arXiv research

The study examines principal directions and curvatures of Lagrangian submanifolds.

In this paper, we characterize and classify all surfaces endowed with canonical principal direction relative to a space-like and light-like, constant direction in Minkowski 3-spaces.

Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful charac…

In this paper, we introduce canonical principal direction (CPD) submanifolds with higher codimension in Euclidean spaces. We obtain the complete classification of surfaces endowed with CPD in the Euclidean 4-space.

Study on discrete surfaces with constant principal curvature for nanocarbon applications.

Survey of SDR methods for high-dimensional regression and embedding.

In the present paper we classify all surfaces in $\E^3$ with a canonical principal direction. Examples of these type of surfaces are constructed. We prove that the only minimal surface with a canonical principal direction in the Euclidean space ${\mathbb{E}}^3$ is the catenoid.

Autoencoders are a deep learning model for representation learning. When trained to minimize the distance between the data and its reconstruction, linear autoencoders (LAEs) learn the subspace spanned by the top principal directions but cannot learn the principal directions themselves. In this paper, we prove that $L_2…

In this paper we characterize and classify surfaces in ${\mathbb{H}}^2\times{\mathbb{R}}$ which have a canonical principal direction. Here ${\mathbb{H}}^2$ denotes the hyperbolic plane. We study some geometric properties such as minimality and flatness. Some examples are given to complete the study.

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show …

A new PCR method using SVD with sparse regularization.

Attention learns PCA on Gaussian data, proving its connection to principal component analysis.

We explore the geometrical interpretation of the PCA based clustering algorithm Principal Direction Divisive Partitioning (PDDP). We give several examples where this algorithm breaks down, and suggest a new method, gap partitioning, which takes into account natural gaps in the data between clusters. Geometric features …

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds…

In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof f…

A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $Π$ is considered. A problem of determination of such surfaces when both principal curvatures are given over a line in $Π$ is formulated and studied. The geometric problem is reduced to th…

Study on null hypersurfaces with constant angle in Lorentzian manifolds.

This research classifies invariant complex structures and Kähler metrics on principal bundles.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the s…

We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometri…

A new construction of a universal connection was given in \cite{BHS}. The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle $E$ over a compact Riemann surface admits a holomorphic connection if and only if the degree of every direct summand of $E$ is degre…

POTD estimates SDR subspace using optimal transport for binary response.

Paper finds principal eigenvalue for infinity Laplacian in metric spaces.

Introduces generalized principal bundles and connections, linking them to standard gauge theories.

The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function $f$ on the plane are studied. We provide a Poincaré-Hopf type formula where the sum over all indices of the principal directions at its umbilic points only depends upon the number of real linear factors…

New Gaussian DPP model reveals directionality in data.

We present a novel algorithm (Principal Sensitivity Analysis; PSA) to analyze the knowledge of the classifier obtained from supervised machine learning techniques. In particular, we define principal sensitivity map (PSM) as the direction on the input space to which the trained classifier is most sensitive, and use anal…

We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic me…

New method prevents deep learning models from forgetting past tasks.

This is a detailed tutorial paper which explains the Principal Component Analysis (PCA), Supervised PCA (SPCA), kernel PCA, and kernel SPCA. We start with projection, PCA with eigen-decomposition, PCA with one and multiple projection directions, properties of the projection matrix, reconstruction error minimization, an…

Improved convergence speed of principal component analysis through modified learning rules.

CPCR mitigates bias in PCR for overparameterized models.

New algorithms detect and estimate rank-one signals with prior directional information.

This paper analyzes how errors accumulate in PCA's deflation method.

A Hopf hypersurface in complex hyperbolic space CH^n is one in which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to the ambient sectional curvature) are studied by means of…

This paper improves Koopman operator approximations by pruning subspaces in RKHS.

Informed by recent work on tensor singular value decomposition and circulant algebra matrices, this paper presents a new theoretical bridge that unifies the hypercomplex and tensor-based approaches to singular value decomposition and robust principal component analysis. We begin our work by extending the principal comp…

Geodesic flows with specific integrals are linked to special 4-webs.

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal minor of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal minor is an Ale…

New method for robust PCA with exponential family distributions.

We introduce a new spatial data structure for high dimensional data called the \emph{approximate principal direction tree} (APD tree) that adapts to the intrinsic dimension of the data. Our algorithm ensures vector-quantization accuracy similar to that of computationally-expensive PCA trees with similar time-complexity…

Enhances interpretability of linear latent spaces through automated clustering and ranking.

Consider a complete Riemannian manifold $M^n$ and let $Σ^n$ be an orientable hypersurface of the product manifold $M\times\mathbb{R}$ endowed with its standard product metric $\langle \,,\, \rangle.$ Let $\nablaξ$ denote the gradient of the height function $ξ$ of $Σ.$ In this note, we characterize the hypersurfaces $Σ$…

Classifies hypersurfaces with specific curvature properties in 4D space.

Dimension reduction of multivariate data supervised by auxiliary information is considered. A series of basis for dimension reduction is obtained as minimizers of a novel criterion. The proposed method is akin to continuum regression, and the resulting basis is called continuum directions. With a presence of binary sup…

Motivated by the Bagging Partial Least Squares (PLS) and Principal Component Analysis (PCA) algorithms, we propose a Principal Model Analysis (PMA) method in this paper. In the proposed PMA algorithm, the PCA and the PLS are combined. In the method, multiple PLS models are trained on sub-training sets, derived from the…

We use foliations and connections on principal Lie groupoid bundles to prove various integrability results for Lie algebroids. In particular, we show, under quite general assumptions, that the semi-direct product associated to an infinitesimal action of one integrable Lie algebroid on another is integrable. This genera…