Non-degenerate Rigid Alignment in a Patch Framework
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912971336187904 |
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| author | Kohli, Dhruv Mishne, Gal Cloninger, Alexander |
| author_facet | Kohli, Dhruv Mishne, Gal Cloninger, Alexander |
| contents | Given a set of overlapping local views (patches) of a dataset, we consider the problem of finding a rigid alignment of the views that minimizes a $2$-norm based alignment error. In general, the views are noisy and a perfect alignment may not exist. In this work, we characterize the non-degeneracy of an alignment in the noisy setting based on the kernel and positivity of a certain matrix. This leads to a polynomial time algorithm for testing the non-degeneracy of a given alignment. Subsequently, we focus on Riemannian gradient descent for minimizing the alignment error, providing a sufficient condition on an alignment for the algorithm to converge (locally) linearly to it. \revadd{Additionally, we provide an exact recovery and noise stability analysis of the algorithm}. In the case of noiseless views, a perfect alignment exists, resulting in a realization of the points that respects the geometry of the views. Under a mild condition on the views, we show that a non-degenerate perfect alignment \revadd{characterizes the infinitesimally rigidity of a realization, and thus the local rigidity of a generic realization}. By specializing the non-degeneracy conditions to the noiseless case, we derive necessary and sufficient conditions on the overlapping structure of the views for \revadd{a perfect alignment to be non-degenerate and equivalently, for the resulting realization to be infinitesimally rigid}. Similar results are also derived regarding the uniqueness of a perfect alignment and global rigidity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_11620 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-degenerate Rigid Alignment in a Patch Framework Kohli, Dhruv Mishne, Gal Cloninger, Alexander Numerical Analysis Differential Geometry Optimization and Control 52C25, 53B21, 53B20, 65K10, 65Y20, 40A05, 05C50 Given a set of overlapping local views (patches) of a dataset, we consider the problem of finding a rigid alignment of the views that minimizes a $2$-norm based alignment error. In general, the views are noisy and a perfect alignment may not exist. In this work, we characterize the non-degeneracy of an alignment in the noisy setting based on the kernel and positivity of a certain matrix. This leads to a polynomial time algorithm for testing the non-degeneracy of a given alignment. Subsequently, we focus on Riemannian gradient descent for minimizing the alignment error, providing a sufficient condition on an alignment for the algorithm to converge (locally) linearly to it. \revadd{Additionally, we provide an exact recovery and noise stability analysis of the algorithm}. In the case of noiseless views, a perfect alignment exists, resulting in a realization of the points that respects the geometry of the views. Under a mild condition on the views, we show that a non-degenerate perfect alignment \revadd{characterizes the infinitesimally rigidity of a realization, and thus the local rigidity of a generic realization}. By specializing the non-degeneracy conditions to the noiseless case, we derive necessary and sufficient conditions on the overlapping structure of the views for \revadd{a perfect alignment to be non-degenerate and equivalently, for the resulting realization to be infinitesimally rigid}. Similar results are also derived regarding the uniqueness of a perfect alignment and global rigidity. |
| title | Non-degenerate Rigid Alignment in a Patch Framework |
| topic | Numerical Analysis Differential Geometry Optimization and Control 52C25, 53B21, 53B20, 65K10, 65Y20, 40A05, 05C50 |
| url | https://arxiv.org/abs/2303.11620 |