Chordal Averaging on Flag Manifolds and Its Applications

• URL: http://arxiv.org/abs/2303.13501v2
• Date: Mon, 17 Jul 2023 18:27:49 GMT
• Title: Chordal Averaging on Flag Manifolds and Its Applications
• Authors: Nathan Mankovich and Tolga Birdal
• Abstract summary: This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric.
• Score: 22.357999963733302
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.

Related papers

• A Flag Decomposition for Hierarchical Datasets [17.424354744499695]
  Flag manifold encode hierarchical nested sequences of subspaces. Current applications are often restricted to extracting flags using common matrix decomposition methods. We propose a novel, flag-based method that decomposes arbitrary hierarchical real-preserving data into a hierarchy-valued flag representation in Stiefel coordinates.
  arXiv  Detail & Related papers  (2025-02-11T18:59:52Z)
• Nested subspace learning with flags [0.03295497357381917]
  We propose a simple trick to enforce nestedness in subspace learning methods. We apply the flag trick to several classical machine learning methods and show that it successfully addresses the nestedness issue.
  arXiv  Detail & Related papers  (2025-02-09T20:29:56Z)
• Categorical Schrödinger Bridge Matching [58.760054965084656]
  The Schr"odinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. We provide a theoretical and algorithmic foundation for solving SB in discrete spaces using the recently introduced Iterative Markovian Fitting (IMF) procedure. This enables us to develop a practical computational algorithm for SB which we call Categorical Schr"odinger Bridge Matching (CSBM)
  arXiv  Detail & Related papers  (2025-02-03T14:55:28Z)
• Disentangled Representation Learning with the Gromov-Monge Gap [65.73194652234848]
  Learning disentangled representations from unlabelled data is a fundamental challenge in machine learning. We introduce a novel approach to disentangled representation learning based on quadratic optimal transport. We demonstrate the effectiveness of our approach for quantifying disentanglement across four standard benchmarks.
  arXiv  Detail & Related papers  (2024-07-10T16:51:32Z)
• Spectral Norm of Convolutional Layers with Circular and Zero Paddings [55.233197272316275]
  We generalize the use of the Gram iteration to zero padding convolutional layers and prove its quadratic convergence. We also provide theorems for bridging the gap between circular and zero padding convolution's spectral norm.
  arXiv  Detail & Related papers  (2024-01-31T23:48:48Z)
• Fun with Flags: Robust Principal Directions via Flag Manifolds [19.034855801255837]
  Principal component analysis (PCA) has been indispensable in computer vision and machine learning. We present a unifying formalism for PCA and its variants, and introduce a framework based on the flags of linear subspaces.
  arXiv  Detail & Related papers  (2024-01-08T18:18:02Z)
• Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold [59.73080197971106]
  This paper presents a first-order conjugate optimization method that converges faster than the steepest descent method. It aims to achieve global convergence over the Stiefel manifold.
  arXiv  Detail & Related papers  (2023-08-21T08:02:16Z)
• The Flag Median and FlagIRLS [1.5943398589344562]
  This work proposes a new subspace prototype, the flag median, and introduces the FlagIRLS algorithm for its calculation. We provide evidence that the flag median is robust to outliers and can be used effectively in algorithms like Linde-Buzo-GreyLBG. We find that using FlagIRLS to compute the flag median converges in $4$ on a synthetic dataset.
  arXiv  Detail & Related papers  (2022-03-08T23:06:58Z)
• Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields [51.07460861448716]
  This paper presents a convex-analytic framework to learn from data. We show that a triangular convexity decomposition is guaranteed by a transform of the corresponding to its upper part.
  arXiv  Detail & Related papers  (2021-09-17T17:46:12Z)
• The flag manifold as a tool for analyzing and comparing data sets [4.864283334686195]
  The shape and orientation of data clouds reflect variability in observations that can confound pattern recognition systems. We show how nested subspace methods, utilizing flag manifold, can help to deal with such additional confounding factors.
  arXiv  Detail & Related papers  (2020-06-24T22:29:02Z)
• Spatial Pyramid Based Graph Reasoning for Semantic Segmentation [67.47159595239798]
  We apply graph convolution into the semantic segmentation task and propose an improved Laplacian. The graph reasoning is directly performed in the original feature space organized as a spatial pyramid. We achieve comparable performance with advantages in computational and memory overhead.
  arXiv  Detail & Related papers  (2020-03-23T12:28:07Z)

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.