  Sum-of-norms clustering is a clustering formulation based on convex
optimization that automatically induces hierarchy. Multiple algorithms have
been proposed to solve the optimization problem: subgradient descent by Hocking
et al., ADMM and ADA by Chi and Lange, stochastic incremental algorithm by
Panahi et al. and semismooth Newton-CG augmented Lagrangian method by Sun et
al. All algorithms yield approximate solutions, even though an exact solution
is demanded to determine the correct cluster assignment. The purpose of this
paper is to close the gap between the output from existing algorithms and the
exact solution to the optimization problem. We present a clustering test that
identifies and certifies the correct cluster assignment from an approximate
solution yielded by any primal-dual algorithm. Our certification validates
clustering for both unit and multiplicative weights. The test may not succeed
if the approximation is inaccurate. However, we show the correct cluster
assignment is guaranteed to be certified by a primal-dual path following
algorithm after sufficiently many iterations, provided that the model parameter
$\lambda$ avoids a finite number of bad values. Numerical experiments are
conducted on Gaussian mixture and half-moon data, which indicate that carefully
chosen multiplicative weights increase the recovery power of sum-of-norms
clustering.
