Important Scientific Presentation Jonathan Doe Department of Electrical Engineering University of Rhode Island
The Initial Data Set • Matrix M t contains c column vectors, m 1 through m c . = [ ] M t m 1 m 2 . . . m c • Taking the SVD of M t gives us � ˆ � Σ t 0 V t ] H M t = [ˆ U t ˜ [ˆ V t ˜ U t ] ˜ 0 Σ t where ˆ U t contains the k left singular vectors of M t cor- responding to its largest singular values, which are the orthonormal basis vectors of the desired subspace. University of Rhode Island present.tex/aj02 1
The First Iteration • Now we create the next matrix M t +1 using the columns of M t , discarding m 1 and using the new column m c +1 . = [ ] M t +1 m 2 m 3 . . . m c m c +1 • What we want are ˆ U t +1 and ˆ Σ t +1 where � ˆ � Σ t +1 0 V t +1 ] H M t +1 = [ˆ U t +1 ˜ [ˆ V t +1 ˜ U t +1 ] ˜ 0 Σ t +1 University of Rhode Island present.tex/aj02 2
The First Iteration • Now we create the next matrix M t +1 using the columns of M t , discarding m 1 and using the new column m c +1 . = [ ] M t +1 m 2 m 3 . . . m c m c +1 • What we want are ˆ U t +1 and ˆ Σ t +1 where � ˆ � Σ t +1 0 V t +1 ] H M t +1 = [ˆ U t +1 ˜ [ˆ V t +1 ˜ U t +1 ] ˜ 0 Σ t +1 University of Rhode Island present.tex/aj02 2-a
Multiple Column Update • More than one column can be added and removed each iteration by adding the portion of all relevant vectors to the orthonormal basis Q . • The matrix Q will be of dimension r × k + 2 n . • The algorithm is otherwise unchanged. University of Rhode Island present.tex/aj02 3
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