Download - March5 gao
ONLINE IDENTIFICATION AND TRACKING OF SUBSPACES FROM HIGHLY INCOMPLETE INFORMATION
Presenter: Ju Gao
Laura Balzano, Robert Nowak and Benjamin Recht
Problem Statement
Given: observations that come from unknown subspace with missing information (e.g. subsampling, compression, noise corruption) Goal: track the underlying signal subspace Remark1: signal subspace might be time varying
Remark2: solution is not unique
Problem Statement Cont’d
1. Update subspace matrix U iteratively using new observation
2. The solution needs to be orthonormal 3. The solution is rotational invariant
Distance between current observation and subspace estimates can be calculated as:
Grassmannian G(n,d) Definition(Grassmannian): the set of all d-dimensional subspace that lies in n-dimensional space is called Grassmannian manifold G(n,d)
Parameterization of Grassmannian manifold: 1. An element S \in G(n,d) can be represented as any n-by-d orthonormal matrices that form the bases for S 2. Quotient space representation V(n,d)/O(d) (Stiefel to orthonormal group)
Solution of Subspace Tracking Problem
Recall the two conditions for the subspace tracking problem
The solution lies in G(n,d)
Gradient descending on G(n,d) gives the solution