Geometric framework for SPD matrices preserving subspace structures.
problem Processing SPD-valued data with preserved subspace structures.
method Thompson geometry of the semidefinite cone, extreme generalized eigenvalues, geodesic space structure.
result Novel inductive mean of SPD matrices based on Thompson geometry.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Study reveals LLM personas have two distinct components: frame-robust aggregated traits and frame-dependent geometric features.
problem Evaluation of LLM personas via psychometric questionnaires discards within-instance correlation structure.
method Constructed within-instance correlation matrices from IPIP-50 responses and analyzed geometry on SPD manifolds under manipulated question orderings.
result Persona expression comprises two dissociable components: aggregated features (Big Five scores) and geometric features (SPD manifold).
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.
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.
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.
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…
The paper uses deep learning to detect financial market regimes from correlation matrices.
problem Detecting financial market regimes from correlation dynamics.
method Representation learning on block hierarchical SPD correlation matrices using SPDNet, SPD-NetBN, and U-SPDNet models.
result Deep learning models overfit in financial market data, misleading performance metrics.
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.
A new algorithm improves sampling for graph learning models.
problem Euclidean proposals struggle near the boundary of PSD matrices.
method ConeMALA, a geometry-aware Langevin algorithm.
result ConeMALA achieves higher ESS/sec and stable diagnostics.
New kernel interprets 3D anisotropic data with rotations and improved predictions.
problem Capturing rotated anisotropy in 3D spatial fields.
method Introduces a Lie-algebraic kernel with three principal length-scales and an explicit rotation.
result Posterior recovers rotated anisotropy and improves prediction over axis-aligned kernels.
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.
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.
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.
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.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued representations often relies on Euclidean approximations or costly geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs.
result Generalizes batch normalization and multinomial logistic regression to broader classes of manifolds.
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.
A tractable pseudo-metric for non-parametric distributions via SPD geometry.
problem Computing distances between non-parametric probability distributions is intractable.
method Two-stage framework: projection onto parametric family, embedding into SPD matrices.
result Closed-form pseudo-metric for two-sample hypothesis testing.
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…
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.
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…
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.
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.
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
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.
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.
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.
We study the online estimation of the optimal policy of a Markov decision process (MDP). We propose a class of Stochastic Primal-Dual (SPD) methods which exploit the inherent minimax duality of Bellman equations. The SPD methods update a few coordinates of the value and policy estimates as a new state transition is obs…
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.
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.
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…
We present a novel framework for domain adaptation, whereby both geometric and statistical differences between a labeled source domain and unlabeled target domain can be integrated by exploiting the curved Riemannian geometry of statistical manifolds. Our approach is based on formulating transfer from source to target …
New k-means method clusters radar image sequences using SPD matrices.
problem Clustering radar image sequences efficiently.
method Developed k-means on SPD matrices for non-Euclidean data. result Effective clustering of radar image sequences via SPD matrices.
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…
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.
ITSPACE improves covariance alignment faster than other methods.
problem Optimizing covariance matrices for machine learning tasks.
method Proximal majorization-minimization method that directly optimizes the Bures-Wasserstein objective.
result ITSPACE achieves lower BW gap solutions faster than other methods.
Noise titration benchmarks time series forecasting models rigorously.
problem Evaluation of time series forecasting models is often flawed due to lack of interventionist methods.
method Interventionist benchmarking using Gaussian noise titration of dynamical systems.
result Fern model outperforms state-of-the-art models in non-stationary conditions.