Filtered Matrix-Vector Products via the Lanczos Algorithm with Applications to Dimension Reduction

This paper discusses an efficient technique for computing filtered matrix-vector (mat-vec) products by exploiting the Lanczos algorithm. The goal of the proposed method, which is the same as that of the truncated singular value decomposition (SVD), is to preserve the quality of the resulting mat-vec product in major singular directions of the matrix. Unlike the SVD-based techniques, the proposed algorithms achieve this goal by using a small number of Lanczos vectors, without explicitly computing the major singular values/vectors of the matrix. The main advantage of the proposed method is its low cost compared with SVDbased techniques. This advantage comes without sacrificing accuracy. The effectiveness of the method is demonstrated on a few sample examples requiring dimension reduction, including information retrieval and face recognition.