New method prevents classifiers from relying on spurious correlations.
problem Group invariant learning fails to prevent classifiers from depending on spurious correlations.
method Statistical independence tests to construct groups and reweight samples by group label proportion.
result New method significantly outperforms existing group invariant learning methods in generalizing to spurious correlation shifts.
This work provides statistical guarantees for GANs that are invariant to certain group symmetries.
problem Learning group-invariant distributions efficiently.
method Study of group-invariant GANs and their performance guarantees.
result Group-invariant GANs require fewer samples and have a reduced discriminator approximation error.
We analyze in this paper a random feature map based on a theory of invariance I-theory introduced recently. More specifically, a group invariant signal signature is obtained through cumulative distributions of group transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as …
Group-invariant neural networks improve approximation accuracy for symmetric functions.
problem Improving approximation accuracy for symmetric functions using neural networks.
method Investigates the generalization error of group-invariant neural networks within the Barron framework.
result Group invariance introduces a factor δ that can significantly improve approximation accuracy when it is small.
Machine learning uses invariant theory to restrict function classes.
problem Creating function classes that respect physical law constraints.
method Using equivariant machine learning and Malgrance's method to parameterize functions.
result Explicitly parameterizes equivariant functions between linear spaces.
A new method for group invariant machine learning using geometric projections.
problem Supervised group invariant and equivariant machine learning.
method Geometric topology approach involving projection of input data into a geometric space parametrizing symmetry group orbits.
result Improvement in accuracy compared to existing methods.
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
Neural networks struggle with extrapolation, but a new framework allows them to learn counterfactual invariances.
problem Neural networks' inability to extrapolate beyond training data distribution.
method Introduces a learning framework that allows neural networks to extrapolate over group transformations based on counterfactual invariances.
result Neural networks can learn counterfactual invariances from a single environment, overcoming their limitations in extrapolation.
Study shows how to reduce data needed for learning under geometric constraints.
problem Learning high-dimensional data with geometric priors.
method Spherical harmonic decompositions and kernel methods for invariance and geometric stability.
result Improvements in sample complexity by leveraging group invariance, with asymptotic behavior depending on spectral properties.
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.
Derives representations invariant under crystallographic groups for functions.
problem Representing and learning functions invariant under crystallographic groups.
method Derives linear and nonlinear representations of functions invariant under crystallographic groups.
result Derives orthonormal crystallographically invariant basis functions and embedding maps.
Universal MLPs with a single hidden layer can learn any function.
problem Learning on various data structures like sequences, images, sets, and graphs.
method Using group theory, the paper proves the universality of a broad class of equivariant MLPs with a single hidden layer.
result Having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance).
SenSeI ensures fair models by enforcing invariance on sensitive groups.
problem Ensuring fair machine learning models that respect sensitive groups.
method Designing a transport-based regularizer to enforce invariance on sensitive sets.
result Certifiably fair ML models trained using SenSeI achieve improved fairness metrics.
Invariance to nuisance transformations is one of the desirable properties of effective representations. We consider transformations that form a \emph{group} and propose an approach based on kernel methods to derive local group invariant representations. Locality is achieved by defining a suitable probability distributi…
Deep learning approximates geometric measures of planar curves.
problem Approximating differential invariants of planar curves.
method Utilizing deep neural networks to estimate geometric measures of planar curves.
result Deep neural networks can learn to overcome instabilities and sampling artifacts.
Study invariant minimizers in convex functions under amenable groups.
problem Finding invariant minimizers in convex functions invariant under amenable groups.
method Analyze smallest closed invariant convex subsets and apply to invariant optimality problem.
result Clarifies relations between equivariant neural networks and statistical theorems.
Our work improves VAE latent space clustering by enforcing invariant and equivariant learning.
problem Current VAEs fail to learn invariant and equivariant clusters in latent space.
method We use a mixture model pdf like Gaussian mixtures to enforce deep, group-invariant learning and separate semantic and equivariant variables.
result Our model effectively learns to disentangle invariant and equivariant representations, improving learning rate and image recognition.
Algorithm improves binary classification of biased grouped data.
problem Improving binary classification for biased, grouped data.
method Assumes partition-projected class-conditional invariance across groups and derives a semi-supervised algorithm to learn a group-aware classifier.
result Demonstrates improved area under the ROC curve compared to baselines.
EquivCNP learns group symmetries for conditional data.
problem Learning conditional models with data symmetries.
method Group equivariant decomposition and Lie group convolutional layers.
result EquivCNP achieves comparable performance and zero-shot generalization.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently p-localization of knot groups and automorphism groups of free nilpotent groups. result Maps from outer automorphism groups yield knot invariants.
The study examines invariants of homology cylinders and their relations to free nilpotent groups.
problem Understanding invariants of homology cylinders and their connections to free nilpotent groups.
method Extensions of Johnson homomorphisms, Milnor invariants, and Orr invariants of links to homology cylinders; establishment of a combined filtration.
result Determination of the image of the filtration under the invariants and investigation of relations among the invariants.
Discover conservation laws from trajectories using a neural network.
problem Finding invariants and conservation laws from large-scale data without prior knowledge.
method ConservNet, a neural network trained with noise-variance loss to discover hidden invariants in grouped multi-dimensional observables.
result Successfully discovers underlying invariants from simulated and real-world systems.
Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
Invariants measure letter interleaving in groups, detecting group dimensions.
problem Detecting group dimensions in arbitrary groups.
method Defining letter-braiding invariants from cochain models of spaces with prescribed fundamental groups.
result Letter-braiding invariants are complete invariants of group dimension series.
Paper introduces threshold invariant fairness to ensure equitable predictions across different groups.
problem Machine learning models can be unfair to certain groups based on sensitive attributes.
method Proposes threshold invariant fairness and uses two approximation methods to equalize risk distributions.
result Demonstrates effectiveness in alleviating threshold sensitivity in fairness models.
Deep networks learn hierarchical data by invariant representations.
problem How many examples are needed for deep networks to learn hierarchical data?
method Random Hierarchy Model: synthetic tasks inspired by language and images hierarchy.
result Deep networks learn by invariant representations and require a detectable number of correlations between low-level features and classes.
We develop methods to learn dictionaries invariant under group symmetries, useful in cryo-EM and tracking.
problem Learning dictionaries invariant under group symmetries.
method Representation theory, non-abelian Fourier analysis, matrix orbitopes, alternating minimization.
result Effective dictionary learning for SO(3) symmetries with guarantees.
Study convolution of invariant valuations on Lie groups.
problem Understanding convolution of valuations on Lie groups.
method Explicit formula for left-invariant valuations, showing existence of smooth bi-invariant valuations, defining convolution on arbitrary Lie groups.
result Unified convolution operations on Lie groups.
New metrics for information geometry and machine learning from Lie groups.
problem Traditional mean methods in data science and machine learning.
method Cartan-Schouten metrics on Lie groups.
result Cartan-Schouten metrics offer advantages over traditional means.
In this study, a novel feature coding method that exploits invariance for transformations represented by a finite group of orthogonal matrices is proposed. We prove that the group-invariant feature vector contains sufficient discriminative information when learning a linear classifier using convex loss minimization. Ba…
The paper explores left invariant k-symplectic structures on Lie groups with bi-invariant metrics.
problem Existence and properties of left invariant k-symplectic structures on Lie groups with bi-invariant metrics.
method Analyzing compact semi-simple Lie groups and specific Lie groups with bi-invariant pseudo-Riemannian metrics.
result Compact semi-simple Lie groups and many Lie groups do not carry left invariant k-symplectic structures, except for specific cases.
The BNS invariant is applied to Kähler groups in new proofs and results.
problem Understanding properties of Kähler groups through the BNS invariant.
method Applications of the Bieri-Neumann-Strebel invariant on Kähler groups.
result Amenable Kähler groups have an empty complement of the BNS invariant.
In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invarian…
New concept SB-generation helps classify transformation groups.
problem Classifying transformation groups through quasi-isometry invariants.
method Identifying SB-generated groups in specific transformation groups.
result SB-generation provides robust extension of finite generation.
The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4-manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.
Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.
problem Proving the uniqueness of Rohlin invariant and extending homology sphere invariants.
method Using the Rohlin invariant's uniqueness, the paper extends invariants from trivial 2-cocycles to those with 2-torsion.
result Generalized invariants of homology spheres with 2-torsion values.
Constraining linear layers in neural networks to respect symmetry transformations from a group G is a common design principle for invariant networks that has found many applications in machine learning. In this paper, we consider a fundamental question that has received little attention to date: Can these networks ap…
Investigates BNSR invariants of link and knot groups, proving specific properties.
problem Characterizing finiteness properties of normal subgroups in link and knot groups.
method Analyzes BNSR invariants of link and knot groups, proving specific properties.
result Proves specific conditions for finiteness properties of link and knot groups.
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.
The paper creates knot invariants using free groups.
problem Invariants of free knots (virtual knots).
method Constructing invariants valued in free groups.
result Series of invariants for free knots.
Develops tests for conditional symmetry under group actions.
problem Testing conditional symmetry in distributions under group actions.
method Nonparametric randomization tests with kernel methods and asymptotic consistency.
result Tests achieve finite-sample Type I error control and power.
Survey on invariant conformal Killing forms on Lie groups.
problem Understanding invariant conformal Killing forms on Lie groups.
method Review of recent results and mention of open questions.
result Discussion of recent findings and open research areas.
ChebLieNet uses Lie groups to create invariant spectral graph networks.
problem Handling anisotropic data in graph neural networks.
method Develops anisotropic convolutional layers on Lie groups with Riemannian metrics.
result Demonstrates the effectiveness of balancing equivariance and invariance.
We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in H3, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such gr…
Defines invariants for reflection groups and connects them to Frobenius structures.
problem Understanding invariants for reflection groups and their relation to Frobenius structures.
method Defines good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants for reflection groups lead to Frobenius structure constants.
Finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of ea…
Frame Averaging makes neural networks invariant or equivariant to new symmetries.
problem Designing neural networks that respect symmetries while being expressive and efficient.
method Introduces Frame Averaging (FA) as a systematic framework to adapt architectures to become invariant or equivariant to new symmetries.
result Frame Averaging guarantees exact invariance or equivariance while being simpler to compute than full group averaging.