Search Results (12)

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. Full article

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. Full article

This paper proposes an optimal beamforming strategy for a downlink multi-user multi-input–multi-output (MIMO) dual-function radar communication (DFRC) system with dirty paper coding (DPC) adopted at the transmitter. We aim to achieve the maximum weighted sum rate of communicating users while adhering to a predetermined transmit covariance constraint for radar performance assurance. To make the intended problem trackable, we leverage the equivalence of the weighted sum rate and the weighted minimum mean squared error (MMSE) to reframe the issue and devise a block coordinate descent (BCD) approach to iteratively calculate transmit and receive beamforming solutions. Through this methodology, we demonstrate that the optimal receive beamforming aligns with the traditional MMSE approach, whereas the optimal transmit beamforming design can be cast into a quadratic optimization problem defined on a complex Stiefel manifold. Based on the majorization–minimization (MM) method, an iterative algorithm is then developed to compute the optimal transmit beamforming design by solving a series of orthogonal Procrustes problems (OPPs) that admit closed-form optimal solutions. Numerical findings serve to validate the efficacy of our scheme. It is demonstrated that our approach can achieve at least 73% higher spectral efficiency than the existing methods in a high signal-to-noise ratio (SNR) regime. Full article

We discuss the rolling, without slipping and without twisting, of Stiefel manifolds equipped with $\alpha$-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. Full article

In this paper, we consider the problem of minimizing a continuously differentiable function on the Stiefel manifold. To solve this problem, we develop a geodesic-free proximal point algorithm equipped with Euclidean distance that does not require use of the Riemannian metric. The proposed method can be regarded as an iterative fixed-point method that repeatedly applies a proximal operator to an initial point. In addition, we establish the global convergence of the new approach without any restrictive assumption. Numerical experiments on linear eigenvalue problems and the minimization of sums of heterogeneous quadratic functions show that the developed algorithm is competitive with some procedures existing in the literature. Full article

In this work, we consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates ${\gamma }_k\left(t\right)$ (via time-dependent spectral density of incoherent photons) for generation of single-qubit gates for a two-level open quantum system which evolves according to the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation with time-dependent coefficients determined by these coherent and incoherent controls. The control problem is formulated as minimization of the objective functional, which is the sum of Hilbert-Schmidt norms between four fixed basis states evolved under the GKSL master equation with controls and the same four states evolved under the ideal gate transformation. The exact expression for the gradient of the objective functional with respect to piecewise constant controls is obtained. Subsequent optimization is performed using a gradient type algorithm with an adaptive step size that leads to oscillating behaviour of the gradient norm vs. iterations. Optimal trajectories in the Bloch ball for various initial states are computed. A relation of quantum gate generation with optimization on complex Stiefel manifolds is discussed. We develop methodology and apply it here for unitary gates as a testing example. The next step is to apply the method for generation of non-unitary processes and to multi-level quantum systems. Full article

Due to the increasing demand for fast data rates and large spectra, millimeter-wave technology plays a vital role in the advancement of 5G communication. The idea behind Mm-Wave communications is to take advantage of the huge and unexploited bandwidth to cope with future multigigabit-per-second mobile data rates, imaging, and multimedia applications. In Mm-Wave systems, digital precoding provides optimal performance at the cost of complexity and power consumption. Therefore, hybrid precoding, i.e., analog–digital precoding, has received significant consideration as a favorable alternative to digital precoding. The conventional methods related to hybrid precoding suffer from low spectral efficiency and large processing time due to nested loops and the number of iterations. A manifold optimization-based algorithm using the gradient method is proposed to increase the spectral efficiency to be near optimal and to speed up the processing speed. A comparison of performances is shown using the simulation outcomes of the proposed work and those of the existing techniques. Full article

Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom. Full article

Deep learning has achieved many successes in different fields but can sometimes encounter an overfitting problem when there are insufficient amounts of labeled samples. In solving the problem of learning with limited training data, meta-learning is proposed to remember some common knowledge by leveraging a large number of similar few-shot tasks and learning how to adapt a base-learner to a new task for which only a few labeled samples are available. Current meta-learning approaches typically uses Shallow Neural Networks (SNNs) to avoid overfitting, thus wasting much information in adapting to a new task. Moreover, the Euclidean space-based gradient descent in existing meta-learning approaches always lead to an inaccurate update of meta-learners, which poses a challenge to meta-learning models in extracting features from samples and updating network parameters. In this paper, we propose a novel meta-learning model called Multi-Stage Meta-Learning (MSML) to post the bottleneck during the adapting process. The proposed method constrains a network to Stiefel manifold so that a meta-learner could perform a more stable gradient descent in limited steps so that the adapting process can be accelerated. An experiment on the mini-ImageNet demonstrates that the proposed method reached a better accuracy under 5-way 1-shot and 5-way 5-shot conditions. Full article

Sparse dictionary learning (SDL) is a classic representation learning method and has been widely used in data analysis. Recently, the ${\ell }_m$ -norm ( $m\ge 3,m\in \mathbb{N}$ ) maximization has been proposed to solve SDL, which reshapes the problem to an optimization problem with orthogonality constraints. In this paper, we first propose an ${\ell }_m$ -norm maximization model for solving dual principal component pursuit (DPCP) based on the similarities between DPCP and SDL. Then, we propose a smooth unconstrained exact penalty model and show its equivalence with the ${\ell }_m$ -norm maximization model. Based on our penalty model, we develop an efficient first-order algorithm for solving our penalty model (PenNMF) and show its global convergence. Extensive experiments illustrate the high efficiency of PenNMF when compared with the other state-of-the-art algorithms on solving the ${\ell }_m$ -norm maximization with orthogonality constraints. Full article

We develop novel multivariate state-space models wherein the latent states evolve on the Stiefel manifold and follow a conditional matrix Langevin distribution. The latent states correspond to time-varying reduced rank parameter matrices, like the loadings in dynamic factor models and the parameters of cointegrating relations in vector error-correction models. The corresponding nonlinear filtering algorithms are developed and evaluated by means of simulation experiments. Full article

Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconis, and Newman: Let the random matrix H_n be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let A_n be a non-random n × n real matrix such that tr (A'_nA_n) = 1. Then, as n→∞, √n tr A_nH_n converges in distribution to the standard normal distribution. Full article