3. Algorithm

SpM program solves the linear equation \bm{G}=K\bm{\rho} with respect to \bm{\rho} for given \bm{G}. Because of ill-conditioned nature of the matrix K, a simple treatment of this equation is numerically unstable. For example, the solution using the Moore-Penrose pseudo-inverse matrix results in NaN. Even if one manages to derive a definite solution, it is quite sensitive to numerical noise and often breaks preconditions that any physical spectra must satisfy. This becomes particularly problematic when \bm{G} is evaluated by quantum Monte Carlo technique.

SpM provides a physical solution which fulfills the equation of concern within a certain accuracy. The solution satisfies the constraints such as sum rule and nonnegativity. The engine of SpM program uses the method of L1-norm regularization to separate relevant information in \bm{G} from irrelevant one which makes the spectrum unphysical. This process is automatically done without hand-tuning parameters.

For details, see the original article

J. Otsuki, M. Ohzeki, H. Shinaoka, K. Yoshimi, “Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data” Phys. Rev. E 95, 061302(R) (2017).