arXiv research

We develop Fourier methods to expand translation-invariant kernels.

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

AON improves neural network generalization by making weights approximately orthogonal.

In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intra…

Let $M$ be a regular Riemann surface with a metric which has constant scalar curvature $ρ$. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles $K_{M}^{m}$. That is, \[\sum_{i=0}^{d_{m}-1}\|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\fracρ{2 m})+O(e^{-\frac{(\log m)^{2}…

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.

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.

We develop a neural network model to classify liver cancer patients into high-risk and low-risk groups using genomic data. Our approach provides a novel technique to classify big data sets using neural network models. We preprocess the data before training the neural network models. We first expand the data using wavel…

The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The classic examples are the Fourier and wavelet transforms that enjoy numerically effic…

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}.

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

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…

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

PolyILR: A Tree-Structured Orthonormal Decomposition of Compositional Data

Two regularization techniques improve GCNN explainability and preference from chemists.

Paper introduces RKHM for more explicit variable structures analysis.

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

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 …

Batch normalization has become ubiquitous in many state-of-the-art nets. It accelerates training and yields good performance results. However, there are various other alternatives to normalization, e.g. orthonormalization. The objective of this paper is to explore the possible alternatives to channel normalization with…

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.…

Bayesian approach improves sparse PCE for high-dimensional problems.

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…

BOOOM optimizes orthonormal matrices without needing gradients.

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…

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…

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…

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

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…

New Ricci curvature means derived from plane curvatures.

Efficiently optimizes CNN and RNN parameters on Stiefel manifold.

We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using…

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.

Unified multi-view learning framework using OPLS with regularization and deep extensions.

New algorithm improves deep learning models' robustness without sacrificing accuracy.

The paper proves fundamental theorems for timelike surfaces in Minkowski 4-space.

New IRL algorithm for continuous state spaces with formal guarantees.

ManifoldFlow relaxes fixed-spectrum Stiefel layers to learn a positive spectrum.

Muon optimizer improves federated learning performance.

Study efficient iterative method for distribution matching using sliced optimal transport.

A special p-form is a p-form which, in some orthonormal basis {e_μ}, has components φ_{μ_1...μ_p} = φ(e_{μ_1},..., e_{μ_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms.