Abstract
This review paper describes the basic concept and technical details of sparse modeling and its applications to quantum many-body problems. Sparse modeling refers to methodologies for finding a small number of relevant parameters that well explain a given dataset. This concept reminds us physics, where the goal is to find a small number of physical laws that are hidden behind complicated phenomena. Sparse modeling extends the target of physics from natural phenomena to data, and may be interpreted as "physics for data". The first half of this review introduces sparse modeling for physicists. It is assumed that readers have physics background but no expertise in data science. The second half reviews applications. Matsubara Green's function, which plays a central role in descriptions of correlated systems, has been found to be sparse, meaning that it contains little information. This leads to (i) a new method for solving the ill-conditioned inverse problem for analytical continuation, and (ii) a highly compact representation of Matsubara Green's function, which enables efficient calculations for quantum many-body systems.
Highlights
A small number of physical laws exist behind apparently complicated behaviors: This is the basic notion of physics
In the second half of this paper, we present the application of sparse modeling to quantum many-body problems
We present the concept of sparse modeling, which opens a new paradigm for solving inverse problems
Summary
A small number of physical laws exist behind apparently complicated behaviors: This is the basic notion of physics. In the second half of this paper, we present the application of sparse modeling to quantum many-body problems. Given that the numerically computed GðÞ contains less information on real-frequency dynamics, we turn our attention to how to extract the relevant information This can be achieved using a new basis set, which is placed as an intermediate representation (IR) between the imaginary and real frequencies. With the IR basis, the sparse-modeling technique leads to a new algorithm for conversion (analytical continuation) from GðÞ to GR ð!Þ for efficient computation in quantum many-body theories. This basis is utilized for a new method for analytical continuation in Sect.
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