  In various subjects, there exist compact and consistent relationships between
input and output parameters. Discovering the relationships, or namely compact
laws, in a data set is of great interest in many fields, such as physics,
chemistry, and finance. While data discovery has made great progress in
practice thanks to the success of machine learning in recent years, the
development of analytical approaches in finding the theory behind the data is
relatively slow. In this paper, we develop an innovative approach in
discovering compact laws from a data set. By proposing a novel algebraic
equation formulation, we convert the problem of deriving meaning from data into
formulating a linear algebra model and searching for relationships that fit the
data. Rigorous proof is presented in validating the approach. The algebraic
formulation allows the search of equation candidates in an explicit
mathematical manner. Searching algorithms are also proposed for finding the
governing equations with improved efficiency. For a certain type of compact
theory, our approach assures convergence and the discovery is computationally
efficient and mathematically precise.
