arXiv research

New method uses almost orthonormal bases to prove low-degree lower bounds in complex statistical models.

We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial …

The paper explains why combining Sobol sequences and polynomials improves LSMC stability.

A new method builds sparse polynomial chaos expansions for models with dependent inputs.

We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively n…

This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ and we associa…

Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. To cope with demanding analysis such as optimization and reliability, surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade.…

A new neural network model uses polynomial chaos theory to improve neural signal processing.

Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, we considered the definition of orthonormal basis in Minkows…

Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, I considered the definition of orthonormal basis in Minkowsk…

Study shows how varying levels of supervision and orthonormality constraints affect generalization errors in subspace fitting.

This paper explores orthonormalization layers as alternatives to batch normalization.

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

Paper introduces efficient orthonormal transformations using Householder reflectors.

We develop Fourier methods to expand translation-invariant kernels.

We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of dimension 5 that has no orthonormal geodesic basis, for any inner product.

Gradient descent learns over-param neural nets better than NTK.

AON improves neural network generalization by making weights approximately orthogonal.

Isometry pursuit identifies orthonormal submatrices from wide matrices.

We generalize the orthonormal basis for the Gaussian RKHS described in \cite{MinhGaussian2010} to an infinite, continuously parametrized, family of orthonormal bases, along with some implications. The proofs are direct generalizations of those in \cite{MinhGaussian2010}.

We show the existence of uniformly bounded sequences of increasing numbers of orthonormal sections of powers $L^k$ of a positive holomorphic line bundle $L$ on a compact Kähler manifold $M$. In particular, we construct for each positive integer $k$, orthonormal sections $s^k_1,\dots,s^k_{n_k}$ in $H^0(M,L^k)$, $n_k\geβ…

PolyILR: A Tree-Structured Orthonormal Decomposition of Compositional Data

Two regularization techniques improve GCNN explainability and preference from chemists.

Guaranteed convergence for tensor factorization using Riemannian gradient descent.

Paper introduces RKHM for more explicit variable structures analysis.

Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.

In the present work we construct a lift of a metric $g$ on a 2-dimensional oriented Riemannian manifold $M$ to a metric $\hat{g}$ on the total space $P$ of the orthonormal frame bundle of $M$. We call this lift the \textit {Wagner lift}. Viktor Vladimirovich Wagner (1908 -1981) proposed a technique to extend a metric d…

The paper equidistributes zeros of random polynomials and sections on manifolds.

Bayesian approach improves sparse PCE for high-dimensional problems.

BOOOM optimizes orthonormal matrices without needing gradients.

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology…

We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…

CODE learns ODE dynamics from sparse data, outperforming neural and kernel methods.

The Frenet frame is generally known an orthonormal vector frame for curves. But, it does not always meet the needs of curve characterizations. In this study, with the help of associated curves of any spatial curve we obtained a new orthonormal frame which has the property that the second vector makes a constant angle w…

We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can decompose a 4-tensor with $n$-dimensional orthonormal components in the…

Study on 3D Lie groups finds all generalized Einstein metrics.

Higher-dimensional Milnor frames are characterized and contrasted with 3D Heisenberg and 4D nilpotent Lie algebras.

Quasi-orthonormal encoding reduces high dimensionality for categorical data.

We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex structures on the Stiefel manifolds of orthonormal 2-frames in C^{n+1}, non compatible…

$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\{…

This paper introduces Haar convolution for GNNs to reduce computational cost.

For any compact Riemannian manifold $(M,g)$ and its heat kernel embedding map $psi_t$ from M into $l^2$ constructed in [BBG], we study the higher derivatives of $psi_t$ with respect to an orthonormal basis at $x$ on $M$. As the heat flow time $t$ goes to 0, it turns out the limiting angles between these derivative vect…

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in ge…

Efficiently optimizes CNN and RNN parameters on Stiefel manifold.

The paper studies horizontal semimartingales on Riemannian manifolds and their connections to Euclidean spaces.

The aim of this paper is to write an explicit orthonormal parallelization for all parallelizable products of spheres, using an explicit isomorphism with a trivial vector bundle.

A machine learning method selects optimal orthonormal bases for functional data analysis.

Graph clustering method uses templates to match vertices and outperforms classical methods.