Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
Defines tangent spaces on causal sets using partial derivatives and metrics.
problem Defining geometric structures on causal sets.
method Using partial derivatives and metrics to define tangent spaces, connection, curvature, parallel transport, and geodesics.
result Approaches expected values for a flat spacetime as density increases.
A novel method for parallel transport and geodesics on submanifolds.
problem Understanding parallel transport and geodesics on submanifolds.
method Rolling tangent space to visualize and analyze parallel transport and geodesics.
result Conditions for parallel transport and geodesics are simplified and visualized in the tangent space.
We equip the whole tangent space TM to a hyperbolic manifold M (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of M extend to isometries of TM by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…
New methods estimate curvature, tangent spaces, and dimension of noisy data.
problem Estimating geometric properties of noisy or sparse data.
method Diffusion geometry tools for Riemannian manifold analysis.
result Significantly outperforms existing methods in noisy or sparse data.
LEGO estimates tangent spaces more robustly than LPCA in noisy data.
problem Estimating tangent spaces in high-noise settings.
method Spectral method using graph Laplacian eigenvectors and gradient orthogonization.
result LEGO yields more robust tangent space estimates than LPCA.
This work introduces the concept of tangent space regularization for neural-network models of dynamical systems. The tangent space to the dynamics function of many physical systems of interest in control applications exhibits useful properties, e.g., smoothness, motivating regularization of the model Jacobian along sys…
New derivations on diffeological spaces are not smooth, expanding tangent space definitions.
problem Lack of smoothness in derivations on diffeological spaces.
method Examined derivations satisfying the Leibniz rule but not smooth with respect to given diffeology.
result Tangent space defined via all derivations is larger than one defined using only smooth derivations.
The tangent space is constructed in sub-Finsler geometry, leading to the failure of the CD condition in 3D-contact manifolds.
problem The failure of the CD condition in sub-Finsler geometry.
method Construction of the tangent space in the measured Gromov-Hausdorff sense, application of nilpotent approximation.
result The CD condition fails in 3D-contact sub-Finsler manifolds.
Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.
problem Combining symmetries in M-theory to extend exceptional tangent spaces.
method Combining known results to show hyperbolic involutory algebra acts on M-algebra through brane-rotating symmetry.
result Extends the hierarchy of exceptional tangent spaces from n ≤ 7 to n = 11.
Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
A new method approximates tangent spaces to simplify neural networks.
problem Efficiency of hierarchical neural networks is hindered by their complexity and training requirements.
method Approximates tangent subspace to enable sparse representation and switch to shallow networks.
result The method improves and sometimes surpasses the performance of original networks after a few epochs.
Two methods for interpolating manifold-valued data are presented.
problem Interpolating manifold-valued functions with derivative constraints.
method Two approaches: weighted Riemannian barycenters and tangent space interpolation.
result Both methods are valid and perform well with numerical examples.
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth cur…
Measures neural network complexity using tangent space diversity.
problem Estimating the true complexity of neural networks.
method Entropy-based measure of tangent spaces from different inputs.
result Captures effective complexity, not just theoretical capacity.
Survey on finite dimensional Lie groups over real numbers.
problem Lack of rigorous proofs for Lie groups using tangent space formalism.
method Formalism of tangent space via chart and vector methods, curve and derivation methods.
result Rigorous proofs of Lie groups facts with this formalism.
Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.
problem Embedding Hermitian symmetric spaces into their tangent spaces.
method Using polarity of the K-action to construct equivariant embeddings.
result Characterizes holomorphic/symplectic embeddings and realizes submanifolds.
For an n-dimensional real hyperbolic manifold M, we calculate the Zariski tangent space of a character variety χ(π1(M),SL(n+1,R)),n>2 at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using rea…
Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections…
We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories o…
The paper examines stability of ReLU networks in tangent space and activation regions.
problem Stability and sensitivity of ReLU networks to small changes.
method Tangent sensitivity measure for ReLU networks, focusing on stability induced by individual examples.
result Tangent sensitivity correlates with the distribution of activation regions and generalization gap.
TSGO optimizes gradients in tensor networks to avoid vanishing/exploding issues.
problem Gradient vanishing and exploding problems in deep learning models.
method TSGO rotates parameters towards gradient direction in tangent space of normalized state.
result TSGO naturally determines learning rate based on angle between parameters and gradient.
A method to fix radius distortion in generative models on curved spaces.
problem Distortion in geodesic radius measurements across different charts on Riemannian manifolds.
method Radial Compensation (RC) adjusts the tangent-space base distribution to match the geodesic radius law, improving model stability and interpretability.
result RC ensures that the model's geodesic radius matches the intended distribution, improving numerical stability and curvature interpretation.
In this note we show that for any proper action of a Banach--Lie group G on a Banach manifold M, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each x∈M, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $…
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
Geometric framework analyzes bias in variational inference for posterior functionals.
problem Analyzing the bias of posterior functionals under variational approximations.
method Developed a geometric framework to evaluate the bias of posterior functionals using the variational tangent space.
result The leading-order bias of a posterior functional is determined by its component orthogonal to the variational tangent space.
Introduces Alexandrov spaces with curvature below, covering various theorems.
problem Understanding spaces with curvature constraints.
method Explains comparison conditions, globalization, tangent spaces, etc.
result Globalization theorem and other theorems established for Alexandrov spaces.
Manifold hypotheses are typically used for tasks such as dimensionality reduction, interpolation, or improving classification performance. In the less common problem of manifold estimation, the task is to characterize the geometric structure of the manifold in the original ambient space from a sample. We focus on the r…
There has been an emerging trend in non-Euclidean statistical analysis of aiming to recover a low dimensional structure, namely a manifold, underlying the high dimensional data. Recovering the manifold requires the noise to be of certain concentration. Existing methods address this problem by constructing an approximat…
We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidea…
The paper introduces a differentially private method for optimization on Riemannian manifolds.
problem Differential privacy in optimization constrained to Riemannian manifolds.
method Adding Gaussian noise to the Riemannian gradient on the tangent space, with privacy and utility guarantees.
result Privacy and utility guarantees for differentially private Riemannian optimization.
Develops a curvature-corrected tangent space method for manifold-valued data.
problem Generalizing real-valued data approximation to manifold-valued data.
method Systematic approach to developing global-geometry aware, computationally feasible approximation schemes.
result Proposes CC-tHOSVD for low-rank approximation of manifold-valued data.
Following recent work by Ghomi, Solomon and Tabachnikov, we study geometry and topology of skew branes. A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any pair of distinct points are not parallel. We prove that if an oriented closed manifold has a non-zero Euler characterist…
DM approximates submanifolds with error bounds.
problem Understanding the accuracy of Diffusion Maps in embedding submanifolds.
method Deriving geometric properties and deriving bounds on embedding errors.
result Error bounds for DM embeddings and tangent spaces.
Frölicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C^\infty(M,G) or \Diff(M), but also projective limits of Lie groups are in a natural way objects of that category, and group operations are morphisms in the category…
A classification theorem for RK-manifolds with linear dependence between invariants of an antiholomorphic plane in the tangent space is proved.
This study proposes sparse estimation methods for the generalized linear models, which run one of least angle regression (LARS) and least absolute shrinkage and selection operator (LASSO) in the tangent space of the manifold of the statistical model. This study approximates the statistical model and subsequently uses e…
Derives key CCM manifold equations for optimization.
problem Lack of rigorous derivation for CCM manifold equations.
method Systematic and rigorous proof of CCM properties.
result Unified reference for CCM Manifold Optimization.
We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L^2 Riemannian metric - so called because it induces an L^2 topology on each tangent space. It turns out that this top…
This paper generalizes optimization techniques to diffeological spaces.
problem Challenges in applying optimization techniques to diffeological spaces due to various tangent space definitions.
method Suitable definition of tangent space, diffeological Riemannian space, diffeological gradient, and diffeological retraction.
result Formulation of an optimization algorithm on diffeological spaces.
Accelerated method finds critical points faster on manifolds.
problem Optimization on non-convex manifolds.
method Accelerated gradient methods on Riemannian manifolds.
result Find approximate first-order critical points faster than regular gradient descent.
Study on manifolds with kinks and Gaussian kernel behavior.
problem Understanding the asymptotic behavior of graph Laplacian on manifolds with singularities.
method Introduced manifolds with kinks, derived asymptotic behavior of Graph Laplacian with Gaussian kernel, and validated results numerically.
result Asymptotic behavior of the Graph Laplacian is determined by the inward sector of the tangent space.
Paper solves robust multi-dimensional scaling with accelerated projections.
problem Localize point locations from noisy pairwise distances.
method Alternating projections with tangent space acceleration.
result Linear convergence of reconstructed points to original points.
The diagnosis of Alzheimer's disease (AD) in routine clinical practice is most commonly based on subjective clinical interpretations. Quantitative electroencephalography (QEEG) measures have been shown to reflect neurodegenerative processes in AD and might qualify as affordable and thereby widely available markers to f…
Study recovers Riemannian quantities from noisy data densities.
problem Recovering geometric structure from noisy data on submanifolds.
method Derive uniform small-noise expansions of noisy density and its derivatives; construct estimators for tangent spaces, intrinsic dimension, and second fundamental form.
result Fundamental Riemannian quantities identifiable from density derivatives.
Paper estimates Wasserstein distance for Ricci shrinkers.
problem Estimating Wasserstein distance for Ricci shrinkers.
method Analyzes Wasserstein distance between measures in tangent spaces.
result Provides upper estimate for Wasserstein distance.
Study equigeodesics on G2-type flag manifolds, splitting tangent spaces.
problem Characterize geodesics in G2-type flag manifolds. method Analyze flag manifolds with G2-type t-roots, split tangent spaces, and classify equigeodesics. result Characterized structural equigeodesic vectors in flag manifolds.
Given 2 points of a smooth hypersurface, their mid-hyperplane is the hyperplane passing through their mid-point and the intersection of their tangent spaces. In this paper we study the envelope of these mid-hyperplanes (EMH) at pairs whose tangent spaces are transversal. We prove that this envelope consists of centers …