arXiv research

Study of tangent spaces in diffeological spaces under Lie group actions.

Defines tangent spaces on causal sets using partial derivatives and metrics.

A novel method for parallel transport and geodesics on submanifolds.

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.

LEGO estimates tangent spaces more robustly than LPCA in noisy data.

New derivations on diffeological spaces are not smooth, expanding tangent space definitions.

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…

The tangent space is constructed in sub-Finsler geometry, leading to the failure of the CD condition in 3D-contact manifolds.

Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.

Paper constructs infinitely many tangent functors on diffeological spaces.

Two methods for interpolating manifold-valued data are presented.

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.

Survey on finite dimensional Lie groups over real numbers.

Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.

For an $n$-dimensional real hyperbolic manifold $M$, we calculate the Zariski tangent space of a character variety $χ(π_1(M),SL(n+1,\mathbb R)), n>2$ 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.

A method to fix radius distortion in generative models on curved spaces.

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 $x \in M$, 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.

Geometric framework analyzes bias in variational inference for posterior functionals.

Introduces Alexandrov spaces with curvature below, covering various theorems.

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.

Develops a curvature-corrected tangent space method for 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…

The gradient-based optimization method for deep machine learning models suffers from gradient vanishing and exploding problems, particularly when the computational graph becomes deep. In this work, we propose the tangent-space gradient optimization (TSGO) for the probabilistic models to keep the gradients from vanishin…

DM approximates submanifolds with error bounds.

A classification theorem for RK-manifolds with linear dependence between invariants of an antiholomorphic plane in the tangent space is proved.

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…

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.

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.

Accelerated method finds critical points faster on manifolds.

Study on manifolds with kinks and Gaussian kernel behavior.

Paper solves robust multi-dimensional scaling with accelerated projections.

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.

Paper estimates Wasserstein distance for Ricci shrinkers.

Study equigeodesics on $G_2$-type flag manifolds, splitting tangent spaces.

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 …

In this paper we show that space of spatial polygons in semi riemann space gives a Kahler manifold. We describe the tangent space and almost complex structure which has many computational advantages.