous Riemannian structures on 3D Lie groups.
New invariants classify shapes of 3-braid closures.
problem Classifying shapes of 3-braid closures using invariants.
method Proved finitely many possible shapes for Rasmussen invariants of 3-braid closures.
result Computed Rasmussen and annular Rasmussen invariants of all 3-braids.
Study homogeneous 8-manifolds with invariant Spin(7)-structures.
problem Classify and describe invariant Spin(7)-structures on homogeneous 8-manifolds.
method Examine canonical presentations G/H, construct invariant Spin(7)-structures, analyze associated connections.
result Explicit examples and classifications of invariant Spin(7)-structures on homogeneous 8-manifolds.
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for .
Learn invariances in neural networks by optimizing over augmentation parameters.
problem Lack of knowledge about present invariances and their extent in data.
method Parameterize a distribution over augmentations and optimize network parameters and augmentation parameters simultaneously.
result Recover correct set and extent of invariances on various tasks from training data alone.
New approach combines invariance and information bottleneck for OOD generalization.
problem OOD generalization failures in classification tasks.
method Revisit linear regression tasks, prove information bottleneck constraint necessary, propose combined approach.
result Combined invariance and information bottleneck approach improves OOD generalization.
Liouville classification of integrable geodesic flows on a projective plane in a potential fieldmath.DG
Classifies geodesic flows on projective plane with potential field.
problem Classifying geodesic flows on a projective plane with a potential field.
method Liouville classification and calculation of Fomenko--Zieschang invariants.
result All Fomenko--Zieschang invariants of the system are calculated.
We use Bott-Chern cohomology to measure the non-Kählerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermit…
Learning invariant representations is an important problem in machine learning and pattern recognition. In this paper, we present a novel framework of transformation-invariant feature learning by incorporating linear transformations into the feature learning algorithms. For example, we present the transformation-invari…
The paper analyzes the tradeoffs between accuracy and invariance in learning representations.
problem Achieving both accuracy and invariance in machine learning models.
method Information theoretic analysis of classification and regression settings.
result Characterization of the accuracy and invariance achievable by any representation of the data.
The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein's Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.
In this paper the notion of an M-th order invariant bilinear differential pairing is introduced and a formal definition is given. If the manifold has an AHS structure, then various first order pairings are constructed. This yields a classification of all first order invariant bilinear differential pairings on homogeneo…
Classifies tight contact structures with special symmetries.
problem Classifying tight contact structures with specific symmetries.
method Proves classification results for tight contact structures in 3-space, ball, and sphere with a new integral torsion.
result New integral torsion dictates a splitting between equivalence classes.
We study -ary commutative superalgebras and -algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their -ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
A new warping-invariant distance improves nearest-neighbor classification efficiency.
problem dtw distance inconsistency and inefficiency in nearest-neighbor classification.
method Showed dtw is not warping-invariant, converted to twi distance.
result twi distance equivalent error rates to dtw, more efficient.
The main goal is to classify 4-dimensional real Lie algebras $\g$ which admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-self-dual metric. Our study is related to the …
Machine learning predicts arithmetic curve invariants with high accuracy.
problem Classifying arithmetic curves based on their invariants.
method Training machine learning algorithms on datasets of elliptic and genus 2 curves.
result High accuracy in classifying curves, including rank, torsion, and integral points.
Complete classification of para-Kähler structures on Lie groups.
problem Classifying para-Kähler structures on Lie groups.
method Complete classification via automorphism consideration.
result Classification of para-Kähler structures on four-dimensional Lie groups.
Study open 3-manifolds as sums of closed ones, finding a classification.
problem Classifying open 3-manifolds that are sums of closed 3-manifolds.
method Introduced topological invariants, classified when finitely many summands up to diffeomorphism.
result Unified classification of open 3-manifolds and closed 3-manifolds.
Proposes FCNs for text classification with size-invariant inputs.
problem Classifying text of varying sizes.
method Uses fully convolutional networks with modifications to attention mechanisms.
result Suboptimal results on ITAmoji task, proposes fixes.