Inverse Mellin Transformation of Continuous Singular Value Decomposition: A Route to Holographic Renormalization

TL;DR

This paper explores the connection between singular value decomposition of matrix data from the 2D Ising model at criticality and holographic renormalization, using inverse Mellin transformation to analyze the spectrum and its relation to correlation length.

Contribution

It introduces a continuous limit of SVD components via inverse Mellin transformation, revealing their relation to two-point correlators and holographic structures.

Findings

SVD components characterized by two-point correlators.

Continuous SVD index corresponds to inverse correlation length.

SVD spectrum exhibits power-law behavior.

Abstract

We examine holographic renormalization by the singular value decomposition (SVD) of matrix data generated by the Monte Carlo snapshot of the 2D classical Ising model at criticality. To take the continuous limit of the SVD enables us to find the mathematical form of each SVD component by the inverse Mellin transformation as well as the power-law behavior of the SVD spectrum. We find that each SVD component is characterized by the two-point spin correlator with a finite correlation length. Then, the continuous limit of the decomposition index in the SVD corresponds to the inverse of the correlation length. These features strongly suggest that the SVD contains mathematical structure the same as the holographic renormalization.