arXiv research

Conformal vector fields on LCP manifolds are orthogonal and Killing.

New differential geometry perspective on orthogonal RNNs.

Local classification of 4D Ricci solitons with specific algebra properties.

Paper derives local Plücker formulas for special orthogonal groups.

A notion of orthogonality in multisymplectic geometry has been developed by Cantrijn, Ibort and de León and used by many authors. In this paper, we review this concept and propose a new type of orthogonality in multisymplectic geometry; we prove a number of results regarding this orthogonality and its associated subspa…

Derives new orthogonal coordinates for evolving surfaces and curves.

In this paper, we study spherical images of the modified orthogonal vector fields and Darboux vector of a regular curve which lies on the unit sphere in Euclidean 3-space.

The paper extends orthogonal decomposition results to hermitian Higgs bundles.

The paper introduces polarizations in symplectic and orthogonal settings.

Deterministic bounds for tensor singular values and vectors, differing from matrix cases.

DFSOS improves sparse discriminant analysis for high-dimensional data.

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curve…

A method for interpreting SVMs using polynomial kernels, revealing model complexity.

Principal component analysis (PCA) is an unsupervised method for learning low-dimensional features with orthogonal projections. Multilinear PCA methods extend PCA to deal with multidimensional data (tensors) directly via tensor-to-tensor projection or tensor-to-vector projection (TVP). However, under the TVP setting, i…

We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field ${\boldsymbol k}$ with geodesic flow, if an integral curve of ${\boldsymbol k}$ is hypersurf…

AuON is a linear-time optimizer that improves upon Muon's performance without approximate orthogonal matrices.

Study null conformal Killing vector fields on complex surfaces.

This paper presents two results conserning real hypersurfaces in CP^{2} and CH^{2}. More precisely, it is proved that real hypersurfaces equipped with structure Jacobi operator satisfying condition $\mathcal{L}_{X}l=\nabla_{X}l$, where \emph{X} is a vector field orthogonal to structure vector field $ξ$, do not exist. A…

New algorithms improve tensor CP decomposition under mild conditions.

We show that $φ$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are all minimal. We prove that an odd-dimensional $φ$-invariant submanifold …

Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.

Inverted file and asymmetric distance computation (IVFADC) have been successfully applied to approximate nearest neighbor search and subsequently maximum inner product search. In such a framework, vector quantization is used for coarse partitioning while product quantization is used for quantizing residuals. In the ori…

SOLAR improves search efficiency and accuracy with sparse, orthogonal embeddings.

The study examines how gamma positivity and PL homeomorphism types affect simplicial spheres.

We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $φ$. For the normal case, we prove that a $φ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $φ$-invariant submanifold $N$ everyw…

Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…

In this paper we explore the "vector semantics" problem from the perspective of "almost orthogonal" property of high-dimensional random vectors. We show that this intriguing property can be used to "memorize" random vectors by simply adding them, and we provide an efficient probabilistic solution to the set membership …

Explores tensor products in hyperdimensional computing.

TGCCA analyzes higher-order tensors using orthogonal rank-R CP decomposition.

Study $Sp(n)$-orbits in complex and $Σ$-complex subspaces of Hermitian quaternionic vector spaces.

Random matrix ensembles yield uniform distributions on manifolds.

The paper explores geometric decompositions for Ricci tensors and their applications.

Study curvature of orthogonal distributions on manifolds.

New ONMF model minimizes KL divergence for better sparse data modeling.

Characterizes spacetimes using doubly torqued vectors.

For homogeneous simply connected Hodge manifolds it is proved that the set of coherent vectors orthogonal to a given one is the divisor responsible for the homogeneous holomorphic line bundle of the coherent vectors. In particular, for naturally reductive spaces, the divisor is the cut locus.

The performance of Orthogonal Matching Pursuit (OMP) for variable selection is analyzed for random designs. When contrasted with the deterministic case, since the performance is here measured after averaging over the distribution of the design matrix, one can have far less stringent sparsity constraints on the coeffici…

In the machine learning field, dimensionality reduction is an important task. It mitigates the undesired properties of high-dimensional spaces to facilitate classification, compression, and visualization of high-dimensional data. During the last decade, researchers proposed many new (non-linear) techniques for dimensio…

The study explores rigid constraints on almost Ricci-Bourguignon solitons on contact metric three-manifolds.

We consider four-dimensional Riemannian manifolds with commuting higher order Jacobi operators defined on two-dimensional orthogonal subspaces (polygons) and on their orthogonal subspaces. More precisely, we discuss higher order Jacobi operator $\mathcal{J}(X)$ and its commuting associated operator $\mathcal{J}(X^{\per…

Develops deformation theory for mapping spaces related to Morse theory and Bott-Thom isomorphism.

Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic suba…

Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.

An algorithm for efficient computation of equivariant neural network layers.

msPCA solves sparse PCA for multiple components efficiently.

We introduce the concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric $\G$ to be scaled by different conformal factors. In particular, we study their infinitesimal version, called bi-conformal vector fields. We show the differential co…

Orthogonal matching pursuit (OMP) is a widely used algorithm for recovering sparse high dimensional vectors in linear regression models. The optimal performance of OMP requires \textit{a priori} knowledge of either the sparsity of regression vector or noise statistics. Both these statistics are rarely known \textit{a p…

The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of gov…