One of the most challenging problems in the domain of 2-D image or 3-D shape is to handle the non-rigid deformation. From the perspective of transformation groups, the conformal transformation is a key part of the diffeomorphism. According to the Liouville Theorem, an important part of the conformal transformation is t…
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48 results for “invariant transformations”
This paper explains how model invariance improves generalization using data transformations.
problem Understanding why model invariance leads to better generalization performance.
method Introducing sample cover induced by transformations and refining generalization bounds.
result The sample covering number can be used to evaluate and select suitable data transformations.
We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We…
Invariance to geometric transformations is a highly desirable property of automatic classifiers in many image recognition tasks. Nevertheless, it is unclear to which extent state-of-the-art classifiers are invariant to basic transformations such as rotations and translations. This is mainly due to the lack of general m…
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…
GT-PCA improves PCA for image and time series data.
problem Lack of robustness to transformations in PCA.
method GT-PCA is a neural network that estimates components invariant to specific transformations.
result GT-PCA outperforms alternative methods in synthetic and real data experiments.
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.
We study the invariance characteristics of pre-trained predictive models by empirically learning transformations on the input that leave the prediction function approximately unchanged. To learn invariant transformations, we minimize the Wasserstein distance between the predictive distribution conditioned on the data i…
Invariants for surfaces up to rigid transformations, with a comeagre subset retrieval algorithm.
problem Identifying compact surfaces up to rigid transformations.
method Degree four polynomials in moments of delta function, effective inversion algorithm.
result Invariants and retrieval algorithm work on a comeagre subset of surfaces.
The Fourier transform of Heegaard Floer d-invariants helps classify 3-manifolds.
problem Classifying 3-manifolds up to integer homology cobordism.
method Exploring the Fourier transform of Heegaard Floer d-invariants.
result Lens spaces are cancellable in the monoid of 3-manifolds up to integer homology cobordism.
We begin an exploration of parametric Backlund transformations for hyperbolic Monge-Ampere systems. We compute invariants for such transformations and explore the behavior of four examples regarding their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for furthe…
New approach learns image transformations directly for clustering.
problem Learning better deep representations for image clustering.
method Directly learns transformations and clusters in image space without abstract features.
result Jointly learns prototypes and transformations using deep learning modules.
Improves contrastive learning invariance with novel training objectives and feature averaging.
problem Contrastive learning's implicit invariance is insufficient for robust performance.
method Introduces a novel training objective and feature averaging approach to enforce invariance.
result Improved performance and robustness to transformations on downstream tasks.
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.
We introduce the spherical phylon group, a subgroup of the group of all formal diffeomorphisms of that fix the origin. The invariant theory of the spherical phylon group is used to understand the invariants of the Laplace transform.
Paper proves Fourier transform for valuations, simplifying previous work.
problem Existence of isomorphism for translation-invariant smooth valuations.
method Directly describes Alesker's isomorphism in terms of Fourier transform on functions.
result Simple proofs of Alesker's Fourier transform properties, including a previously conjectured result.
The paper studies third-order PDEs invariant under affine transformations and connects them to the Fubini-Pick invariant.
problem Investigating third-order PDEs invariant under affine transformations.
method Using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant PDEs on homogeneous manifolds].
result Derives third-order PDEs from the Fubini-Pick invariant.
Convolutional neural networks (CNNs) have achieved state-of-the-art results on many visual recognition tasks. However, current CNN models still exhibit a poor ability to be invariant to spatial transformations of images. Intuitively, with sufficient layers and parameters, hierarchical combinations of convolution (matri…
Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are …
This work tackles OOD generalization by leveraging causal invariance without needing to recover causal features.
problem Learning models that perform well on out-of-distribution (OOD) data.
method Causal invariant transformations to modify non-causal features while preserving causal parts.
result Theoretical and practical methods to learn a minimax optimal model across domains using single domain data.
One of the fundamental problems in supervised classification and in machine learning in general, is the modelling of non-parametric invariances that exist in data. Most prior art has focused on enforcing priors in the form of invariances to parametric nuisance transformations that are expected to be present in data. Le…
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.
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.
Representations in the auditory cortex might be based on mechanisms similar to the visual ventral stream; modules for building invariance to transformations and multiple layers for compositionality and selectivity. In this paper we propose the use of such computational modules for extracting invariant and discriminativ…
This paper studies the generalization error of invariant classifiers. In particular, we consider the common scenario where the classification task is invariant to certain transformations of the input, and that the classifier is constructed (or learned) to be invariant to these transformations. Our approach relies on fa…
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.
Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms IImath.GT
Study proves Volume Conjecture for Reshetikhin-Turaev invariants.
problem Volume Conjecture for Reshetikhin-Turaev invariants.
method Hyperbolic cone metrics and discrete Fourier transforms.
result Proves Volume Conjecture for most figure-8 knot configurations.
Discussing moving frames for curve and surface invariants.
problem Identifying differential invariants of curves and surfaces.
method Using moving frames for Euclidean, affine, and conformal transformations.
result Determine differential invariants of curves and surfaces.
This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank- Bäcklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a clas…
For many tasks and data types, there are natural transformations to which the data should be invariant or insensitive. For instance, in visual recognition, natural images should be insensitive to rotation and translation. This requirement and its implications have been important in many machine learning applications, a…
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 prove the homotopy invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L^2 cohomology of the covering space vanishes, the homotopy invariance was establi…
The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.
problem Calculating Witten-Reshetikhin-Turaev invariants for Seifert fibered homology 3-spheres.
method Explicit modular transformation formulas of homological blocks.
result New proof of Witten asymptotic conjecture for Seifert fibered homology 3-spheres.
Proposes a measure to predict generalization in non-matching environments.
problem Characterizing and comparing generalization of machine learning models in non-matching environments.
method Neighborhood invariance measure, calculating invariance as the largest fraction of transformed points classified into the same class.
result Strong and robust correlation between neighborhood invariance and actual out-of-domain generalization.
IsoGCNs learn invariant and equivariant graph features for efficient simulations.
problem Learning isometric transformation invariant and equivariant features in graphs for simulations.
method Transformation invariant and equivariant Graph Convolutional Networks (IsoGCNs).
result IsoGCNs outperform state-of-the-art methods on geometrical and physical simulation tasks.
The paper finds transformation formulas for quaternionic complex structures.
problem Quaternionic projective invariance of -Cauchy-Fueter complex.
method Explicit transformation formulae under .
result Quaternionic projectively invariant operator and defining density.
Symmetry transformations induce invariances which are frequently described with deep latent variable models. In many complex domains, such as the chemical space, invariances can be observed, yet the corresponding symmetry transformation cannot be formulated analytically. We propose to learn the symmetry transformation …
Relativistic Quantum Financeq-fin.MF
Employing the Klein-Gordon equation, we propose a generalized Black-Scholes equation. In addition, we found a limit where this generalized equation is invariant under conformal transformations, in particular invariant under scale transformations. In this limit, we show that the stock prices distribution is given by a C…
The paper connects quantum -symbols to tetrahedra volumes via discrete Fourier transforms.
problem Understanding the asymptotic behavior of quantum -symbols and their relation to 3-manifold invariants.
method Proposing and proving a conjecture linking discrete Fourier transforms of quantum -symbols to the volumes of deeply truncated tetrahedra.
result Supporting evidence for the conjecture in specific cases, with numerical calculations for larger dihedral angles.
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.
A new invariant captures geometric features of circle embeddings.
problem Capturing geometric features of circle embeddings invariantly.
method Chordal distance transform and persistent homology.
result Persistent homology of chordal distance transform is invariant.
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms Imath.GT
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
problem Volume conjecture for Reshetikhin-Turaev invariants of 3-manifolds with links.
method Volume conjecture, hyperbolic cone metrics, discrete Fourier transforms, change-of-pair operations.
result Volume conjecture proven for specific cases, provides approach to solving Volume Conjecture for hyperbolic 3-manifolds.
We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional defined on exact divergence-free vector fields of class on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mat…
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…
The study characterizes real flag manifolds with invariant generalized almost complex structures.
problem Characterizing real flag manifolds with invariant generalized almost complex structures.
method Characterization through invariant -transformations and classification of structures.
result No -maximal real flag manifolds admit integrable invariant generalized almost complex structures.
Learning to Exploit Invariances in Clinical Time-Series Data using Sequence Transformer Networkscs.LG
Recently, researchers have started applying convolutional neural networks (CNNs) with one-dimensional convolutions to clinical tasks involving time-series data. This is due, in part, to their computational efficiency, relative to recurrent neural networks and their ability to efficiently exploit certain temporal invari…
The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into .
result Alexander polynomial as the singular locus of a coherent sheaf.