A fast SVD for multilevel block Hankel matrices with minimal memory storage

Abstract

Motivated by the Cadzow filtering in seismic data processing, this paper presents a fast SVD method for multilevel block Hankel matrices. A seismic data presented as a multidimensional array is first transformed into a two dimensional multilevel block Hankel (MBH) matrix. Then the Lanczos process is applied to reduce the MBH matrix into a bidiagonal or tridiagonal matrix. Finally, the SVD of the reduced matrix is computed using the twisted factorization method. To achieve high efficiency, we propose a novel fast MBH matrix-vector multiplication method for the Lanczos process. In comparison with existing fast Hankel matrix-vector multiplication methods, our method applies 1-D, instead of multidimensional, FFT and requires minimum storage. Moreover, a partial SVD is performed on the reduced matrix, since complete SVD is not required by the Caszow filtering. Our numerical experiments show that our fast MBH matrix-vector multiplication method significantly improves both the computational cost and storage requirement. Our fast MBH SVD algorithm is particularly efficient for large size multilevel block Hankel matrices.

Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory

“…Table 3 provides a summary of experiments that contrast the computational costs of Algorithm 3, both with and without use of fast matrix-matrix multiplication, FBHMRSVD, and RSVD, and for direct calculation using the partial SVD. FBHMVM can also be realized using the 1DFFT [15], and these experiments are also reported. They demonstrate that using the 1DFFT generally increases the computational costs.…”

“…In this paper, a fast block Hankel matrix randomized SVD (FBHMRSVD) algorithm that requires minimal storage is proposed. FBHMRSVD is based on fast block Hankel matrix-vector multiplications (FBHMVM) [27,15] and the use of a randomized SVD (RSVD) [12,11,25]. A fast non-convex low-rank matrix decomposition for potential field separation (FNCLRMD PFS) based on the FBHMRSVD is obtained.…”

“…While the RSVD has been introduced to improve the efficiency in general for finding a high accuracy partial SVD, [11], there are other options for efficiently calculating a high accuracy partial SVD of a block Hankel matrix, e.g. [15]. But, it is not hard to show that the RSVD is more efficient even for the block Hankel case.…”

Reply to letter to the editor “Low‐rank seismic denoising with optimal rank selection for Hankel matrices” by S. Trickett

“…The computational burden may cause an issue when say filtering in two spatial dimensions simultaneously, in which case is a block Hankel matrix of size if each spatial tie is in the order of trace by trace. In this case, by using Lanczos process and fast Hankel matrix‐vector multiplication (Trickett, 2003; Lu et al ., 2015), the computational complexity of estimating the entire singular values and the first singular values are and , respectively. Thus, computing could be more efficient than , though we note that the window size is usually not too large in practice (such as 15 in T15) so that the events can be regarded as linear.…”

Fast and Robust Low-Rank Approximation for Five-Dimensional Seismic Data Reconstruction

“…Firstly, high computational cost during SVD process is the biggest problem we encounter especially for 5D dataset [54], [55]. Since our target multi-level block Hankel/Toeplitz matrix or tensor array tends to be extremely large, the truncated SVD process consumes huge amounts of time.…”