Symmetric Positive Semi-definite Riemannian Geometry with Application to Domain Adaptation

• URL: http://arxiv.org/abs/2007.14272v2
• Date: Tue, 4 Aug 2020 16:00:03 GMT
• Title: Symmetric Positive Semi-definite Riemannian Geometry with Application to Domain Adaptation
• Authors: Or Yair, Almog Lahav, and Ronen Talmon
• Abstract summary: We present new results on the geometry of symmetric positive semi-definite (SPSD) matrices. We propose an algorithm for Domain Adaptation (DA) and demonstrate its performance in two applications: fusion of hyper-spectral images and motion identification.
• Score: 7.126737403006778
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we present new results on the Riemannian geometry of symmetric positive semi-definite (SPSD) matrices. First, based on an existing approximation of the geodesic path, we introduce approximations of the logarithmic and exponential maps. Second, we present a closed-form expression for Parallel Transport (PT). Third, we derive a canonical representation for a set of SPSD matrices. Based on these results, we propose an algorithm for Domain Adaptation (DA) and demonstrate its performance in two applications: fusion of hyper-spectral images and motion identification.

Related papers

• Relative Representations: Topological and Geometric Perspectives [53.88896255693922]
  Relative representations are an established approach to zero-shot model stitching. We introduce a normalization procedure in the relative transformation, resulting in invariance to non-isotropic rescalings and permutations. Second, we propose to deploy topological densification when fine-tuning relative representations, a topological regularization loss encouraging clustering within classes.
  arXiv  Detail & Related papers  (2024-09-17T08:09:22Z)
• Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry [63.694184882697435]
  Global Covariance Pooling (GCP) has been demonstrated to improve the performance of Deep Neural Networks (DNNs) by exploiting second-order statistics of high-level representations.
  arXiv  Detail & Related papers  (2024-07-15T07:11:44Z)
• Regularized Projection Matrix Approximation with Applications to Community Detection [1.3761665705201904]
  This paper introduces a regularized projection matrix approximation framework designed to recover cluster information from the affinity matrix. We investigate three distinct penalty functions, each specifically tailored to address bounded, positive, and sparse scenarios. Numerical experiments conducted on both synthetic and real-world datasets reveal that our regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.
  arXiv  Detail & Related papers  (2024-05-26T15:18:22Z)
• The Fisher-Rao geometry of CES distributions [50.50897590847961]
  The Fisher-Rao information geometry allows for leveraging tools from differential geometry. We will present some practical uses of these geometric tools in the framework of elliptical distributions.
  arXiv  Detail & Related papers  (2023-10-02T09:23:32Z)
• Adaptive Log-Euclidean Metrics for SPD Matrix Learning [73.12655932115881]
  We propose Adaptive Log-Euclidean Metrics (ALEMs), which extend the widely used Log-Euclidean Metric (LEM) The experimental and theoretical results demonstrate the merit of the proposed metrics in improving the performance of SPD neural networks.
  arXiv  Detail & Related papers  (2023-03-26T18:31:52Z)
• Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals [24.798859309715667]
  We propose a new method to deal with distributions of covariance matrices. We show that it is an efficient surrogate to the Wasserstein distance in domain adaptation for Brain Computer Interface applications.
  arXiv  Detail & Related papers  (2023-03-10T09:08:46Z)
• GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition [2.817412580574242]
  We propose a novel method for representation and classification of two-person interactions from 3D skeleton sequences. We show that the proposed method achieves competitive results in two-person interaction recognition on three benchmarks for 3D human activity understanding.
  arXiv  Detail & Related papers  (2021-11-25T13:57:43Z)
• On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization [5.876141028192136]
  This paper proposes a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. We observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization. Results provide a few new theoretical insights to unanswered questions in the literature.
  arXiv  Detail & Related papers  (2021-10-23T03:13:56Z)
• Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization [8.59387261480044]
  We consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We show the sandwich relation can be used to transfer more quantitative geometric properties from one formulation to another.
  arXiv  Detail & Related papers  (2021-08-03T22:14:01Z)
• q-Paths: Generalizing the Geometric Annealing Path using Power Means [51.73925445218366]
  We introduce $q$-paths, a family of paths which includes the geometric and arithmetic mixtures as special cases. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference.
  arXiv  Detail & Related papers  (2021-07-01T21:09:06Z)
• Improving Metric Dimensionality Reduction with Distributed Topology [68.8204255655161]
  DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets.
  arXiv  Detail & Related papers  (2021-06-14T17:19:44Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.