Approximate minimum norm subspace projection of least squares weights without an SVD

A QR based technique is presented for estimating the approximate numerical rank and corresponding signal subspace of a matrix together with the subspace projection of the least squares weights. Theoretical difficulties associated with conventional QR factorisation are overcome by applying the technique of row-zeroing QR to the covariance matrix. Thresholding is simplified compared with the use of the data matrix as the diagonal value spectrum is sharpened and the subspace estimate is improved. An approximation to the minimum norm solution for the projection of the least squares weight onto the signal subspace of the data is obtained simply, without performing an SVD