We propose a data-driven framework for learning reduced-order moment dynamics from partial differential equation (PDE)-governed systems using neural Ordinary Differential Equations (neural ODEs). In contrast to derivative-based methods, such as SINDy (Sparse Identification of Nonlinear Dynamics), which necessitate densely sampled data and are sensitive to noise, our approach based on neural ODEs directly models moment trajectories, enabling robust learning from sparse and potentially irregular time series. Using as an application platform the nonlinear Schrödinger equation, the framework accurately recovers governing moment dynamics when closure is available, even from limited, irregular, and noisy observations. For systems without analytical closure, we introduce a data-driven coordinate transformation strategy based on Stiefel manifold optimization, enabling the discovery of low-dimensional representations in which the moment dynamics become closed, facilitating interpretable and reliable modeling. We also explore cases where a closure model is not known, such as a Fisher-Kolmogorov-Petrovskii-Piskounov reaction-diffusion system. Here, we demonstrate that neural ODEs can still effectively approximate the unclosed moment dynamics and achieve superior extrapolation accuracy compared to physical-expert-derived ODE models. This advantage remains robust even under sparse and irregular sampling, highlighting the method's robustness in data-limited settings. Our results highlight the neural ODE framework as a powerful and flexible tool for learning interpretable, low-dimensional moment dynamics in complex PDE-governed systems.

The Tverberg–Vrećica conjecture claims a broad generalization of Tverberg’s classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this manuscript, we establish complex analogues of these results, where the corresponding transversals are complex affine spaces. The proofs of the complex Tverberg–Vrećica conjecture and its optimal colorful version rely on the non-vanishing of an equivariant Euler class. Furthermore, we obtain new Borsuk–Ulam-type theorems on complex Stiefel manifolds. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments by Axelrod-Freed and Soberón, and provide a direct proof of the complex central transversal theorem.

Multi-task learning is a widely used technique for harnessing information from various tasks. Recently, the sparse orthogonal factor regression (SOFAR) framework, based on the sparse singular value decomposition (SVD) within the coefficient matrix, was introduced for interpretable multi-task learning, enabling the discovery of meaningful latent feature-response association networks across different layers. However, conducting precise inference on the latent factor matrices has remained challenging due to the orthogonality constraints inherited from the sparse SVD constraints. In this article, we suggest a novel approach called the high-dimensional manifold-based SOFAR inference (SOFARI), drawing on the Neyman near-orthogonality inference while incorporating the Stiefel manifold structure imposed by the SVD constraints. By leveraging the underlying Stiefel manifold structure that is crucial to enabling inference, SOFARI provides easy-to-use bias-corrected estimators for both latent left factor vectors and singular values, for which we show to enjoy the asymptotic mean-zero normal distributions with estimable variances. We introduce two SOFARI variants to handle strongly and weakly orthogonal latent factors, where the latter covers a broader range of applications. We illustrate the effectiveness of SOFARI and justify our theoretical results through simulation examples and a real data application in economic forecasting. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Constraint Coordinate-Momentum Phase Space Formulations for Finite-State Quantum Systems: The Relation between Commutator Variables and Complex Stiefel Manifolds

Abstract Let M i {M}_{i} , for i = 1 i=1 , 2, be a Kähler manifold, and let G G be a compact Lie group acting on M i {M}_{i} by Kähler isometries. Suppose that the action admits a momentum map μ i {\mu }_{i} , and let N i ≔ μ i − 1 ( 0 ) {N}_{i}:= {\mu }_{i}^{-1}\left(0) be a regular-level set. When the action of G G on N i {N}_{i} is proper and free, the Meyer-Marsden-Weinstein quotient P i ≔ N i ∕ G {P}_{i}:= {N}_{i}/G is a Kähler manifold and π i : N i → P i {\pi }_{i}:{N}_{i}\to {P}_{i} is a principal fiber bundle with base P i {P}_{i} and characteristic fiber G G . In this article, we define an almost-complex structure on the manifold N 1 × N 2 {N}_{1}\times {N}_{2} and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N 1 × N 2 {N}_{1}\times {N}_{2} . As applications, we consider a nonintegrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds S 2 n + 1 × S ( ℋ ) {{\mathbb{S}}}^{2n+1}\times S\left({\mathcal{ {\mathcal H} }}) , for n ≥ 1 n\ge 1 , where S ( ℋ ) S\left({\mathcal{ {\mathcal H} }}) denotes the unit sphere of an infinite-dimensional complex Hilbert space ℋ {\mathcal{ {\mathcal H} }} .

Abstract We study invariant Einstein metrics and Einstein–Randers metrics on the Stiefel manifold . We use a characterization for (nonflat) homogeneous Einstein–Randers metrics as pairs of (nonflat) homogeneous Einstein metrics and invariant Killing vector fields. It is well known that, for Stiefel manifolds the isotropy representation contains equivalent summands, so a complete description of invariant metrics is difficult. We prove, by assuming additional symmetries, that the Stiefel manifolds and admit at least four and six invariant Einstein metrics, respectively. Two of them are Jensen's metrics and the other two and four are new metrics. Also, we prove that admit at least two invariant Einstein metrics, which are Jensen's metrics. Finally, we show that the previous mentioned Stiefel manifolds and admit a certain number of non–Riemmanian Einstein–Randers metrics.

Direct minimization on the complex Stiefel manifold in Kohn-Sham density functional theory for finite and extended systems

This work investigates the problem of efficiently learning discriminative low-dimensional (LD) representations of multiclass image objects. We propose a generic end-to-end approach that jointly optimizes sparse dictionary and convolutions for learning LOW-dimensional discriminative image representations, named SparConvLow, taking advantage of convolutional neural networks (CNNs), dictionary learning, and orthogonal projections. The whole learning process can be summarized as follows. First, a CNN module is employed to extract high-dimensional (HD) preliminary convolutional features. Second, to avoid the high computational cost of direct sparse coding on HD CNN features, we learn sparse representation (SR) over a task-driven dictionary in the space with the feature being orthogonally projected. We then exploit the discriminative projection on SR. The whole learning process is consistently treated as an end-to-end joint optimization problem of trace quotient maximization. The cost function is well-defined on the product of the CNN parameters space, the Stiefel manifold, the Oblique manifold, and the Grassmann manifold. By using the explicit gradient delivery, the cost function is optimized via a geometrical stochastic gradient descent (SGD) algorithm along with the chain rule and the backpropagation. The experimental results show that the proposed method can achieve a highly competitive performance with the state-of-the-art (SOTA) image classification, object categorization, and face recognition methods, under both supervised and semi-supervised settings. The code is available at https://github.com/MVPR-Group/SparConvLow.

Quantum channels, complex Stiefel manifolds, and optimization

A Semidefinite Relaxation for Sums of Heterogeneous Quadratic Forms on the Stiefel Manifold

Meta-learning, or "learning to learn," aims to enable models to quickly adapt to new tasks with minimal data. While traditional methods like Model-Agnostic Meta-Learning (MAML) optimize parameters in Euclidean space, they often struggle to capture complex learning dynamics, particularly in few-shot learning scenarios. To address this limitation, we propose Stiefel-MAML, which integrates Riemannian geometry by optimizing within the Stiefel manifold, a space that naturally enforces orthogonality constraints. By leveraging the geometric structure of the Stiefel manifold, we improve parameter expressiveness and enable more efficient optimization through Riemannian gradient calculations and retraction operations. We also introduce a novel kernel-based loss function defined on the Stiefel manifold, further enhancing the model’s ability to explore the parameter space. Experimental results on benchmark datasets—including Omniglot, Mini-ImageNet, FC-100, and CUB—demonstrate that Stiefel-MAML consistently outperforms traditional MAML, achieving superior performance across various few-shot learning tasks. Our findings highlight the potential of Riemannian geometry to enhance meta-learning, paving the way for future research on optimizing over different geometric structures.

Quantum resource theories (QRTs) provide a comprehensive and practical framework for the analysis of diverse quantum phenomena. A fundamental task within QRTs is the quantification of resources inherent in a given quantum state. In this work, we introduce a unified computational framework for a class of widely utilized quantum resource measures, derived from convex roof extensions. We establish that the computation of these convex roof resource measures can be reformulated as an optimization problem over a Stiefel manifold, which can be further unconstrained through polar projection. Compared to existing methods employing semi-definite programming (SDP), gradient-based techniques or seesaw strategy, our approach not only demonstrates satisfying computational efficiency but also maintains applicability across various scenarios within a unified framework. We substantiate the efficacy of our method by applying it to several key quantum resources, including entanglement, coherence, and magic states. Moreover, our methodology can be readily extended to other convex roof quantities beyond the domain of resource theories, suggesting broad applicability in the realm of quantum information theory.

An Efficient Algorithm for the Riemannian Logarithm on the Stiefel Manifold for a Family of Riemannian Metrics

Numerical range of real-valued linear mapping on the complex Stiefel manifold: Convexity and application

For a quasi-two-dimensional nonlinear sigma model on the real Stiefel manifolds with a generalized (anisotropic) metric, the equations of a two-charge renormalization group (RG) for the homothety and anisotropy of the metric as effective couplings are obtained in a one-loop approximation. Normal coordinates and the curvature tensor are exploited for the renormalization of the metric. The RG trajectories are investigated and the presence of a fixed point common to four critical lines or four phases (tetracritical point) in the general case, or its absence in the case of an Abelian structure group, is established. For the tetracritical point, the critical exponents are evaluated and compared with those known earlier for a simpler particular case.

Correction: Totally Geodesic Submanifolds and Polar Actions on Stiefel Manifolds

High Curvature Means Low Rank: On the Sectional Curvature of Grassmann and Stiefel Manifolds and the Underlying Matrix Trace Inequalities

This paper focuses on the H 2 optimal model reduction problem of positive systems. According to the coefficient matrices of the positive system, the nonnegative orthonormal matrix is taken as the projection matrix, and the H 2 optimal model reduction problem is developed. Since the projection matrix is orthonormal and nonnegative, the H 2 optimal model reduction problem is reformulated as a constrained optimization problem defined on the Stiefel manifold, and further regarded as a constrained optimization problem defined on the oblique manifold. By the augmented Lagrangian function, the constrained optimization problem defined on the oblique manifold is tackled by employing the Dai-Yuan-type conjugate gradient method to solve a series of unconstrained optimization subproblems. When the objective function of a subproblem satisfies some conditions, the iterative sequence produced by the conjugate gradient method is convergent. Finally, numerical experiments illustrate the efficiency of the proposed model reduction method.

Non-negative Matrix Factorization (NMF) has gained popularity due to its effectiveness in clustering and feature selection tasks. It is particularly valuable for managing high-dimensional data by reducing dimensionality and providing meaningful semantic representations. However, traditional NMF methods may encounter challenges when dealing with noisy data, outliers, or when the underlying manifold structure of the data is overlooked. This paper introduces an innovative approach called SGRiT, which employs Stiefel manifold optimization to enhance the extraction of latent features. These learned features have been shown to be highly informative for clustering tasks. The method leverages a spectral decomposition criterion to obtain a low-dimensional embedding that captures the intrinsic geometric structure of the data. Additionally, this paper presents a solution for addressing the Stiefel manifold problem and utilizes a Riemannian-based trust region algorithm to optimize the loss function. The outcome of this optimization process is a new representation of the data in a transformed space, which can subsequently serve as input for the NMF algorithm. Furthermore, this paper incorporates a novel subspace graph regularization term that considers high-order geometric information and introduces a sparsity term for the factor matrices. These enhancements significantly improve the discrimination capabilities of the learning process. This paper conducts an impartial analysis of several essential NMF algorithms. To demonstrate that the proposed approach consistently outperforms other benchmark algorithms, four clustering evaluation indices are employed.

A method for computing a [Formula: see text]-curve with given initial and final velocities interpolating a finite number of points on a reductive homogeneous space is presented. Here the reductive homogeneous space is assumed to be embedded into some manifold in a suitable way making the proposed approach very general. Building on the notion of intrinsic rolling, the method presented here offers a solution of the interpolation problem in closed form. This is illustrated on the example of matrix Lie groups. Moreover, this method is applied to the (compact) Stiefel manifold, where an efficient algorithm for solving the interpolation problem is also obtained.