An Analytical Formula for Spectrum Reconstruction

TL;DR

This paper provides a theoretical validation and error analysis of an approximation formula for spectrum reconstruction, which is crucial for eigenvalue-based techniques like PCA, especially in high-dimensional settings.

Contribution

We prove the validity of an existing approximation formula for spectrum reconstruction and determine its error order in high-dimensional asymptotics.

Findings

The approximation formula is mathematically justified.

Error order depends on the ratio of dimension to number of features.

Results are applicable when both dimension and features grow large.

Abstract

We study the spectrum reconstruction technique. As is known to all, eigenvalues play an important role in many research fields and are foundation to many practical techniques such like PCA(Principal Component Analysis). We believe that related algorithms should perform better with more accurate spectrum estimation. There was an approximation formula proposed, however, they didn't give any proof. In our research, we show why the formula works. And when both number of features and dimension of space go to infinity, we find the order of error for the approximation formula, which is related to a constant $c$-the ratio of dimension of space and number of features.