Multidimensional NMR Inversion without Kronecker Products: Multilinear Inversion

TL;DR

This paper introduces a memory-efficient, multilinear inversion method for multidimensional NMR that overcomes limitations of traditional Kronecker product approaches, enabling flexible, high-dimensional analysis with arbitrary regularization.

Contribution

The authors develop a novel multilinear inversion technique that avoids kernel compression and Kronecker products, allowing for scalable, flexible multidimensional NMR inversion with arbitrary regularization.

Findings

Requires less than 0.1% memory of traditional methods

Extends easily to arbitrary dimensions and non-separable kernels

Simplifies implementation with only a cost function and its derivative

Abstract

Multidimensional NMR inversion using Kronecker products poses several challenges. First, kernel compression is only possible when the kernel matrices are separable, and in recent years, there has been an increasing interest in NMR sequences with non-separable kernels. Second, in three or more dimensions, the singular value decomposition is not unique; therefore kernel compression is not well-defined for higher dimensions. Without kernel compression, the Kronecker product yields matrices that require large amounts of memory, making the inversion intractable for personal computers. Finally, incorporating arbitrary regularization terms is not possible using the Lawson-Hanson (LH) or the Butler-Reeds-Dawson (BRD) algorithms. We develop a minimization-based inversion method that circumvents the above problems by using multilinear forms to perform multidimensional NMR inversion without using kernel compression or Kronecker products. The new method is memory efficient, requiring less than 0.1% of the memory required by the LH or BRD methods. It can also be extended to arbitrary dimensions and adapted to include non-separable kernels, linear constraints, and arbitrary regularization terms. Additionally, it is easy to implement because only a cost function and its first derivative are required to perform the inversion.