Realtime finite element modeling of bridges assists modern structural health monitoring systems by providing comprehensive insights into structural integrity. This capability is essential for ensuring the safe operation of bridges and preventing sudden catastrophic failures. However, FEM computational cost and the need for realtime analysis pose significant challenges. Additionally, the input data is a 7 dimensional vector, while the output is a 1017 dimensional vector, making accurate and efficient analysis particularly difficult. In this study, we propose a novel hybrid quantum classical Multilayer Perceptron pipeline leveraging Symmetric Positive Definite matrices and Riemannian manifolds for effective data representation. To maintain the integrity of the qubit structure, we utilize SPD matrices, ensuring data representation is well aligned with the quantum computational framework. Additionally, the method leverages polynomial feature expansion to capture nonlinear relationships within the data. The proposed pipeline combines classical fully connected neural network layers with quantum circuit layers to enhance model performance and efficiency. Our experiments focused on various configurations of such hybrid models to identify the optimal structure for accurate and efficient realtime analysis. The best performing model achieved a Mean Squared Error of 0.00031, significantly outperforming traditional methods.
Department of Information Engineering and Computer Science, University of Trento（信息工程与计算机科学系，特伦托大学）；Computing and Mathematical Sciences, Caltech（计算与数学科学系，加州理工学院）；School of Artificial Intelligence and Computer Science, Jiangnan University（人工智能与计算机科学学院，江南大学）
Riemannian Networks over Full-Rank Correlation Matrices
Department of Information Engineering and Computer Science, University of Trento, Trento, Italy（特伦托大学信息工程与计算机科学系）；School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, China（江南大学人工智能与计算机科学学院）
CY Cergy Paris Université（赛尔齐-蓬图瓦兹大学）；ENSEA（法国高等电力学院）；CNRS（法国国家科学研究中心）
In this paper, we propose a novel method for representation and classification of two-person interactions from 3D skeleton sequences. The key idea of our approach is to use Gaussian distributions to capture statistics on R n and those on the space of symmetric positive definite (SPD) matrices. The main challenge is how to parametrize those distributions. Towards this end, we develop methods for embedding Gaussian distributions in matrix groups based on the theory of Lie groups and Riemannian symmetric spaces. Our method relies on the Riemannian geometry of the underlying manifolds and has the …
Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry
School of Artificial Intelligence and Computer Science, Jiangnan University（江南大学人工智能与计算机学院）；Department of Information Engineering and Computer Science, University of Trento（特伦托大学信息工程与计算机科学系）
SPDAlign: Interpretable Riemannian Alignment for EEG Forward Modeling Shifts
BIFOLD（柏林智能数据与机器学习研究所）；Technical University of Berlin（柏林工业大学）；Advanced Telecommunications Research Institute International（国际电气通信基础技术研究所）；RIKEN AIP（理化学研究所先进智能研究中心）；Nanyang Technological University（南洋理工大学）
In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry with the popular Affine-Invariant (AI) geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold. Our study begins with an observation that the BW metric has a linear dependence on SPD matrices in contrast to the quadratic dependence of the AI metric. We build on this to show that the BW metric is a more suitable and robust choice for several Riemannian optimization problems over ill-conditioned SPD matrices. We show that the BW geometry has a non-negative curvature, which further im…
Towards Robust EEG Decoding Based on Riemannian Self-Attention
School of Artificial Intelligence and Computer Science（人工智能与计算机科学学院）；Jiangnan University（江南大学）；Department of Information Engineering and Computer Science（信息工程与计算机科学系）；University of Trento（特伦托大学）；Centre for Vision, Speech and Signal Processing (CVSSP)（视觉、语音与信号处理中心）；University of Surrey（萨里大学）
CNRS（法国国家科学研究中心）；University of Lille（里尔大学）；Centrale Lille（里尔中央理工学院）；IMT Nord Europe（法国北部高等矿业电信学院）；Institut Mines-Télécom（法国矿业电信学院）；INSERM（法国国家健康与医学研究院）
Existing multimodal stress/pain recognition approaches generally extract features from different modalities independently and thus ignore cross-modality correlations. This paper proposes a novel geometric framework for multimodal stress/pain detection utilizing Symmetric Positive Definite (SPD) matrices as a representation that incorporates the correlation relationship of physiological and behavioural signals from covariance and cross-covariance. Considering the non-linearity of the Riemannian manifold of SPD matrices, well-known machine learning techniques are not suited to classify these mat…
Technion, Israel Institute of Technology（以色列理工学院）；Technical University of Munich（慕尼黑工业大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
In this paper, we address the problem of Domain Adaptation (DA) using Optimal Transport (OT) on Riemannian manifolds. We model the difference between two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for DA in a well-defined sense, up to a volume preserving map. We then focus on the manifold of Symmetric and Positive-Definite (SPD) matrices, whose structure provided a useful context in recent applications. We demonstrate the polar factorization theorem on this manifold. Due to the uniqueness of the weighted Riemannian mean, and by exploi…
Technion – Israel Institute of Technology（以色列理工学院）；Yale University（耶鲁大学）
In this paper, we consider data acquired by multimodal sensors capturing complementary aspects and features of a measured phenomenon. We focus on a scenario in which the measurements share mutual sources of variability but might also be contaminated by other measurement-specific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear data-driven dimension reduction methods, with the well-known Riemannian geometry of symmetric and positive-definite (SPD) matrices. Manifold learning typically includes the spectral analysis of a kernel built …
A Unified SPD Token Transformer Framework for EEG Classification: Systematic Comparison of Geometric Embeddings
Independent Researcher（独立研究者）；Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan（国家阳明交通大学电子物理系）；Department of Mathematics, Boise State University, Boise, ID, USA（数学系，博伊州立大学）
Recognition of electroencephalographic (EEG) signals highly affect the efficiency of non-invasive brain-computer interfaces (BCIs). While recent advances of deep-learning (DL)-based EEG decoders offer improved performances, the development of geometric learning (GL) has attracted much attention for offering exceptional robustness in decoding noisy EEG data. However, there is a lack of studies on the merged use of deep neural networks (DNNs) and geometric learning for EEG decoding. We herein propose a manifold attention network (mAtt), a novel geometric deep learning (GDL)-based model, featurin…
Emory University（埃默里大学）；University of Chicago（芝加哥大学）；Georgia Institute of Technology（佐治亚理工学院）；University of International Business and Economics（对外经济贸易大学）
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）
Covariance data as represented by symmetric positive definite (SPD) matrices are ubiquitous throughout technical study as efficient descriptors of interdependent systems. Euclidean analysis of SPD matrices, while computationally fast, can lead to skewed and even unphysical interpretations of data. Riemannian methods preserve the geometric structure of SPD data at the cost of expensive eigenvalue computations. In this paper, we propose a geometric method for unsupervised clustering of SPD data based on the Thompson metric. This technique relies upon a novel "inductive midrange" centroid computa…
Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds
Nanyang Technological University, Singapore（南洋理工大学，新加坡）；Indian Institute of Technology Bombay, India（印度理工学院班加罗尔，印度）；University of Sydney, Australia（悉尼大学，澳大利亚）；Microsoft India（微软印度）
Monash University Malaysia（莫纳什大学马来西亚分校）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；University College London（伦敦大学学院）
Common measures of brain functional connectivity (FC) including covariance and correlation matrices are semi-positive definite (SPD) matrices residing on a cone-shape Riemannian manifold. Despite its remarkable success for Euclidean-valued data generation, use of standard generative adversarial networks (GANs) to generate manifold-valued FC data neglects its inherent SPD structure and hence the inter-relatedness of edges in real FC. We propose a novel graph-regularized manifold-aware conditional Wasserstein GAN (GR-SPD-GAN) for FC data generation on the SPD manifold that can preserve the globa…
Ferdowsi University of Mashhad（马什哈德菲尔多西大学）；University of Waterloo（滑铁卢大学）；University of Tehran（德黑兰大学）
This work concentrates on optimization on Riemannian manifolds. The Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm is a commonly used quasi-Newton method for numerical optimization in Euclidean spaces. Riemannian LBFGS (RLBFGS) is an extension of this method to Riemannian manifolds. RLBFGS involves computationally expensive vector transports as well as unfolding recursions using adjoint vector transports. In this article, we propose two mappings in the tangent space using the inverse second root and Cholesky decomposition. These mappings make both vector transport and adjoin…
Mitsubishi Electric Research Labs (MERL)（三菱电机研究实验室）；Australian Centre for Robotic Vision, Australian National University（澳大利亚国立大学澳大利亚机器人视觉中心）；University of Minnesota（明尼苏达大学）；Research School of Engineering, The Australian National University（澳大利亚国立大学工程研究学院）；Monash University（莫纳什大学）；Data61-CSIRO（澳大利亚联邦科学与工业研究组织数据61）
Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There exist several similarity measures for comparing SPD matrices with documented benefits. However, selecting an appropriate measure for a given problem remains a challenge and in most cases, is the result of a trial-and-error process. In this paper, we propose to learn similarity measures in a data-driven manner. To this end, we capitalize on the αβ-log-det diverg…
IBM Research（IBM研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）
Connectomics has emerged as a powerful tool in neuroimaging and has spurred recent advancements in statistical and machine learning methods for connectivity data. Despite connectomes inhabiting a matrix manifold, most analytical frameworks ignore the underlying data geometry. This is largely because simple operations, such as mean estimation, do not have easily computable closed-form solutions. We propose a geometrically aware neural framework for connectomes, i.e., the mSPD-NN, designed to estimate the geodesic mean of a collections of symmetric positive definite (SPD) matrices. The mSPD-NN i…
