We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…
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48 results for “orthogonal representation theory”
Types of Symmetries of Knotsmath.GT
Classifies symmetries of knots using group actions and orthogonal representation theory.
problem Classifying symmetries of knots in 3-sphere.
method Using geometrization and orthogonal representation theory, constructing examples, and distinguishing symmetries.
result Cyclic and dihedral families of symmetries of prime knots and composite knots.
Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence between such manifold submetries and maximal Laplacian alge…
We show how Ramond free neutral Fermi fields lead to a -function theory of BKP type which describes iso-orthogonal deformations of systems of ortogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation.
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
Batch normalization makes deep neural networks' representations increasingly orthogonal.
problem Orthogonality of deep neural network representations.
method Random linear transformations in successive batch-normalizations.
result Orthogonality of representations improves SGD performance.
Racah matrices and higher -symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the re…
In current paper we refer to the geometrical classification of the Einstein equations which has been developed by one of the authors of this paper. This classification was based on the classical theory for decomposition of the tensor product of representations into irreducible components, which is studied in the elemen…
The paper introduces polarizations in symplectic and orthogonal settings.
problem Understanding polarizations in symplectic and orthogonal contexts.
method Exposition of symplectic and orthogonal polarizations, emphasizing symmetry.
result Grassmannians of polarizations in symplectic and orthogonal settings.
Develops deformation theory for mapping spaces related to Morse theory and Bott-Thom isomorphism.
problem Studying weak homotopy equivalences and decompositions of vector bundles.
method Morse theory on path spaces, deformation theory, Clifford representations, Bott-Thom isomorphism.
result Stable decompositions of vector bundles over sphere bundles derived from Clifford representations.
Special orthogonal representations from octonions have geometric properties linked to binary cubics.
problem Understanding geometric properties of special orthogonal representations from octonions.
method Using octonions and their derivations, spinors, and covariants to show geometric properties.
result Covariants and Mathews identities of these representations are related to the Fano plane and .
New representation theory for surface groups to SO0(2,3).
problem Understanding representations of surface groups into special orthogonal groups.
method Non-maximal Anosov representations via Higgs bundles and non-Abelian Hodge correspondence.
result Generalization of Filip's result on weight 3 variation of Hodge structures.
New measures on orbit spaces for orthogonal groups identified.
problem Characterizing measures on orbit spaces of orthogonal groups.
method Constructing Hilbert measures on orbit spaces of coregular representations of orthogonal groups.
result Hilbert measures have singularities if and only if the number of copies equals the dimension.
We investigate a certain class of solvable metric Lie algebras. For this purpose a theory of twofold extensions associated to an orthogonal representation of an abelian Lie algebra is developed. Among other things, we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre an…
In this paper, we present new results on using orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries for complex cases (i.e., complex measurement vector, complex dictionary and complex additive white Gaussian noise (CAWGN)). A sufficient condition that OMP can recover …
Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the representation space. We say that representations with this property are of class O^2. O…
Generalized Gaussian Bridgesmath.PR
A generalized bridge is the law of a stochastic process that is conditioned on N linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the underlying process. Thus, future knowledge of the …
Conformal Autoencoders infer intrinsic dimensionality and impose invariance.
problem Detecting intrinsic dimensionality and imposing invariance in nonlinear manifold data.
method Imposing orthogonality conditions on latent variables to infer intrinsic dimensionality and build coordinate invariance.
result The method can infer intrinsic dimensionality and build coordinate invariance on submanifolds.
Unified framework for debiased machine learning using Riesz representer and Bregman divergence.
problem Estimating causal and structural parameters in machine learning.
method Generalized Riesz regression for fitting Riesz representer via Bregman divergence minimization.
result Automatic covariate balancing and Neyman orthogonality properties for debiased estimation.
This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on represen…
Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli …
Explores tensor products in hyperdimensional computing.
problem Understanding tensor products in hyperdimensional computing.
method Generalized results from graph embeddings to vector symbolic architectures and hyperdimensional computing.
result Tensor product is the most general and expressive representation with errorless unbinding and detection.
Novel prior for orthogonal functions improves functional component estimation.
problem Improving orthogonality in functional principal component analysis.
method Sequential adaptive priors for orthogonal functions using hierarchical conditionally normal distributions.
result Proposed prior leads to nearly orthogonal posterior estimates.
Study connects curvature to graph theory and reveals differences.
problem Exploring differences between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
method Real (1,1)--forms and Weitzenböck curvature operator used to represent graph Dirichlet energy.
result Curvature differences illuminated between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
Paper improves Monte Carlo sampling with new theoretical insights and methods.
problem Improving Monte Carlo sampling for variance reduction.
method Theoretical analysis of negatively dependent random variables and novel extensions using number theory and particle algorithms.
result Near-Orthogonal Monte Carlo (NOMC) consistently outperforms Orthogonal Monte Carlo (OMC) in various applications.
Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariantsmath.GT
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about t…
New optimization algorithms on orthogonal group for machine learning.
problem Efficient optimization on the orthogonal group for machine learning tasks.
method Stochastic geometric algorithms on Lie groups.
result Strong performance on diverse machine learning tasks.
The Shapley value theory is used for risk allocation in non-orthogonal risk factors.
problem Risk allocation among non-orthogonal risk factors in financial portfolios.
method Using Shapley value from cooperative game theory to allocate risk contributions.
result Explicit formulas and numerical algorithms for calculating risk allocations are derived.
We give a geometric characterisation of the topological invariants associated to SO(m,m+1)-Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m,m+1)-Higg…
Orthogonal deep models defend against black-box attacks by ensuring internal representations are nearly orthogonal.
problem Vulnerability of deep learning models to black-box adversarial attacks.
method Introduce a gradient regularization scheme to encourage deep models' internal representations to be orthogonal to another model's.
result Orthogonal deep models significantly boost robustness against transferable black-box adversarial attacks.
Corrected whitening restores orthogonality in high-dimensional spherical Gaussian mixtures.
problem In high-dimensional data, standard whitening fails to preserve orthogonality of mixture means.
method Derived exact limits for whitened means dot products using random matrix theory, constructed a corrected whitening matrix.
result Corrected whitening allows for improved estimation of spherical Gaussian mixtures in the large-dimensional regime.
Paper optimizes tensor deflation for non-orthogonal signals.
problem Recovering low-rank signals from noisy tensors with correlated components.
method Developed an asymptotic analysis and optimized deflation procedure using random tensor theory.
result Proposed an efficient tensor deflation algorithm that optimizes a parameter introduced in the deflation mechanism.
Multi-head attention mechanism is capable of learning various representations from sequential data while paying attention to different subsequences, e.g., word-pieces or syllables in a spoken word. From the subsequences, it retrieves richer information than a single-head attention which only summarizes the whole sequen…
We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expa…
Extends Fried's result to arbitrary representations of compact hyperbolic manifolds.
problem Behavior of dynamical zeta functions at the origin for compact hyperbolic manifolds.
method Uses complex-valued torsion instead of Ray-Singer analytic torsion.
result Holomorphicity and value at s=0 for twisted Ruelle zeta function for arbitrary representations.
Different neural networks trained on the same dataset often learn similar input-output mappings with very different weights. Is there some correspondence between these neural network solutions? For linear networks, it has been shown that different instances of the same network architecture encode the same representatio…
Improved Gaussian process models for interpretable predictions.
problem Complex responses require high-dimensional interaction terms in additive Gaussian processes.
method Orthogonal additive kernel (OAK) with orthogonality constraint on additive functions.
result OAK models achieve similar or better predictive performance with fewer terms, retaining interpretability.
Enhances Gaussian processes with spherical features for better scalability and flexibility.
problem Lack of representation learning in Gaussian processes compared to deep neural networks.
method Introduces spherical inter-domain features to improve GP approximation and scalability.
result The method alleviates limitations and improves scalability compared to alternative strategies.
Geometrically, Kostant's Convexity Theorem is extended to submetries with a fat section.
problem Extending Kostant's Convexity Theorem to a broader class of representations.
method Introducing a new concept of 'fat section' and proving the theorem for submetries with this property.
result Kostant's Convexity Theorem is partially extended to submetries with a fat section.
Anti-Transfer Learning for Task Invariance in Convolutional Neural Networks for Speech Processingcs.LG
Anti-transfer learning prevents misleading representations for speech tasks.
problem Misleading representations learned from orthogonal tasks in speech processing.
method Penalizes similarity between activations of a network and another trained on an orthogonal task.
result Improves classification accuracy and invariance to the orthogonal task.
Let G be a finite group. The unit sphere in a finite-dimensional orthogonal G-representation motivates the definition of homotopy representations, due to tom Dieck. We introduce an algebraic analogue, and establish its basic properties including the Borel-Smith conditions and realization by finite G-CW-complexes.
This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
problem Proving the asymptotic freeness of layerwise Jacobians in multilayer perceptrons (MLPs).
method Replacing each layer's parameter matrix with itself multiplied by a Haar orthogonal matrix, and using the invariance of the MLP.
result Proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
Study finds Calabi-Yau models' operator spectra match random matrix theory.
problem Understanding spectra of Calabi-Yau sigma models.
method Numerical methods for Ricci-flat metrics, averaging over complex structure moduli space.
result Spectrum matches Gaussian orthogonal ensemble of random matrix theory.
The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.
ScoreMatchingRiesz improves debiased machine learning and policy effects estimation.
problem Improving debiased machine learning and policy effects estimation.
method Score matching and Riesz representer estimation.
result Estimates policy path for continuous treatments, improving interpretability.
We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra , as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras and .
We develop a mean-field theory for multi-component ICA in high dimensions.
problem Understanding multi-component ICA in high-dimensional settings.
method Asymptotically exact mean-field theory for multi-component online ICA.
result Explicit learnability boundaries and competition conditions linking step size, data moments, and initialization.
In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of int…