  The Schr\"odinger equation is at the heart of modern quantum mechanics. Since
exact solutions of the ground state are typically intractable, standard
approaches approximate Schr\"odinger equation as forms of nonlinear generalized
eigenvalue problems $F(V)V = SV\Lambda$ in which $F(V)$, the matrix to be
decomposed, is a function of its own top-$k$ smallest eigenvectors $V$, leading
to a "self-consistency problem". Traditional iterative methods heavily rely on
high-quality initial guesses of $V$ generated via domain-specific heuristics
methods based on quantum mechanics. In this work, we eliminate such a need for
domain-specific heuristics by presenting a novel framework, Self-consistent
Gradient-like Eigen Decomposition (SCGLED) that regards $F(V)$ as a special
"online data generator", thus allows gradient-like eigendecomposition methods
in streaming $k$-PCA to approach the self-consistency of the equation from
scratch in an iterative way similar to online learning. With several critical
numerical improvements, SCGLED is robust to initial guesses, free of
quantum-mechanism-based heuristics designs, and neat in implementation. Our
experiments show that it not only can simply replace traditional
heuristics-based initial guess methods with large performance advantage
(achieved averagely 25x more precise than the best baseline in similar wall
time), but also is capable of finding highly precise solutions independently
without any traditional iterative methods.
