Academic literature on the topic 'Hierarchical orthogonality'

Since polynomials of higher than fourth degree, which is the algebraic counterpart of generic geometric dimension, are insolvable in general, then presumably no more than just four mutually orthogonal geometric dimensions can be placed within a single geometric space if it is expected to be fully operational. Hence a hierarchical notion of dimension is needed in order to ensure that at least virtual orthogonality is respected, which in turn implies presence of certain hierarchically organized multispatial structures. It is shown that the operational constraint on physical spaces implies of necessity presence of both: 4-dimensional (4D) spacetime and a certain 4D timespace.

Dictionary Learning (DL) plays a crucial role in numerous machine learning tasks. It targets at finding the dictionary over which the training set admits a maximally sparse representation. Most existing DL algorithms are based on solving an optimization problem, where the noise variance and sparsity level should be known as the prior knowledge. However, in practice applications, it is difficult to obtain these knowledge. Thus, non-parametric Bayesian DL has recently received much attention of researchers due to its adaptability and effectiveness. Although many hierarchical priors have been used to promote the sparsity of the representation in non-parametric Bayesian DL, the problem of redundancy for the dictionary is still overlooked, which greatly decreases the performance of sparse coding. To address this problem, this paper presents a novel robust dictionary learning framework via Bayesian inference. In particular, we employ the orthogonality-promoting regularization to mitigate correlations among dictionary atoms. Such a regularization, encouraging the dictionary atoms to be close to being orthogonal, can alleviate overfitting to training data and improve the discrimination of the model. Moreover, we impose Scale mixture of the Vector variate Gaussian (SMVG) distribution on the noise to capture its structure. A Regularized Expectation Maximization Algorithm is developed to estimate the posterior distribution of the representation and dictionary with orthogonality-promoting regularization. Numerical results show that our method can learn the dictionary with an accuracy better than existing methods, especially when the number of training signals is limited.

AbstractWe construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Legendre and Jacobi polynomials. For both the mass matrix and the quasi-stiffness matrix, better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle [Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709-6733].

AbstractBy employing the infinite multilevel representation of the residual, we derive computable bounds to estimate the distance of finite element approximations to the solution of the Poisson equation. If the finite element approximation is a Galerkin solution, the derived error estimator coincides with the standard element and edge based estimator. If Galerkin orthogonality is not satisfied, then the discrete residual additionally appears in terms of the BPX preconditioner. As a by-product of the present analysis, conditions are derived such that the hierarchical error estimation is reliable and efficient.

A Weibull-model-based approach is examined to handle under- and overdispersed discrete data in a hierarchical framework. This methodology was first introduced by Nakagawa and Osaki (1975, IEEE Transactions on Reliability, 24, 300–301), and later examined for under- and overdispersion by Klakattawi et al. (2018, Entropy, 20, 142) in the univariate case. Extensions to hierarchical approaches with under- and overdispersion were left unnoted, even though they can be obtained in a simple manner. This is of particular interest when analysing clustered/longitudinal data structures, where the underlying correlation structure is often more complex compared to cross-sectional studies. In this article, a random-effects extension of the Weibull-count model is proposed and applied to two motivating case studies, originating from the clinical and sociological research fields. A goodness-of-fit evaluation of the model is provided through a comparison of some well-known count models, that is, the negative binomial, Conway–Maxwell–Poisson and double Poisson models. Empirical results show that the proposed extension flexibly fits the data, more specifically, for heavy-tailed, zero-inflated, overdispersed and correlated count data. Discrete left-skewed time-to-event data structures are also flexibly modelled using the approach, with the ability to derive direct interpretations on the median scale, provided the complementary log–log link is used. Finally, a large simulated set of data is created to examine other characteristics such as computational ease and orthogonality properties of the model, with the conclusion that the approach behaves best for highly overdispersed cases.

3D (three-dimensional) structured light scanning system is widely used in the field of reverse engineering, quality inspection, and so forth. Camera calibration is the key for scanning precision. Currently, 2D (two-dimensional) or 3D fine processed calibration reference object is usually applied for high calibration precision, which is difficult to operate and the cost is high. In this paper, a novel calibration method is proposed with a scale bar and some artificial coded targets placed randomly in the measuring volume. The principle of the proposed method is based on hierarchical self-calibration and bundle adjustment. We get initial intrinsic parameters from images. Initial extrinsic parameters in projective space are estimated with the method of factorization and then upgraded to Euclidean space with orthogonality of rotation matrix and rank 3 of the absolute quadric as constraint. Last, all camera parameters are refined through bundle adjustment. Real experiments show that the proposed method is robust, and has the same precision level as the result using delicate artificial reference object, but the hardware cost is very low compared with the current calibration method used in 3D structured light scanning system.

A biologically plausible low-order model (LOM) of biological neural networks is proposed. LOM is a recurrent hierarchical network of models of dendritic nodes and trees; spiking and nonspiking neurons; unsupervised, supervised covariance and accumulative learning mechanisms; feedback connections; and a scheme for maximal generalization. These component models are motivated and necessitated by making LOM learn and retrieve easily without differentiation, optimization, or iteration, and cluster, detect, and recognize multiple and hierarchical corrupted, distorted, and occluded temporal and spatial patterns. Four models of dendritic nodes are given that are all described as a hyperbolic polynomial that acts like an exclusive-OR logic gate when the model dendritic nodes input two binary digits. A model dendritic encoder that is a network of model dendritic nodes encodes its inputs such that the resultant codes have an orthogonality property. Such codes are stored in synapses by unsupervised covariance learning, supervised covariance learning, or unsupervised accumulative learning, depending on the type of postsynaptic neuron. A masking matrix for a dendritic tree, whose upper part comprises model dendritic encoders, enables maximal generalization on corrupted, distorted, and occluded data. It is a mathematical organization and idealization of dendritic trees with overlapped and nested input vectors. A model nonspiking neuron transmits inhibitory graded signals to modulate its neighboring model spiking neurons. Model spiking neurons evaluate the subjective probability distribution (SPD) of the labels of the inputs to model dendritic encoders and generate spike trains with such SPDs as firing rates. Feedback connections from the same or higher layers with different numbers of unit-delay devices reflect different signal traveling times, enabling LOM to fully utilize temporally and spatially associated information. Biological plausibility of the component models is discussed. Numerical examples are given to demonstrate how LOM operates in retrieving, generalizing, and unsupervised and supervised learning.

Dans un modèle qui peut s'avérer complexe et fortement non linéaire, les paramètres d'entrée, parfois en très grand nombre, peuvent être à l'origine d'une importante variabilité de la sortie. L'analyse de sensibilité globale est une approche stochastique permettant de repérer les principales sources d'incertitude du modèle, c'est-à-dire d'identifier et de hiérarchiser les variables d'entrée les plus influentes. De cette manière, il est possible de réduire la dimension d'un problème, et de diminuer l'incertitude des entrées. Les indices de Sobol, dont la construction repose sur une décomposition de la variance globale du modèle, sont des mesures très fréquemment utilisées pour atteindre de tels objectifs. Néanmoins, ces indices se basent sur la décomposition fonctionnelle de la sortie, aussi connue soue le nom de décomposition de Hoeffding. Mais cette décomposition n'est unique que si les variables d'entrée sont supposées indépendantes. Dans cette thèse, nous nous intéressons à l'extension des indices de Sobol pour des modèles à variables d'entrée dépendantes. Dans un premier temps, nous proposons une généralisation de la décomposition de Hoeffding au cas où la forme de la distribution des entrées est plus générale qu'une distribution produit. De cette décomposition généralisée aux contraintes d'orthogonalité spécifiques, il en découle la construction d'indices de sensibilité généralisés capable de mesurer la variabilité d'un ou plusieurs facteurs corrélés dans le modèle. Dans un second temps, nous proposons deux méthodes d'estimation de ces indices. La première est adaptée à des modèles à entrées dépendantes par paires. Elle repose sur la résolution numérique d'un système linéaire fonctionnel qui met en jeu des opérateurs de projection. La seconde méthode, qui peut s'appliquer à des modèles beaucoup plus généraux, repose sur la construction récursive d'un système de fonctions qui satisfont les contraintes d'orthogonalité liées à la décomposition généralisée. En parallèle, nous mettons en pratique ces différentes méthodes sur différents cas tests<br>A mathematical model aims at characterizing a complex system or process that is too expensive to experiment. However, in this model, often strongly non linear, input parameters can be affected by a large uncertainty including errors of measurement of lack of information. Global sensitivity analysis is a stochastic approach whose objective is to identify and to rank the input variables that drive the uncertainty of the model output. Through this analysis, it is then possible to reduce the model dimension and the variation in the output of the model. To reach this objective, the Sobol indices are commonly used. Based on the functional ANOVA decomposition of the output, also called Hoeffding decomposition, they stand on the assumption that the incomes are independent. Our contribution is on the extension of Sobol indices for models with non independent inputs. In one hand, we propose a generalized functional decomposition, where its components is subject to specific orthogonal constraints. This decomposition leads to the definition of generalized sensitivity indices able to quantify the dependent inputs' contribution to the model variability. On the other hand, we propose two numerical methods to estimate these constructed indices. The first one is well-fitted to models with independent pairs of dependent input variables. The method is performed by solving linear system involving suitable projection operators. The second method can be applied to more general models. It relies on the recursive construction of functional systems satisfying the orthogonality properties of summands of the generalized decomposition. In parallel, we illustrate the two methods on numerical examples to test the efficiency of the techniques

The biology of cell adhesion and migration has traditionally been studied on 2D glass or plastic surfaces. While such studies have shed light on the molecular mechanisms governing these processes [1], current knowledge is limited by the dissimilarity between the flat surfaces conventionally employed and the topographically complex extracellular matrix (ECM) cells routinely navigate within the body. On ECM-coated flat surfaces, cells are presented with an unlimited expanse of adhesive ligand and can spread and migrate freely. Conversely, the availability of ligand in vivo is generally restricted to ECM structures, forcing cells to form adhesions in prescribed locations distributed through 3D space depending on the geometry and organization of the surrounding matrix [2]. These physical constraints on cell adhesion likely have profound consequences on intracellular signaling and resulting migration, and calls into question whether the mechanisms and modes of cell motility observed on flat substrates are truly reflective of the in vivo scenario [3]. The topographies of ECMs found in vivo are varied but largely fibrillar, ranging from the tightly crosslinked fibers that form the sheet-like basement membrane, to the structure of fibrin-rich clots and collagenous connective tissues. Collagen comprises approximately 25% of the human body by mass, and as such, purified collagen has served as a popular setting for the study of cell migration within a fibrillar context for many decades [4]. However, a major limitation to the use of these gels is the inability to orthogonally dictate key structural features that impact cell behavior. For example, in contrast to the large range of fiber diameters found in vivo within connective tissue resulting from hierarchical collagen assembly and multiple types of collagens [3], collagen gels are limited to fibril diameters of ∼500nm. Furthermore, recreating the structural anisotropy common to connective tissues in collagen gels is technically challenging [5]. Thus, there remains a significant need for engineered fibrillar materials that afford precise and independent control of architectural and mechanical features for application in cell biology. In this work, we develop two approaches to fabricating fibrillar ECMs in order to study cell adhesion and migration in vitro.