Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
The paper explores totally geodesic submanifolds in SPD matrices and their properties.
problem Characterizing and understanding totally geodesic submanifolds in SPD matrices.
method Detailed geometric analysis and projection properties of SPD matrices.
result A non-linear projection on totally geodesic submanifolds has the minimizing property.
New k k k -means method clusters radar image sequences using SPD matrices.
problem Clustering radar image sequences efficiently.
method Developed k k k -means on SPD matrices for non-Euclidean data. result Effective clustering of radar image sequences via SPD matrices.
We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all posi…
New geometric structures defined on SPD matrices for better understanding.
problem Understanding SPD matrices and their geometric properties.
method Introducing Finslerian and dual information-geometric structures on James' bicone domain.
result Geodesics correspond to straight lines in coordinate systems, and new dissimilarities generalize existing ones.
SpodNet learns SPD matrices with structural constraints.
problem Estimating SPD matrices with additional structural constraints.
method Introduces SpodNet, a neural network module that guarantees SPD outputs and supports structural constraints.
result SpodNet learns SPD and sparse matrices effectively.
Researchers approximate partition functions on Riemannian spaces in the large N limit.
problem Computing normalization factors (partition functions) on Riemannian symmetric spaces is challenging.
method Approximation techniques in the large N limit, including saddle-point equations.
result Formulas for leading order terms in the large N limit of SPD matrices and related spaces.
Paper proposes SPD-DDPM for SPD matrices, improving on previous discriminative models.
problem Challenges in handling large-scale SPD matrix data for discriminative models.
method Introduces a generative model using Gaussian distribution in SPD space, allowing unconditional and conditional predictions.
result Effective fitting of data distribution and accurate predictions on both conditional and unconditional data.
Improved method for computing Fréchet means on SPD matrices.
problem Computing Fréchet means on the manifold of SPD matrices.
method Random matrix theory-based approach for estimating Fréchet means.
result Significantly outperforms state-of-the-art methods in experiments.
We introduce a wrapped Gaussian for SPD matrices, enhancing data analysis.
problem Handling circular and non-flat data distributions on SPD manifolds.
method Introduced a non-isotropic wrapped Gaussian using the exponential map, derived theoretical properties, and proposed a maximum likelihood framework.
result Demonstrated the robustness and flexibility of the wrapped Gaussian model on synthetic and real-world datasets.
A new geometric method for clustering SPD data improves upon Euclidean and Riemannian approaches.
problem Skewed interpretations of SPD data in Euclidean analysis and computational inefficiency of Riemannian methods.
method Proposes a geometric method based on the Thompson metric for unsupervised clustering of SPD data.
result Demonstrates improved clustering results using inductive midrange centroid computation.
Researchers developed a new Riemannian manifold for SPD matrix-valued optimal transport problems.
problem Optimal transport between SPD matrix-valued measures.
method Formulated as a generalized optimal transport problem with block SPD matrices, endowed with a novel Riemannian manifold structure.
result The novel Riemannian manifold allows solving SPD matrix-valued optimal transport problems using Riemannian optimization.
A new mechanism for differentially private Fréchet mean on SPD matrices.
problem Privacy-preserving statistical summaries for SPD matrices.
method Tangent Gaussian mechanism for log-Euclidean metric.
result Significantly better utility and computational efficiency.
We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating t…
Structured regularizers enable faster optimization on SPD manifolds with constraints.
problem Optimizing SPD matrices with additional constraints.
method Structured regularizers based on symmetric gauge functions.
result Structured regularizers can preserve or induce desirable structure like convexity.
FedSPDnet improves federated learning for SPD matrices, outperforming existing methods.
problem Federated learning for SPD matrices with orthogonality constraints.
method Two efficient aggregation strategies: ProjAvg and RLAvg, preserving geometric structure.
result FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation.
The paper presents a probabilistic framework for SPD matrices in machine learning.
problem Machine learning on SPD matrices is fragmented; this paper aims to unify it.
method Unified probabilistic framework using Gaussian distributions and Bayes classifiers.
result Different SPD machine learning tools can be reinterpreted and extended using Gaussian distributions.
Python package for SPD matrix distances, reproducible and extensible.
problem Computing distances between SPD matrices for various applications.
method Unified, extensible framework supporting multiple SPD metrics.
result Reproducible and accessible SPD matrix comparison tool.
New metrics defined on SPD matrices link to divergences and curvature.
problem Defining and characterizing metrics on SPD matrices.
method Developed a principle of deformed metrics and introduced balanced bilinear forms.
result Introduce Mixed-Euclidean metrics with negative sectional curvature.
Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent space…
New scalable geometric framework for SPD matrices.
problem Costly spectral computations in SPD matrix analysis.
method Efficient computation of extreme generalized eigenvalues through Hilbert and Thompson geometries of the semidefinite cone.
result Existence and uniqueness of a novel iterative mean of SPD matrices.
Efficiently clusters data on manifolds using Fréchet maps.
problem Clustering on high-dimensional, non-Euclidean manifolds is computationally challenging.
method Introduces p p p -Fréchet map to embed manifold data into Euclidean space for k-means clustering. result Significant performance gains in runtime and accuracy compared to existing methods.
Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the choice of one particular metric for a given application. Invariance under the acti…
Proposes a neural network for recognizing 3D skeleton-based interactions.
problem Recognizing two-person interactions from 3D skeleton sequences.
method Uses Gaussian distributions and Riemannian geometry of SPD matrices and matrix groups.
result Achieves competitive results on three benchmarks for 3D human activity understanding.
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.
problem Limited coverage of O(n)-invariant metrics by kernel metrics.
method Characterization of O(n)-invariant metrics, intermediate classes construction.
result Introduction of cometric-stability as a key property for geodesics.
A new method compares unaligned datasets using log-Euclidean signatures of SPD matrices.
problem Efficiently comparing datasets with unknown alignment.
method Diffusion operators, Riemannian geometry, log-Euclidean metric.
result LES distance recovers meaningful structural differences, outperforming existing methods.
A new coordinate system for SPD matrices simplifies computations and generative modeling.
problem Computing and modeling SPD matrices
method Reverse telescoping coordinate system
result Significantly reduces computational complexity and facilitates generative modeling.
The paper uses spectral flow on SPD matrices to analyze multimodal data.
problem Analyzing data from multiple sensors with shared and unique sources.
method Combines manifold learning with Riemannian geometry of SPD matrices.
result Spectral analysis of kernels on SPD manifold reveals common and unique components.
Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle …
The present paper proposes generalized Gaussian kernel adaptive filtering, where the kernel parameters are adaptive and data-driven. The Gaussian kernel is parametrized by a center vector and a symmetric positive definite (SPD) precision matrix, which is regarded as a generalization of the scalar width parameter. These…
This paper studies geometric properties of Wasserstein metric on SPD(n).
problem Understanding the geometry of symmetric positive-definite matrices under Wasserstein metric.
method Using fiber bundles, the paper derives explicit geometric quantities and proves global properties.
result The manifold is globally geodesic convex with non-negative curvatures but no conjugate pair and cut locus.
The paper proposes methods to find a shared active subspace for multivariate vector-valued functions.
problem Minimizing the deviation between function evaluations in the original and reconstructed spaces.
method Manipulating gradients or SPD matrices to identify a shared structure.
result Summing SPD matrices often identifies the best shared active subspace.
Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the Riemannian manifold of symmetric positive definite (SPD) matrices they belong to. In…
Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
Paper builds neural networks on matrix manifolds using gyrovector spaces.
problem Lack of concepts in gyrovector spaces for matrix manifolds.
method Generalized gyrovector space concepts for SPD and Grassmann manifolds, proposing new neural network models.
result Demonstrated effectiveness in human action recognition and knowledge graph completion.
Researchers develop geodesics for a new metric on correlation matrices.
problem Lack of intrinsic tools for statistical analyses of correlation matrices.
method Developed geodesics for the quotient-affine metric on full-rank correlation matrices.
result Provided fundamental Riemannian operations for the quotient-affine metric.
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. …
New method optimizes on curved manifolds without curvature dependence.
problem Curvature-dependent regret in online optimization on Hadamard manifolds.
method Riemannian online gradient descent for h-convex functions.
result Established O ( T ) O(\sqrt{T}) O ( T ) and O ( log ( T ) ) O(\log(T)) O ( log ( T )) regret guarantees, curvature-independent. GOPSA optimizes EEG data for cross-site age prediction, improving performance on multiple metrics.
problem Predictive shifts in EEG data from different sites and participants.
method Geodesic Optimization for Predictive Shift Adaptation (GOPSA) on the SPD manifold.
result Significantly higher performance on age prediction metrics compared to state-of-the-art methods.
New SPD metrics improve stability and efficiency in neural networks.
problem Designing stable and efficient Riemannian metrics on SPD manifolds.
method Cholesky decomposition to derive SPD metrics.
result Proposed metrics provide closed-form operators, computational efficiency, and improved numerical stability.
New algorithm reduces complexity for SPD manifold optimization.
problem Efficiently minimize functions over SPD manifold.
method Low-complexity Riemannian subspace descent with sparse updates.
result Innovative updates avoid costly matrix operations.
Riemannian geometry has been applied to Brain Computer Interface (BCI) for brain signals classification yielding promising results. Studying electroencephalographic (EEG) signals from their associated covariance matrices allows a mitigation of common sources of variability (electronic, electrical, biological) by constr…
This research simplifies Riemannian LBFGS for SPD manifolds.
problem Optimization on Riemannian manifolds, especially SPD.
method Two mappings for tangent space, making vector transports and adjoint vector transports identity.
result RLBFGS becomes less computationally expensive and easier to analyze.
Enhanced EEG classification improves motor imagery detection with less computation.
problem Improving classification accuracy of motor imagery EEG signals.
method Integrates Block-Toeplitz structure into augmented covariance matrices and uses Siegel metric.
result Significantly reduces computational time without compromising classification accuracy.
Designs neural networks on matrix manifolds for improved performance in tasks like human action recognition.
problem Designing neural networks for tasks on non-Euclidean manifolds.
method Develops fully-connected and convolutional layers for SPD manifolds, and MLR on SPSD manifolds.
result Demonstrates improved performance in human action recognition and node classification tasks.
This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.
problem Understanding which mean to use for symmetrizing Bregman divergences on positive definite matrices.
method Axiomatic definition of mean functionals and variational principles over the cone of positive definite matrices.
result The arithmetic mean is canonical for forward symmetrization, and the arithmetic, log-Euclidean, and harmonic means for reverse symmetrization.
Dictionary leaning (DL) and dimensionality reduction (DR) are powerful tools to analyze high-dimensional noisy signals. This paper presents a proposal of a novel Riemannian joint dimensionality reduction and dictionary learning (R-JDRDL) on symmetric positive definite (SPD) manifolds for classification tasks. The joint…