The goal of this paper is to show how different machine learning tools on the Riemannian manifold $\mathcal{P}_d$ of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several Gaussian distributions defined on $\mathcal{P}_d$. We will show how popular classifiers on $\mathcal{P}_d$ can be reinterpreted as Bayes Classifiers using these Gaussian distributions. These distributions will also be used for outlier detection and dimension reduction. By showing that those distributions are pervasive in the tools used on $\mathcal{P}_d$, we allow for other machine learning tools to be extended to $\mathcal{P}_d$.
Differential privacy has become crucial in the real-world deployment of statistical and machine learning algorithms with rigorous privacy guarantees. The earliest statistical queries, for which differential privacy mechanisms have been developed, were for the release of the sample mean. In Geometric Statistics, the sample Fréchet mean represents one of the most fundamental statistical summaries, as it generalizes the sample mean for data belonging to nonlinear manifolds. In that spirit, the only geometric statistical query for which a differential privacy mechanism has been developed, so far, …
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 …
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…
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…
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）
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 …
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）
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）
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…
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）
Shenyang Institute of Automation, Chinese Academy of Sciences（中国科学院沈阳自动化研究所）；Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences（中国科学院机器人与智能制造创新研究院）；University of Chinese Academy of Sciences（中国科学院大学）；University of Birmingham（伯明翰大学）；Beijing Normal University（北京师范大学）；Zhejiang University（浙江大学）
In this paper, we develop a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. In many classification scenarios, the data can be naturally represented by symmetric positive definite matrices, which are inherently points that live on a curved Riemannian manifold. Due to the non-Euclidean geometry of Riemannian manifolds, traditional Euclidean machine learning algorithms yield poor results on such data. In this paper, we generalize the probabilistic learning vector quantization algorithm for data points living on the manifold of sym…
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…
Bosch Center for Artificial Intelligence（博世人工智能中心）；University of Leoben（莱奥本大学）
Learning complex robot motions necessarily demands to have models that are able to encode and retrieve full-pose trajectories when tasks are defined in operational spaces. Probabilistic movement primitives (ProMPs) stand out as a principled approach that models trajectory distributions learned from demonstrations. ProMPs allow for trajectory modulation and blending to achieve better generalization to novel situations. However, when ProMPs are employed in operational space, their original formulation does not directly apply to full-pose movements including rotational trajectories described by q…
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）
SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges
Inria（法国国家信息与自动化研究所）；CEA（法国国家原子能与替代能源委员会）；Université Paris-Saclay（巴黎萨克雷大学）；Advanced Telecommunications Research Institute International (ATR)（国际先进电信研究机构）；RIKEN Artificial Intelligence Project（理化学研究所人工智能项目）；College of Computing and Data Science (CCDS) and Centre of AI in Medicine (C-AIM)（计算与数据科学学院和医学人工智能中心）
Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope
本文提出了一种新的几何和概率方法，用于在多个对象或图像集合之间同步对应关系。核心方法包括基于Birkhoff-Riemannian L-BFGS优化放松后的循环一致性损失，以及基于Birkhoff-Riemannian Langevin Monte Carlo生成Birkhoff多面体样本并估计解的置信度。
Deep Riemannian Networks for End-to-End EEG Decoding
University of Freiburg（弗莱堡大学）；Medical Center - University of Freiburg（弗莱堡大学医学中心）；Faculty of Medicine, University of Freiburg（弗莱堡大学医学院）；Institute for Machine-Brain Interfacing Technology（机器脑接口技术研究所）
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…
Center for Data Science, Peking University（数据科学中心，北京大学）；Zuse Institute Berlin（柏林Zuse研究所）；Institute of Mathematics, Freie Universität Berlin（柏林自由大学数学研究所）；LMAM and School of Mathematical Sciences, Center for Machine Learning Research and Center for Data Science, Peking University（LMAM和数学系，机器学习研究中心和数据科学中心，北京大学）
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…
