Dynamic Subspace Estimation from Undersampled Data using Grassmannian Geodesics

Abstract:

This work considers recovering a sequence of low-rank matrices from undersampled measurements, where the underlying subspace varies across samples over time. Existing works involve concatenating all of the samples from each time point to recover the underlying matrix
under the assumption that the data are well-approximated by a single, static subspace. However, this assumption is inappropriate for applications where the best low-rank approximations vary over time. To address this issue, we propose a Riemannian block majorize minimization algorithm that constrains the time-varying subspaces as a geodesic along the Grassmann manifold. Our proposed method can faithfully estimate the best-fit subspaces at each time point, even when there are fewer samples at each time point than the subspace dimension. Theoretically, we show that our algorithm enjoys a monotonically non-increasing objective function while converging to an $\epsilon$-stationary point within $\widetilde{\mathcal{O}}(\epsilon^{-2})$ iterations. We demonstrate the effectiveness of our algorithm on synthetic, dynamic fMRI, and video data, where the samples at each time are either compressed or partially missing.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Mathieu_Salzmann1

Submission Number: 7752