Constructs tangent groupoid for filtered manifolds without coordinates.
problem No specific problem stated; intrinsic construction of tangent groupoid.
method Intrinsic construction of tangent groupoid.
result Intrinsic construction of tangent groupoid for filtered manifolds.
Paper proves convergence of Kalman filter on Stiefel manifolds with measurement errors.
problem Filtering constant particle with measurement errors on Stiefel manifolds.
method Extended Kalman filter applied to Stiefel manifold-valued observations.
result Convergence of the extended Kalman filter proved for constant system process.
Improved Kalman filter for Stiefel manifold measurements.
problem Improving accuracy in measurements on Stiefel manifolds.
method Generalization of extended Kalman filter for Stiefel manifold-valued measurements.
result Significant improvement over raw measurements.
The study extends hypoellipticity to filtered manifolds and applies it to BGG sequences.
problem Analyzing hypoellipticity on general filtered manifolds.
method Extending Rockland criterion to pseudodifferential calculus, constructing parametrix, generalizing BGG machinery.
result Generalized BGG sequences are Rockland in a graded sense.
We find a splitting in a special cohomology theory for complex manifolds.
problem Finding a splitting in a specific cohomology theory.
method Construct Hodge filtered function spaces and show they satisfy an unstable splitting.
result Obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology.
We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family o…
Geometric model for Hodge filtered complex cobordism constructed.
problem Constructing a geometric model for Hodge filtered complex cobordism.
method Refinement of Pontryagin-Thom construction to create an explicit isomorphism.
result Explicit isomorphism between geometric and abstract models for complex manifolds.
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered co…
Defines Wodzicki residue using groupoids and fibered distributions.
problem Defining and understanding the Wodzicki residue in noncommutative geometry.
method Using groupoid language and filtered manifolds, defining the residue and showing its properties.
result The groupoidal residue is a trace on pseudodifferential operators and matches the usual residue in certain cases.
Latent FxLMS accelerates ANC by adapting along low-dimensional filter weights.
problem Improving active noise control with neural adaptive filters.
method Training an auto-encoder on filter coefficients, constraining weights to latent variables, and updating in latent space.
result Latent FxLMS converges in fewer steps with comparable error to standard FxLMS.
In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct L2 metric as opposed to the usual Hellinger distance and the related Fisher Information metric. We explain how this apparatus can be used…
New complexes derived from any filtered cochain complex compute the same cohomology.
problem Constructing cohomologically equivalent subcomplexes from filtered cochain complexes.
method Presenting a general construction that produces subcomplexes from any filtered cochain complex of finite depth.
result The construction of subcomplexes depends only on the filtration up to isomorphism.
Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.
problem Defining operators with varying orders on filtered manifolds.
method Using generalized fixed point algebras and nilpotent Lie groups, they construct a new calculus.
result They establish a new calculus that reflects the behavior of differential operators on filtered manifolds.
A new approach to symbol calculus on filtered manifolds using C∗-algebras.
problem Symbol calculus on filtered manifolds with local isomorphism to stratified Lie groups.
method Establishing a surjective ∗-homomorphism between a C∗-algebra bundle and the algebra of bounded continuous sections. result Existence of a surjective ∗-homomorphism sym_M: Π_M → C_b(E_hom) with specific kernel properties. MFCNs use sparse graphs to approximate manifold convergence.
problem Understanding manifold neural networks (MNNs).
method Sparse graph approximation for manifold convergence.
result Method converges to continuum limit as data points increase.
Paper constructs subcomplexes from filtered Riemannian manifolds.
problem Understanding subcomplexes on filtered Riemannian manifolds.
method General construction of subcomplexes from two distinct complexes.
result Aligns with Rumin complex on regular subRiemannian manifolds.
Isomorphism found between filtered calculus and crossed products.
problem Tackles isomorphism in filtered calculus and crossed products.
method Uses natural R-action and structure result for C*-algebra of graded nilpotent Lie groups.
result Found isomorphism between kernel of tangent groupoid and crossed product.
This research explores canonical Cartan connections for filtered G-structures.
problem Determining canonical Cartan connections for filtered G-structures.
method Generalization of parabolic geometries, Lie algebra valued forms, and explicit characterization.
result Existence of canonical Cartan connections for filtered G-structures, with specific features.
Kähler information manifolds for signal filters in weighted Hardy spaces are explored.
problem Developing a geometric framework for signal processing filters in weighted Hardy spaces.
method Introducing weighted Hardy spaces and smooth transformations of transfer functions, demonstrating the Kähler manifold structure.
result The Riemannian geometry of weighted Hardy norms for transfer functions forms a Kähler manifold.
Introduces MFCNs for better understanding manifold neural networks.
problem Understanding manifold neural networks (MNNs).
method Filter-combine framework on high-dimensional point clouds, approximating manifold by sparse graph.
result Method converges to continuum limit as data points increase.
Adaptive Gaussian kernel filtering with updated parameters.
problem Improving kernel adaptive filtering for better performance.
method Adaptive updating of Gaussian kernel parameters on an SPD manifold.
result Validation of the proposed method through experimental results.
Knot lattice homology invariant is preserved under certain 3-manifold diffeomorphisms.
problem Preserving knot lattice homology invariants under 3-manifold diffeomorphisms.
method Examined filtered lattice chain homotopy types of negative-definite forests with one unframed vertex.
result Filtered lattice chain homotopy type is an invariant of the diffeomorphism type of resulting 3-manifolds.
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show th…
We generalize Fulton and MacPherson's configuration space construction to weighted filtered manifolds.
problem Infinitesimal collision data in filtered manifolds with higher-order compatibility.
method Generalizing Fulton and MacPherson's blow-up approach to weighted arrangements of submanifolds.
result Smoothness of the weighted blow-up under reasonable assumptions.
New method for predicting portfolio dynamics using non-Euclidean geometry.
problem Predicting efficient portfolios with geometric structure.
method Non-Euclidean conditional expectation and filtering equations.
result Accurate numerical forecasts of portfolio dynamics.
Graph neural networks are found to be primarily low-pass filters, not manifold learners.
problem Improving performance and scalability of graph neural networks for graph-structured data.
method Developed a theoretical framework based on graph signal processing.
result Graph neural networks only perform low-pass filtering on feature vectors and do not have non-linear manifold learning property.
Paper solves complex signal processing problem efficiently.
problem Learning an unknown filter from multiple sparse convolutions.
method Nonconvex optimization over the sphere manifold using manifold gradient descent.
result Manifold gradient descent provably recovers the filter under random data model.
Study light ray transform on Lorentzian manifolds without conjugate points.
problem Recovering spacelike singularities from weighted light ray transforms.
method Fourier Integral Operator analysis and filtered back-projection.
result Recovery of spacelike singularities from weighted light ray transforms without conjugate points.
The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that th…
New PTC convolution preserves properties of Euclidean convolutions on manifolds.
problem Lack of generalizable convolutions on curved domains with desirable properties.
method Parallel transport convolution (PTC) on Riemannian manifolds.
result PTC preserves compactly supported filters and directionality.
Study of BGG sequences on foliated manifolds with transverse parabolic geometry.
problem Analysis of BGG sequences on foliated manifolds with transverse parabolic structures.
method Filtered calculus and transversal index theory for filtered manifolds.
result Derived curved BGG sequences for foliated manifolds with transverse parabolic geometry.
Proves correspondence between harmonic and Higgs bundles.
problem Connecting harmonic and Higgs bundles for study.
method Kobayashi-Hitchin correspondence for polystable bundles.
result Establishes correspondence between good wild harmonic bundles and polystable good filtered λ-flat bundles. The paper develops a heat kernel expansion for Rockland operators on filtered manifolds.
problem Analyzing heat kernel expansions for non-commutative geometries.
method Established a universal heat kernel expansion for Rockland operators on closed filtered manifolds using a new calculus.
result Implications of the heat expansion for complex powers, heat trace asymptotics, and eigenvalue asymptotics are generalized to this new calculus.
New algebra defined for Legendrian submanifolds, preserving key invariants.
problem Defining a new algebra to preserve invariants of Legendrian submanifolds.
method Combining string topology techniques with combinatorial methods to count holomorphic disks.
result The new algebra PDA is a filtered, differential graded algebra that captures invariants of Legendrian submanifolds. We construct geometric shrinkage priors for Kählerian signal filters. Based on the characteristics of Kähler manifolds, an efficient and robust algorithm for finding superharmonic priors which outperform the Jeffreys prior is introduced. Several ansätze for the Bayesian predictive priors are also suggested. In particul…
Researchers study Killing superalgebras in 2D manifolds.
problem Characterizing Killing superalgebras in 2D pseudo-Riemannian manifolds.
method Computing Spencer cohomology and filtered deformations of superalgebras.
result Killing superalgebras arise as solutions for geometric and skew-Killing spinors.
A new method routes EEG covariance matrices across domains using adaptive subspace selection.
problem Challenges in cross-domain EEG decoding due to distinct SPD manifold regions.
method Dynamic Stiefel routing with expert filters and cross-attention for adaptive subspace projection.
result Consistent gains across three datasets: balanced accuracy improves from 0.773 to 0.823, 0.757 to 0.809, and 0.801 to 0.839.
Kernel mixture network estimates complex conditional densities.
problem Nonparametric estimation of complex conditional densities.
method Neural network with kernel mixture model.
result Kernel mixture network outperforms existing methods in filtering and generative modeling.
This study shows that certain cohomology groups of symplectic manifolds are always even-dimensional.
problem Understanding the cohomology structure of symplectic manifolds.
method Constructing and deforming a skew-adjoint operator to prove the vanishing property.
result The even dimensionality of even-degree cohomology groups in (4n+2)-dimensional symplectic manifolds.
Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.
problem Enhance graph representation learning by leveraging spectral and curvature signals.
method Proposes Spectro-Riemannian Graph Neural Networks (CUSP) that combines spectral and curvature insights.
result Empirical evaluation shows CUSP outperforms state-of-the-art models by up to 5.3%.
Develops optimal low-dimensional approximations to high-dimensional SDEs.
problem Approximating solutions to high-dimensional SDEs in a low-dimensional space.
method Introduces Ito-vector and Ito-jet projections for optimal approximation.
result Optimal projection filters yield better approximations than Stratonovich projection.
A new method reduces high-dimensional filtering to quadratic complexity.
problem High-dimensional dynamical systems inference and simulation.
method Low-rank Kalman filtering using dynamical low-rank integrator.
result The method reproduces exact Kalman filter in low-rank limit.
We extend results of Pachner and Casali to give finite sets of moves relating triangulations of PL manifolds respecting filtrations by locally flat manifolds and stratifications in which a finite family of simple local models exists for neighborhoods of strata.
Geometric wavelet scattering on manifolds improves neural network understanding.
problem Improving neural network understanding on manifold and graph domains.
method Defining a geometric scattering transform based on wavelet filters and nonlinearities.
result Generalizes deformation stability and local translation invariance to manifolds.
Identifies directed graphs from node measurements using polynomial filters.
problem Inferring directed network topology from nodal measurements.
method System identification of graph convolutional filter followed by topology inference.
result Effective recovery of directed graphs from measurements.
Nonlinear filtering extracts relevant variables from multimodal sleep data.
problem Recover relevant variables from multiple sensor data.
method Diffusion-based manifold learning for nonlinear filtering.
result Method gives robust data-driven representation correlated with sleep process.
We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of Z-graded subalgebras with maximum odd dimension of the N=1 Poincaré superalgebra in four dimensions. Part of this calcula…
This paper contains the technical foundations from stochastic differential geometry for the construction of geometrically intrinsic nonlinear recursive filters. A diffusion X on a manifold N is run for a time interval T, with a random initial condition. There is a single observation consisting of a nonlinear function o…