Orthonormal Research Articles

On semi-orthogonal matrices with row vectors of equal lengths

When does a rectangular matrix with an orthonormal set of column vectors have row vectors of equal lengths? The column spaces of such matrices are multidimensional generalizations of the projection plane used in isometric perspective. We show that in the absence of unexpected linear relations, any rectangular matrix can be row-scaled so that if we were to orthonormalize the column vectors, the row vectors would attain equal lengths in the process. We use Grassmann coordinates to reduce the question into an instance of the famous matrix scaling problem, and with the help of existing theory introduce simple numerical solutions.

Derivation, characterization, and application of complete orthonormal sequences for representing general three-dimensional states of residual stress

Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent any sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives.

Targeted optimization in small-scale atomic structure calculations: application to Au I

The lack of reliable atomic data can be a severe limitation in astrophysical modelling, in particular of events such as kilonovae that require information on all neutron-capture elements across a wide range of ionization stages. Notably, the presence of non-orthonormalities between electron orbitals representing configurations that are close in energy can introduce significant inaccuracies in computed energies and transition probabilities. Here, we propose an explicit targeted optimization (TO) method that can effectively circumvent this concern while retaining an orthonormal orbital basis set. We illustrate this method within the framework of small-scale atomic structure models of Au I, using the Grasp2018 multiconfigurational Dirac–Hartree–Fock atomic structure code. By comparing to conventional optimization schemes we show how a TO approach improves the energy level positioning and ordering. TO also leads to better agreement with experimental data for the strongest E1 transitions. This illustrates how small-scale models can be significantly improved with minor computational costs if orbital non-orthonormalities are considered carefully. These results should prove useful to multi-element atomic structure calculations in, for example, astrophysical opacity applications involving neutron-capture elements.

Post-processing for Bayesian analysis of reduced rank regression models with orthonormality restrictions

Orthonormality constraints are common in reduced rank models. They imply that matrix-variate parameters are given as orthonormal column vectors. However, these orthonormality restrictions do not provide identification for all parameters. For this setup, we show how the remaining identification issue can be handled in a Bayesian analysis via post-processing the sampling output according to an appropriately specified loss function. This extends the possibilities for Bayesian inference in reduced rank regression models with a part of the parameter space restricted to the Stiefel manifold. Besides inference, we also discuss model selection in terms of posterior predictive assessment. We illustrate the proposed approach with a simulation study and an empirical application.

Orthonormal eigenfunction expansions for sixth-order boundary value problems

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. To solve such problems, we first derive a complete set of odd and even orthonormal eigenfunctions — resembling trigonometric sines and cosines, as well as the so-called “beam” functions. These functions intrinsically satisfy boundary conditions (BCs) of relevance to thin-film flows, since they are the solutions of a self-adjoint sixth-order Sturm–Liouville BVP with the same BCs. Next, we propose a Galerkin spectral approach for sixth-order problems; namely the sought function as well as all its derivatives and terms appearing in the differential equation are expanded into an infinite series with respect to the derived complete orthonormal (CON) set of eigenfunctions. The unknown coefficients in the series expansion are determined by solving the algebraic system derived by taking successive inner products with each member of the CON set of eigenfunctions. The proposed method and its convergence are demonstrated by solving two model sixth-order BVPs.

Christov expansion method for nonlocal nonlinear evolution equations

Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

Basis scaling and double pruning for efficient inference in network-based transfer learning

Network-based transfer learning allows the reuse of deep learning features with limited data, but the resulting models can be unnecessarily large. Although network pruning can improve inference efficiency, existing algorithms usually require fine-tuning that may not be suitable for small datasets. In this paper, using the singular value decomposition, we decompose a convolutional layer into two layers: a convolutional layer with the orthonormal basis vectors as the filters, and a “BasisScalingConv” layer which is responsible for rescaling the features and transforming them back to the original space. As the filters in each decomposed layer are linearly independent, when using the proposed basis scaling factors with the Taylor approximation of importance, pruning can be more effective and fine-tuning individual weights is unnecessary. Furthermore, as the numbers of input and output channels of the original convolutional layer remain unchanged after basis pruning, it is applicable to virtually all architectures and can be combined with existing pruning algorithms for double pruning to further increase the pruning capability. When transferring knowledge from ImageNet pre-trained models to different target domains, with less than 1% reduction in classification accuracies, we can achieve pruning ratios up to 74.6% for CIFAR-10 and 98.9% for MNIST in model parameters.

Uncertainty principles for the short‐time Fourier transform on the lattice

AbstractIn this paper, we study a few versions of the uncertainty principle for the short‐time Fourier transform on the lattice . In particular, we establish the uncertainty principle for orthonormal sequences, Donoho–Stark's uncertainty principle, Benedicks‐type uncertainty principle, Heisenberg‐type uncertainty principle, and local uncertainty inequality for this transform on . Also, we obtain the Heisenberg‐type uncertainty inequality using the ‐entropy of the short‐time Fourier transform on .

Relative acceleration of orthonormal basis vectors for the geometric conduction blocks of the cardiac electric signal propagation on anisotropic curved surfaces

Geometric conduction blocks stop cardiac electric propagation due to the shape or conductivity properties of the domain. The blocks are considered to cause many abnormal cardiac electric propagations, leading to cardiac electrophysiological pathologies, such as cardiac fibrillation and arrhythmia. Locating such multidimensional conduction blocks is challenging, particularly in a complex domain with a complex shape and strong anisotropy, such as the heart. To address this problem, we propose a novel mathematical model of the geometric conduction block using the relative acceleration adopted from space-time physics. An efficient numerical scheme for the mathematical model is also proposed to predict the unidirectional conduction block effectively, even in a complex domain. The relative acceleration in the cardiac electric propagation corresponds to the sink-source relationship between the excited (after repolarization) and excitable (before depolarization) cardiac cells, representing the geometric growth rate of the volume of metric balls. The trajectory is constructed from the wavefront of diffusion-reaction equations by aligning orthonormal basis vectors along the gradient of the action potential. Relative acceleration is computed along the propagational direction from the connection 1-form of the basis vectors. The proposed mathematical model and numerical scheme are applied to demonstrate geometric conduction blocks in two-dimensional (2D) simple curved domains with strong anisotropy.

Optical dipole potential energy of a two-level atom interacting with a tightly focused truncated optical Bessel beam

Truncated optical Bessel (TOB) beams are an orthonormal set of optical vortex beams that possess orbital angular momentum (OAM). It is characterized by a significant longitudinal electric field component at the sub-wavelength scales. The photons of a circularly polarized TOB beam also carry spin angular momentum (SAM) and can interact with a two-level atom, in the far-off resonance regime, giving rise to an optical dipole potential energy that involves a spin–orbit term that stems from the inclusion of the longitudinal field component. When a two-level atom is simultaneously irradiated by two TOB beams, the optical dipole potential energy, in some special cases, is entirely dependent on the OAM–SAM coupling. We explore the cold-atom trapping options enabled by the use of TOB beams, either single ones or combinations thereof. We put emphasis on case in which the OAM–SAM coupling is the dominant contribution. The similarities and differences between the results obtained for the TOB beams and those for the Laguerre–Gaussian beams are also discussed.

Hofstadter-like spectrum and magnetization of artificial graphene constructed with cylindrical and elliptical quantum dots

A comparative study of the electronic and magnetic properties of quasi-two-dimensional electrons in an artificial graphene-like superlattice composed of circular and elliptical quantum dots is presented. A complete orthonormal set of basis functions has been implemented for calculation of the energy dispersions, Hofstadter spectra, density of states and orbital magnetization of the considered systems. Our calculations indicate a topological change in the miniband structure due to the ellipticity of the quantum dots, and non-trivial modifications of the electron energy dispersion surfaces with the change of the magnetic flux per unite cell. The ellipticity of the quantum dots leads to an opening of a gap and considerable modifications of the Hofstadter spectrum. The orbital magnetization is shown to reveal significant oscillations with the change of the magnetic flux. The ellipticity of quantum dots has a qualitative impact on the dependencies of the magnetization on both the magnetic flux and the temperature.

Uncertainty principles for the windowed Opdam–Cherednik transform

In this paper, we study a few versions of the uncertainty principle for the windowed Opdam–Cherednik transform. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho–Stark's uncertainty principle, Benedicks‐type uncertainty principle, Heisenberg‐type uncertainty principle, and local uncertainty inequality for this transform. We also obtain the Heisenberg‐type uncertainty inequality using the ‐entropy of the windowed Opdam–Cherednik transform.

Complete and orthonormal sets of exponential-type orbitals with non-integer quantum numbers

Atomic and molecular orbitals show exponential decrease at long range. Complete orthonormal basis sets for atoms should satisfy this criterion. A number of such bases have been used in physics (e.g. Coulomb Sturmians). The challenge of this work is first adapting Slater type Orbitals for this role, as they are not radially orthogonal. Even more important is their generalization to non-integer quantum numbers that have applications for configuration interaction. This generalization requires the whole apparatus of non-integer calculus that is presented using the Riemann–Liouville approach.

Using deep LSD to build operators in GANs latent space with meaning in real space.

Generative models rely on the idea that data can be represented in terms of latent variables which are uncorrelated by definition. Lack of correlation among the latent variable support is important because it suggests that the latent-space manifold is simpler to understand and manipulate than the real-space representation. Many types of generative model are used in deep learning, e.g., variational autoencoders (VAEs) and generative adversarial networks (GANs). Based on the idea that the latent space behaves like a vector space Radford et al. (2015), we ask whether we can expand the latent space representation of our data elements in terms of an orthonormal basis set. Here we propose a method to build a set of linearly independent vectors in the latent space of a trained GAN, which we call quasi-eigenvectors. These quasi-eigenvectors have two key properties: i) They span the latent space, ii) A set of these quasi-eigenvectors map to each of the labeled features one-to-one. We show that in the case of the MNIST image data set, while the number of dimensions in latent space is large by design, 98% of the data in real space map to a sub-domain of latent space of dimensionality equal to the number of labels. We then show how the quasi-eigenvectors can be used for Latent Spectral Decomposition (LSD). We apply LSD to denoise MNIST images. Finally, using the quasi-eigenvectors, we construct rotation matrices in latent space which map to feature transformations in real space. Overall, from quasi-eigenvectors we gain insight regarding the latent space topology.

TWISTED SHIFT-INVARIANT SYSTEM IN

AbstractWe consider a general twisted shift-invariant system, $V^{t}(\mathcal {A})$ , consisting of twisted translates of countably many generators and study the problem of obtaining a characterization for the system $V^{t}(\mathcal {A})$ to form a frame sequence or a Riesz sequence. We illustrate our theory with some examples. In addition to these results, we study a dual twisted shift-invariant system and also obtain an orthonormal sequence of twisted translates from a given Riesz sequence of twisted translates.

Topological two-body bands in a multiband Hubbard model

In a multiband Hubbard model the self-consistency relations for the two-body bound-state bands are in the form of a nonlinear eigenvalue problem. Assuming that the resultant eigenvectors form an orthonormal set, e.g., in the strong-binding regime, here we reformulate their Berry curvatures and the associated Chern numbers. As an illustration we solve the two-body problem in a Haldane-Hubbard model with attractive on-site interactions and analyze its topological phase diagrams from weak to strong couplings, i.e., by keeping track of the gap closings in between the low-lying two-body bands. The resultant Chern numbers are consistent with the lobe structure of the phase diagrams in the strong-coupling regime.

A novel representation of electromagnetic field interaction with vertically stratified structures

PurposeThe purpose of this study is to develop an efficient model to solve the electromagnetic forward problem using a novel semi-analytical approach to compute the electromagnetic fields because of the presence of a scatterer.Design/methodology/approachThe proposed model involves a novel formulation of a complete orthonormal set of radiating/nonradiating polarization currents. Furthermore, an integral equation-based representation is derived, and the appropriate boundary conditions are imposed to get the scattered electromagnetic field. An error term is introduced to evaluate the obtained results.FindingsThe proposed model was tested using several examples at different frequencies. The results of this study show that the novel representation exhibits fast convergence behavior and achieves highly accurate results, when compared to the results provided by the transmission line method.Originality/valueThe derived formulations presented in this study are significant in the electromagnetic forward modelling field because of the meaningful physical representation they provide. This is an important aspect that leads to precise calculation of electromagnetic fields for various applications.

Orthonormal product quantization network for scalable face image retrieval

Existing deep quantization methods provided an efficient solution for large-scale image retrieval. However, the significant intra-class variations, like pose, illumination, and expressions in face images, still pose a challenge. In light of this, face image retrieval requires sufficiently powerful learning metrics, which are absent in current deep quantization works. Moreover, to tackle the growing unseen identities in the query stage, face image retrieval drives more demands regarding model generalization and scalability than general image retrieval tasks. This paper integrates product quantization with orthonormal constraints into an end-to-end deep learning framework to effectively retrieve face images. Specifically, we propose a novel scheme that uses predefined orthonormal vectors as codewords to enhance the quantization informativeness and reduce codewords’ redundancy. A tailored loss function maximizes discriminability among identities in each quantization subspace for both the quantized and original features. An entropy-based regularization term is imposed to reduce the quantization error. Experiments are conducted on four commonly-used face datasets under both seen and unseen identity retrieval settings. Our method outperforms all the compared state-of-the-art under both settings. The proposed orthonormal codewords consistently boost both models’ standard retrieval performance and generalization ability, demonstrating the superiority of our method for scalable face image retrieval.

Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime

For ( t , x ) ∈ ( 0 , ∞ ) × T D (t,x) \in (0,\infty )\times \mathbb {T}^{\mathfrak {D}} , the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as t ↓ 0 t \downarrow 0 , i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents q ~ 1 , ⋯ , q ~ D ∈ R \widetilde {q}_1,\cdots ,\widetilde {q}_{\mathfrak {D}} \in \mathbb {R} , which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at { t = 1 } \lbrace t = 1 \rbrace , as long as the exponents are “sub-critical” in the following sense: max I , J , B = 1 , ⋯ , D I > J { q ~ I + q ~ J − q ~ B } > 1 \underset {\substack {I,J,B=1,\cdots , \mathfrak {D}\\ I > J}}{\max } \{\widetilde {q}_I+\widetilde {q}_J-\widetilde {q}_B\}>1 . Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with D = 3 \mathfrak {D}= 3 and q ~ 1 ≈ q ~ 2 ≈ q ~ 3 ≈ 1 / 3 \widetilde {q}_1 \approx \widetilde {q}_2 \approx \widetilde {q}_3 \approx 1/3 , which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for D ≥ 38 \mathfrak {D}\geq 38 with max I = 1 , ⋯ , D | q ~ I | > 1 / 6 \underset {I=1,\cdots ,\mathfrak {D}}{\max } |\widetilde {q}_I| > 1/6 . In this paper, we prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the 1 + D 1+\mathfrak {D} -dimensional Einstein-scalar field system for all D ≥ 3 \mathfrak {D}\geq 3 and the 1 + D 1+\mathfrak {D} -dimensional Einstein-vacuum equations for D ≥ 10 \mathfrak {D}\geq 10 ; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in 1 + 3 1+3 dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized U ( 1 ) U(1) -symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized U ( 1 ) U(1) -symmetric solutions. Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to t t , and to handle this difficulty, we use t t -weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the t t -weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.

A new characterization of Aminov surface with regards to its Gauss map in E^4

In this work, we focus on Aminov surface with regard to its Gauss map in E^4. Firstly, we write the covariant derivatives according to linear combinations of orthonormal vectors and separate the equalities using Gauss and Weingarten formulas. Then, we get the laplace of the Gauss map. After giving some conditions, we yield as main results: Aminov surfaces can not have harmonic Gauss map and can not have pointwise one-type Gauss map of I. kind in E^4. Further, we give an example of helical cylinder which is also congruent to an Aminov surface. Lastly, we obtain the conditions of having pointwise one-type Gauss map of II. kind.