Typical reconstruction limit and phase transition of maximum entropy method

We investigate the dependence of the maximum entropy method (MEM) reconstruction performance on the default model. The maximum entropy method is a reconstruction technique that utilizes prior information, referred to as the default model, to recover original signals from observed data, and it is widely used in underdetermined systems. The broad applications have been reported in fields ranging from the analysis of observational data in seismology and astronomy, to large-scale computations in quantum chemistry, and even in social sciences such as linguistics. However, a known drawback of MEM is that its results depend on the assumed default model. In this study, we employ the replica method to elucidate how discrepancies in the default model affect the reconstruction of signals with specific distributions. We report that for binary signals, even small discrepancies can induce phase transitions, leading to reconstruction failure. Additionally, by comparing MEM with reconstruction based on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi>L</a:mi> <a:mn>1</a:mn> </a:msub> </a:math> -norm regularization, a method proposed in recent years, we demonstrate that MEM exhibits lower reconstruction accuracy under certain conditions.