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We use recent extensions of the Borsuk–Ulam theorem for Stiefel manifolds to generalize the ham sandwich theorem to mass assignments. A k-dimensional mass assignment continuously imposes a measure on each k-dimensional affine subspace of R d . Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces S k - 1 ⊂ S k ⊂ ... ⊂ S d - 1 ⊂ R d such that S i bisects all the mass assignments on S i + 1 for every i. We show it is possible to do so whenever the number of mass assignments of dimensions (k , ... , d) is a permutation of (k , ... , d) . We extend previous work on mass assignments and the central transversal theorem. We also study the problem of halving several families of (d - k) -dimensional affine spaces of R d using a (k - 1) -dimensional affine subspace contained in some translate of a fixed k-dimensional affine space. For k = d - 1 , there results can be interpreted as dynamic ham sandwich theorems for families of moving points. [ABSTRACT FROM AUTHOR]

This paper computes the Fadell–Husseini index of Stiefel manifolds in the case where the group acts via permutations of the orthogonal vectors. The computations are carried out in the case of elementary Abelian p-groups. The results are shown to imply certain generalizations of the Kakutani–Yamabe–Yujobo theorem. [ABSTRACT FROM AUTHOR]

Abstract In complex systems, it’s crucial to uncover latent mechanisms and their context-dependent relationships. This is especially true in medical research, where identifying unknown cancer mechanisms and their impact on phenomena like drug resistance is vital. Directly observing these mechanisms is challenging due to measurement complexities, leading to an approach that infers latent mechanisms from observed variable distributions. Despite machine learning advancements enabling sophisticated generative models, their black-box nature complicates the interpretation of complex latent mechanisms. A promising method for understanding these mechanisms involves estimating latent factors through linear projection, though there’s no assurance that inferences made under specific conditions will remain valid across contexts. We propose a novel solution, suggesting data, even from systems appearing complex, can often be explained by sparse dependencies among a few common latent factors, regardless of the situation. This simplification allows for modeling that yields significant insights across diverse fields. We demonstrate this with datasets from finance, where we capture societal trends from stock price movements, and medicine, where we uncover new insights into cancer drug resistance through gene expression analysis.

In complex systems, it's crucial to uncover latent mechanisms and their context-dependent relationships. This is especially true in medical research, where identifying unknown cancer mechanisms and their impact on phenomena like drug resistance is vital. Directly observing these mechanisms is challenging due to measurement complexities, leading to an approach that infers latent mechanisms from observed variable distributions. Despite machine learning advancements enabling sophisticated generative models, their black-box nature complicates the interpretation of complex latent mechanisms. A promising method for understanding these mechanisms involves estimating latent factors through linear projection, though there's no assurance that inferences made under specific conditions will remain valid across contexts. We propose a novel solution, suggesting data, even from systems appearing complex, can often be explained by sparse dependencies among a few common latent factors, regardless of the situation. This simplification allows for modeling that yields significant insights across diverse fields. We demonstrate this with datasets from finance, where we capture societal trends from stock price movements, and medicine, where we uncover new insights into cancer drug resistance through gene expression analysis. [ABSTRACT FROM AUTHOR]

We generalize the Yao–Yao partition theorem by showing that for any smooth measure in R d there exist equipartitions using (t + 1) 2 d - 1 convex regions such that every hyperplane misses the interior of at least t regions. In addition, we present tight bounds on the smallest number of hyperplanes whose union contains the boundary of an equipartition of a measure into n regions. We also present a simple proof of a Borsuk–Ulam type theorem for Stiefel manifolds that allows us to generalize the central transversal theorem and prove results bridging the Yao–Yao partition theorem and the central transversal theorem. [ABSTRACT FROM AUTHOR]

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space G/H with reductive decomposition g = h ⊕ m , we consider rollings of m over G/H without slip and without twist, where G/H is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space Q which is tangent to a certain distribution. By considering an H-principal fiber bundle π ¯ : Q ¯ → Q over the configuration space equipped with a suitable principal connection, rollings of m over G/H can be expressed in terms of horizontally lifted curves on Q ¯ . The total space of π ¯ : Q ¯ → Q is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of m over G/H as solutions of an explicit, time-variant ordinary differential equation (ODE) on Q ¯ , the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in G/H is the projection of a one-parameter subgroup in G. Lie groups and Stiefel manifolds are discussed as examples. [ABSTRACT FROM AUTHOR]

To any V in the Grassmannian Gr k (R n) of k-dimensional vector subspaces in R n one can associate the diagonal entries of the ( n × n ) matrix corresponding to the orthogonal projection of R n to V. One obtains a map Gr k (R n) → R n (the Schur–Horn map). The main result of this paper is a criterion for pre-images of vectors in R n to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill et al. (SIAM J Appl Algebra Geom 1:38–72, 2017). [ABSTRACT FROM AUTHOR]

We discuss the rolling, without slipping and without twisting, of Stiefel manifolds equipped with α -metrics, from an intrinsic and an extrinsic point of view. We, however, start with a more general perspective, namely, by investigating the intrinsic rolling of normal naturally reductive homogeneous spaces. This gives evidence as to why a seemingly straightforward generalization of the intrinsic rolling of symmetric spaces to normal naturally reductive homogeneous spaces is not possible, in general. For a given control curve, we derive a system of explicit time-variant ODEs whose solutions describe the desired rolling. These findings are applied to obtain the intrinsic rolling of Stiefel manifolds, which is then extended to an extrinsic one. Moreover, explicit solutions of the kinematic equations are obtained, provided that the development curve is the projection of a not necessarily horizontal one-parameter subgroup. In addition, our results are put into perspective with examples of the rolling Stiefel manifolds known from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we compute the $BP$ -cohomology of complex projective Stiefel manifolds. The method involves the homotopy fixed point spectral sequence, and works for complex oriented cohomology theories. We also use these calculations and $BP$ -operations to prove new results about equivariant maps between Stiefel manifolds. [ABSTRACT FROM AUTHOR]

The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative examples. [ABSTRACT FROM AUTHOR]

We discuss the rolling, without slipping and without twisting, of Stiefel manifolds equipped with α-metrics, from an intrinsic and an extrinsic point of view. We, however, start with a more general perspective, namely, by investigating the intrinsic rolling of normal naturally reductive homogeneous spaces. This gives evidence as to why a seemingly straightforward generalization of the intrinsic rolling of symmetric spaces to normal naturally reductive homogeneous spaces is not possible, in general. For a given control curve, we derive a system of explicit time-variant ODEs whose solutions describe the desired rolling. These findings are applied to obtain the intrinsic rolling of Stiefel manifolds, which is then extended to an extrinsic one. Moreover, explicit solutions of the kinematic equations are obtained, provided that the development curve is the projection of a not necessarily horizontal one-parameter subgroup. In addition, our results are put into perspective with examples of the rolling Stiefel manifolds known from the literature.

This work aims to interpolate parametrized reduced order model (ROM) basis constructed via the proper orthogonal decomposition (POD) to derive a robust ROM of the system's dynamics for an unseen target parameter value. A novel non-intrusive space-time (ST) POD basis interpolation scheme is proposed, for which we define ROM spatial and temporal basis curves on compact Stiefel manifolds. An interpolation is finally defined on a mixed part encoded in a square matrix directly deduced using the spacial part, the singular values and the temporal part, to obtain an interpolated snapshot matrix, keeping track of accurate space and temporal eigenvectors. Moreover, in order to establish a well-defined curve on the compact Stiefel manifold, we introduce a new procedure, the so-called oriented SVD. Such an oriented SVD produces unique right and left eigenvectors for generic matrices, for which all singular values are distinct. It is important to notice that the ST POD basis interpolation does not require the construction and the subsequent solution of a reduced-order FEM model as classically is done. Hence it is avoiding the bottleneck of standard POD interpolation which is associated with the evaluation of the nonlinear terms of the Galerkin projection on the governing equations. As a proof of concept, the proposed method is demonstrated with the adaptation of rigid-thermoviscoplastic finite element ROMs applied to a typical nonlinear open forging metal forming process. Strong correlations of the ST POD models with respect to their associated high-fidelity FEM counterpart simulations are reported, highlighting its potential use for near real-time parametric simulations using off-line computed ROM POD databases. [ABSTRACT FROM AUTHOR]

Gaussian variational approximation is a popular methodology to approximate posterior distributions in Bayesian inference, especially in high-dimensional and large data settings. To control the computational cost, while being able to capture the correlations among the variables, the low rank plus diagonal structure was introduced in the previous literature for the Gaussian covariance matrix. For a specific Bayesian learning task, the uniqueness of the solution is usually ensured by imposing stringent constraints on the parameterized covariance matrix, which could break down during the optimization process. In this article, we consider two special covariance structures by applying the Stiefel manifold and Grassmann manifold constraints, to address the optimization difficulty in such factorization architectures. To speed up the updating process with minimum hyperparameter-tuning efforts, we design two new schemes of Riemannian stochastic gradient descent methods and compare them with other existing methods of optimizing on manifolds. In addition to fixing the identification issue, results from both simulation and empirical experiments prove the ability of the proposed methods of obtaining competitive accuracy and comparable converge speed in both high-dimensional and large-scale learning tasks. for this article are available online. [ABSTRACT FROM AUTHOR]

For a smooth manifold N denote by E m (N) the set of smooth isotopy classes of smooth embeddings N → ℝ m . A description of the set E m (S p × S q ) was known only for p = q = 0 or for p = 0, m ≠ q + 2 or for 2 m ⩾ 2(p + q) + max{ p , q } + 4. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For m ⩾ 2 p + q + 3 we introduce an abelian group structure on E m (S p × S q ) and describe this group 'up to an extension problem'. This result has corollaries which, under stronger dimension restrictions, more explicitly describe E m (S p × S q ). The proof is based on relations between sets E m (N) for different N and m , in particular, on a recent exact sequence of M. Skopenkov. [ABSTRACT FROM AUTHOR]

By analogy with complex K –theory and K –theory with reality, there are theories of unitary functor calculus and unitary functor calculus with reality, both of which are generalisations of Weiss' orthogonal calculus. In this paper we show that unitary functor calculus can be completely recovered from the unitary functor calculus with reality, in analogy to how complex topological K –theory is completely recovered from K –theory with reality via forgetting the C 2 –action. [ABSTRACT FROM AUTHOR]

In this paper, we design the precoding matrix with broad coverage for massive multi-input multi-output public channels. In order to guarantee the efficient use of power amplifiers and improve the achievable ergodic rate, equal transmit power per-antenna and semi-unitary constraints on the precoding matrix are considered simultaneously. Within the framework of manifold optimization, the precoding matrix design under the above two constraints becomes an optimization problem over the intersection of the oblique manifold and the Stiefel manifold. We propose to use the steepest descent method on the intersection of these two manifolds to obtain the optimal solution. By using the alternating projections method, the search direction of the steepest descent method is derived. Meanwhile, the convergence analysis for the proposed approach is also provided. The proposed approach can be used for both omnidirectional and sector-shaped power pattern design. Simulation results show that the power pattern of our designed precoder has less variation in different spatial directions within the cell coverage compared with the existing Zadoff–Chu scheme. [ABSTRACT FROM AUTHOR]

When optimizing on a Riemannian manifold, it is important to use an efficient retraction, which maps a point on a tangent space to a point on the manifold. In this paper, we prove a map based on the QR factorization to be a retraction on the generalized Stiefel manifold. In addition, we propose an efficient implementation of the retraction based on the Cholesky QR factorization. Numerical experiments show that the proposed retraction is more efficient than the existing one based on the polar factorization. [ABSTRACT FROM AUTHOR]

We study the problem of constructing a graph Fourier transform (GFT) for directed graphs (digraphs), which decomposes graph signals into different modes of variation with respect to the underlying network. Accordingly, to capture low, medium, and high frequencies we seek a digraph (D)GFT such that the orthonormal frequency components are as spread as possible in the graph spectral domain. To that end, we advocate a two-step design whereby we 1) find the maximum directed variation (i.e., a novel notion of frequency on a digraph) a candidate basis vector can attain and 2) minimize a smooth spectral dispersion function over the achievable frequency range to obtain the desired spread DGFT basis. Both steps involve non-convex, orthonormality-constrained optimization problems, which are efficiently tackled via a feasible optimization method on the Stiefel manifold that provably converges to a stationary solution. We also propose a heuristic to construct the DGFT basis from Laplacian eigenvectors of an undirected version of the digraph. We show that the spectral-dispersion minimization problem can be cast as supermodular optimization over the set of candidate frequency components, whose orthonormality can be enforced via a matroid basis constraint. This motivates adopting a scalable greedy algorithm to obtain an approximate solution with quantifiable worst-case spectral dispersion. We illustrate the effectiveness of our DGFT algorithms through numerical tests on synthetic and real-world networks. We also carry out a graph-signal denoising task, whereby the DGFT basis is used to decompose and then low pass filter temperatures recorded across the United States. [ABSTRACT FROM AUTHOR]

Abstract We prove that the tangent bundle of a generic space of planar n -gons with specified side lengths, identified under isometry, plus a trivial line bundle is isomorphic to (n − 2) times a canonical line bundle. We then discuss consequences for orientability, cobordism class, immersions, and parallelizability. [ABSTRACT FROM AUTHOR]

In this paper, the H2 optimal model order reduction method for the large-scale multiple-input multiple-output (MIMO) discrete system is investigated. First, the MIMO discrete system is resolved into a number of single-input single-output (SISO) subsystems, and the H2 norm of the original MIMO discrete system is expressed by the cross Gramian of each subsystem. Then, the retraction and the vector transport on the Stiefel manifold are introduced, and the geometric conjugate gradient model order reduction method is proposed. The reduced system of the original MIMO discrete system is generated by using the proposed method. Finally, two numerical examples show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpℍn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpℍn ≅ Sp(n)/Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view Vpℍn as a total space over the generalized Wallach space Sp(n)/(Sp(k1)×Sp(k2)×Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)). [ABSTRACT FROM AUTHOR]

Stiefel manifolds V q R n of all orthonormal q -frames in R n are diffeomorphic to the homogeneous space SO ( n ) ∕ SO ( n − q ) . In this paper, we prove that there exist at least four homogeneous Einstein–Randers metrics on SO ( n ) ∕ SO ( n − 4 ) ( n ≥ 6 ) , Moreover, we get at least six homogeneous Einstein–Randers metrics on SO ( 7 ) ∕ SO ( 2 ) . [ABSTRACT FROM AUTHOR]

Randall and others [2] used the intrinsic join product to express the index of a k -field with finite singularities on the total space of a smooth fiber bundle F ↪ M ⟶ B , for 1 < k ≤ 4 . The k -field with finite singularities on the total space M arises from k -fields with finite singularities given on the fiber F and on the base B . The index of the k -field on M is the join product of indices of the k -fields with finite singularities defined on F and on B . In this work we extend the results of [2] for the cases where k = 5 and 6. We present tables for the join product of generators of π m − 1 ( V m , k ) for k = 5 and 6, where V m , k denotes the real Stiefel manifold of k -frames in R m . The index of a k -field with finite singularities on a smooth, closed, oriented manifold, for k = 5 and 6, is given precisely in terms of the characteristic numbers of the manifold as coefficients of the generators of π m − 1 ( V m , k ) . [ABSTRACT FROM AUTHOR]

In this paper, we propose two feasible methods based on projections using a curvilinear search for solving optimization problems with orthogonality constraints. In one of them we apply a projected Adams-Moulton-like update scheme. All our algorithms compute the SVD decomposition in each iteration to preserve feasibility. Additionally, we present some convergence results. Finally, we perform numerical experiments with simulated problems; and analyze the performance of the proposed methods compared with state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

In the existing literature, the numerical estimation of multivariate normalizing constant is considered highly non-trivial, intractable or a big barrier to Bayesian approach, and hence non-parametric and many other techniques are adopted as an alternative for inferential problems, i.e., without the use of normalizing constant. This problem is a substantial one, and is mainly due to the difficult normalizing constant requiring accurate and efficient calculation for parametric models. The core objective of this paper is to estimate the numerical value of multivariate normalizing constant in Fisher–Bingham parametric model of distributions in the most general context. The ultimate goal is then the practicability of multivariate Fisher–Bingham model with the inclusion of normalizing constant for Bayesian inference via the standard Maximum Likelihood Estimation (MLE) algorithm. The normalizing constant is evaluated with the method of Saddle-Point Approximation (SPA) via Taylor Series expansion, where the first four terms of the series are considered the most crucial for obtaining the threshold value of the multivariate normalizing constant. Furthermore, it plays the most important role for the practicability of multivariate parametric models with manifold valued data being more useful for statistical inferences. We evaluate the numerical value of multivariate normalizing constant with best accuracy estimates. The numerical value is then used for statistical inferences with a simple platform of MLE for experiments on one synthetic dataset on Stiefel manifolds and three publicly available real world datasets with consideration of Grassmannian points on the Fisher–Bingham model. The results obtained via numerical experiments are compared with results of existing related techniques, where our algorithm outperforms or at least is best comparable to the results of existing techniques currently available in literature. [ABSTRACT FROM AUTHOR]

We derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if Ω is a skew-Hermitian matrix and t is a sufficiently small scalar, then there exists a polynomial of degree n in tΩ (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of etΩ with error O(t2n+1). We prove this fact and then leverage it to derive high-order approximations of the Riemannian exponential map on the Grassmannian and Stiefel manifolds. Along the way, we derive related results concerning the supercloseness of the geometric and arithmetic means of unitary matrices. [ABSTRACT FROM AUTHOR]

Estimating 3D shape of an object from 2D observations in a monocular image is fundamentally an inverse problem due to the ambiguity of the projection from 3D to 2D and becomes more challenging when there are undesirable outliers in the observations. In this paper, we develop a robust model to estimate 3D shape from 2D landmarks with an unknown camera pose. The 3D shape of the object is assumed as a linear combination of a group of prior shape bases. At the same time, we explicitly model the outliers as sparse noises to handle severely contaminated observations. The objective function is nonconvex and nonsmooth constrained on Stiefel manifold, where the coupling of underdetermined shape representation coefficients and camera pose makes it more difficult to solve. We first propose a numerical algorithm based on alternating direction method of multipliers for the no-outlier case. We set the orthogonality constraints into the smooth subproblem, which admits a closed-form solution, and the other subproblems are all well known and can be easily solved. We then extend this algorithm to the proposed robust model. The proposed algorithms can achieve convergence rapidly. The experimental results on both synthetic data and real data show that the proposed method outperforms the other methods. [ABSTRACT FROM PUBLISHER]

Direct search methods are mainly designed for use in problems with no equality constraints. However, there are many instances where the feasible set is of measure zero in the ambient space and no mesh point lies within it. There are methods for working with feasible sets that are (Riemannian) manifolds, but not all manifolds are created equal. In particular, reductive homogeneous spaces seem to be the most general space that can be conveniently optimized over. The reason is that a ‘law of motion’ over the feasible region is also given. Examples include Rn and its linear subspaces, Lie groups, and coset manifolds such as Grassmannians and Stiefel manifolds. These are important arenas for optimization, for example, in the areas of image processing and data mining. We demonstrate optimization procedures over general reductive homogeneous spaces utilizing maps from the tangent space to the manifold. A concrete implementation of the probabilistic descent direct search method is shown. This is then extended to a procedure that works solely with the manifold elements, eliminating the need for the use of the tangent space. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of optimization problems with orthogonality constraints, the feasible region of which is called the Stiefel manifold. Our new framework combines a function value reduction step with a correction step. Different from the existing approaches, the function value reduction step of our algorithmic framework searches along the standard Euclidean descent directions instead of the vectors in the tangent space of the Stiefel manifold, and the correction step further reduces the function value and guarantees a symmetric dual variable at the same time. We construct two types of algorithms based on this new framework. The first type is based on gradient reduction including the gradient re ection (GR) and the gradient projection (GP) algorithms. The other one adopts a columnwise block coordinate descent (CBCD) scheme with a novel idea for solving the corresponding CBCD subproblem inexactly. We prove that both GR/GP with a fixed step size and CBCD belong to our algorithmic framework, and any clustering point of the iterates generated by the proposed framework is a first-order stationary point. Preliminary experiments illustrate that our new framework is of great potential. [ABSTRACT FROM AUTHOR]

The Cayley transform for orthogonal groups is a well known construction with applications in real and complex analysis, linear algebra and computer science. In this work, we construct Cayley transforms on Stiefel manifolds. Applications to the Lusternik–Schnirelmann category and optimization problems are presented. [ABSTRACT FROM AUTHOR]

Quaternionic elements in orbital mechanics are usually related to the Kustaanheimo-Stiefel transformation or to the definition of the orbital plane. The new set of regular elements presented in this paper stems from the form of the equations of motion of a rotating solid, which model the evolution of a quaternion defining the orientation of a bodyfixed frame and the change in the angular velocity of such frame. By replacing the body-fixed frame with a special orbital frame and accounting for the radial motion separately, an equivalent solution to orbital motion can be constructed. The variation of parameters technique furnishes a new set of elements that is independent from the orbital plane. A second-order Sundman transformation introduces a fictitious time that replaces the physical time as the independent variable. This technique improves the numerical performance of the method and simplifies the derivation. The use of a time clement yields an even smoother evolution of the orbital elements under perturbations. Once the Lagrange and Poisson brackets are obtained, the most general nonosculating version of the set of elements is presented. Regarding the performance, numerical experiments show that the method is comparable to other formulations involving similar stabilization and regularization techniques. [ABSTRACT FROM AUTHOR]

This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument. In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifolds. This is achieved using Pizzetti formulas that express the integrals in terms of differential operators. [ABSTRACT FROM AUTHOR]

An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds. Suppose that H is an infinite-dimensional separable Hilbert space. Let S⊂H be the sphere, p∈S. Let μ be the push forward of a Gaussian measure γ from TpS onto S using the exponential map. Let v∈TpS be a Cameron--Martin vector for γ; let R be a rotation of S in the direction v, and ν=R#μ be the rotated measure. Then μ,ν are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron--Martin direction produces equivalent measures. Let γ be a Gaussian measure on H; then there exists a smooth closed manifold M⊂H such that the projection of H to the nearest point on M is not well defined for points in a set of positive γ measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability. [ABSTRACT FROM AUTHOR]

We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the existing optimization-based approach, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm converges locally and exhibits a linear rate of convergence. [ABSTRACT FROM AUTHOR]

In this paper we propose a new Riemannian conjugate gradient method for optimization on the Stiefel manifold. We introduce two novel vector transports associated with the retraction constructed by the Cayley transform. Both of them satisfy the Ring-Wirth nonexpansive condition, which is fundamental for convergence analysis of Riemannian conjugate gradient methods, and one of them is also isometric. It is known that the Ring-Wirth nonexpansive condition does not hold for traditional vector transports as the differentiated retractions of QR and polar decompositions. Practical formulae of the new vector transports for low-rank matrices are obtained. Dai's nonmonotone conjugate gradient method is generalized to the Riemannian case and global convergence of the new algorithm is established under standard assumptions. Numerical results on a variety of low-rank test problems demonstrate the effectiveness of the new method. [ABSTRACT FROM AUTHOR]

The Stiefel manifold Vp,d is the space of all d × p orthonormal matrices, with the d-1 hypersphere and the space of all orthogonal matrices constituting special cases. In modeling data lying on the Stiefel manifold, parametric distributions such as the matrix Langevin distribution are often used; however, model misspecification is a concern and it is desirable to have nonparametric alternatives. Current nonparametric methods are mainly Fréchet-mean based. We take a fully generative nonparametric approach, which relies on mixing parametric kernels such as the matrix Langevin. The proposed kernel mixtures can approximate a large class of distributions on the Stiefel manifold, and we develop theory showing posterior consistency. While there exists work developing general posterior consistency results, extending these results to this particular manifold requires substantial new theory. Posterior inference is illustrated on a dataset of near-Earth objects. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to propose a new method to generate smooth interpolating curves on Stiefel manifolds. This method is obtained from a modification of the geometric Casteljau algorithm on manifolds and is based on successive quasi-geodesic interpolation. The quasi-geodesics introduced here for Stiefel manifolds have constant speed, constant covariant acceleration and constant geodesic curvature, and in some particular circumstances they are true geodesics. [ABSTRACT FROM AUTHOR]

In this article, by slightly modifying the search direction of the nonmonotone Hestenes–Stiefel method, a variant Hestenes–Stiefel conjugate gradient method is proposed that satisfies the sufficient descent condition independent of any line search. This algorithm also possesses information about the gradient value and the function value. We establish the global convergence of our methods without the assumption that the steplength is bounded away from zero. Numerical results illustrate that our method can efficiently solve the test problems, and therefore is promising. [ABSTRACT FROM AUTHOR]

For a symplectic manifold with an anti-symplectic involution having non-empty fixed locus, we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first Stiefel–Whitney class. As a corollary, we obtain a large number of examples, which include all odd-dimensional projective spaces and many complete intersections, for which many types of real moduli spaces are orientable. For these manifolds, we define open Gromov–Witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points; a WDVV-type recursion obtained in a sequel computes these invariants for many real symplectic manifolds. If there are boundary marked points, we define the invariants under some restrictions on the allowed boundary constraints, even though the moduli spaces are not orientable in these cases. [ABSTRACT FROM AUTHOR]

The mod 2 cohomology of the real flag manifolds is known to be isomorphic to a polynomial algebra modulo a certain ideal. In this paper reduced Gröbner bases for these ideals are obtained in the case of manifolds F(1,..., 1, 2,..., 2,n). As an application of this result, the appropriate Stiefel-Whitney classes are calculated and some new non-embedding and non-immersion theorems for some manifolds of this type are obtained. [ABSTRACT FROM AUTHOR]

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier–Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the Navier–Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. The reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem. [ABSTRACT FROM AUTHOR]