Approximate Regularization Path for Nuclear Norm Based H2 Model Reduction
This work provides a tool for finding the appropriate trade-off in model reduction for dynamical systems, though it is incremental as it extends existing regularization path concepts to the nuclear norm case.
The paper tackles the problem of model reduction for dynamical systems by using nuclear norm regularization to balance model fit and complexity, and presents a method to efficiently compute the entire regularization path up to a specified tolerance by solving a fixed number of optimization problems.
This paper concerns model reduction of dynamical systems using the nuclear norm of the Hankel matrix to make a trade-off between model fit and model complexity. This results in a convex optimization problem where this trade-off is determined by one crucial design parameter. The main contribution is a methodology to approximately calculate all solutions up to a certain tolerance to the model reduction problem as a function of the design parameter. This is called the regularization path in sparse estimation and is a very important tool in order to find the appropriate balance between fit and complexity. We extend this to the more complicated nuclear norm case. The key idea is to determine when to exactly calculate the optimal solution using an upper bound based on the so-called duality gap. Hence, by solving a fixed number of optimization problems the whole regularization path up to a given tolerance can be efficiently computed. We illustrate this approach on some numerical examples.