Motivated by the observation that the ability of the $\ell_1$ norm in promoting sparsity in graphical models with Laplacian constraints is much weakened, this paper proposes to learn graph Laplacian with a non-convex penalty: minimax concave penalty (MCP).
The sparsity and clustering structure of the concentration matrix is enforced to reduce model complexity and describe inherent regularities.
We prove that the least squares estimator is computable via solving a constrained convex quadratic programming (QP) problem with $(n+1)d$ variables and at least $n(n-1)$ linear inequality constraints, where $n$ is the number of data points.
In this paper, we consider high-dimensional nonconvex square-root-loss regression problems and introduce a proximal majorization-minimization (PMM) algorithm for these problems.
In addition, we derive the corresponding HS-Jacobian to the proximal mapping and analyze its structure --- which plays an essential role in the efficient computation of the PPA subproblem via applying a semismooth Newton method on its dual.
The perfect recovery properties of the convex clustering model with uniformly weighted all pairwise-differences regularization have been proved by Zhu et al. (2014) and Panahi et al. (2017).
When the support points of the barycenter are pre-specified, this problem can be modeled as a linear programming (LP) problem whose size can be extremely large.
Clustering may be the most fundamental problem in unsupervised learning which is still active in machine learning research because its importance in many applications.
To seek a solution of high recovery quality beyond the reach of the nuclear norm, in this paper, we propose a rank-corrected procedure using a nuclear semi-norm to generate a new estimator.