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

Proposed Method

Ordinary gradient:

Remember distance function:

Proposed Method

Ordinary gradient:

Remember distance function:

Proposed Method

Ordinary gradient:

Remember distance function:

Natural gradient [Edelman 1998]:

Proposed Method

Ordinary gradient:

Remember distance function:

Natural gradient [Edelman 1998]:

Gradient Descend in Grassmannian [Edelman 1998]:

SVD of natural gradient:

Updating rule:

GROUSE Algorithm

Simulation Results

Simulation Results