We consider recent work by Haber and Ruthotto, where deep learning neural networks are viewed as discretisations of an optimal control problem. We review the first order conditions for optimality, and the conditions ensuring optimality after discretization. This leads to a class of algorithms for solving the discrete optimal control problem which guarantee that the corresponding discrete necessary conditions for optimality are fulfilled. We discuss two different deep learning algorithms and make a preliminary analysis of the ability of the algorithms to generalize.
joint work with Martin Benning, Carola-Bibiane Schönlieb, Evren Yarman, Ozan Öktem
In inverse problems it is common to include a convex optimisation problem which enforces some sparsity on the reconstruction, e.g. atomic scale Electron tomography. One draw-back to this, in the era of big data, is that we are still required to store and compute using each of those coefficients we expect to be zero. In this talk, we address this by parametrising the reconstruction as a sparse Gaussian Mixture Model and fitting, on a continuous grid, to the raw data. We will discuss the pros (computational complexity) and overcoming the cons (introduction of non-convexity) of such an approach.
joint work with Martin Burger, Elena Resmerita
In this talk, we discuss iterative regularisation of continuous, linear ill-posed problems via an entropic projection method. We present a detailed convergence analysis of the scheme that respects the continuous nature of the underlying problem and we present numerical results for continuous and ill-posed inverse problems to support the theory.