EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > Representer Theorems


 CONTENTS
 Home Page
 News & Events
 Members
 Publications
 Tutorials and Reviews
 Research
 Demos
 Download Algorithms

 DOWNLOAD
 PDF not available
 PS not available
 All BibTeX References

Representer Theorems for the Design of Deep Neural Networks and the Resolution of Continuous-Domain Inverse Problems

M. Unser

Ninth International Conference on Machine Learning in Medical Imaging (MLMI'18), Granada, Kingdom of Spain, September 16, 2018, in press.

Please do not bookmark the "In Press" papers as content and presentation may differ from the published version.


Our purpose in this talk is to reenforce the (deep) connection between splines and learning techniques. To that end, we first describe a recent representer theorem that states that the extremal points of a broad class of linear inverse problems with generalized total-variation regularization are adaptive splines whose type is linked to the underlying regularization operator L. For instance, when L is n-th derivative (resp., Laplacian) operator, the optimal reconstruction is a non-uniform polynomial (resp., polyharmonic) spline with the smallest possible number of adaptive knots.

The crucial observation is that such continuous-domain solutions are intrinsically sparse, and hence compatible with the kind of formulation (and algorithms) used in compressed sensing.

We then make the link with current learning techniques by applying the theorem to optimize the shape of individual activations in a deep neural network. By selecting the regularization functional to be the 2nd-order total variation, we obtain an "optimal" deep-spline network whose activations are piece-linear splines with a few adaptive knots. Since each spline knot can be encoded with a ReLU unit, this provides a variational justification of the popular ReLU architecture. It also suggests some new computational challenges for the determination of the optimal activations involving linear combinations of ReLUs.

References

  1. M. Unser, J. Fageot, J.P. Ward, "Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization," SIAM Review, vol. 59, no. 4, pp. 769-793, December 2017.

  2. M. Unser, "A Representer Theorem for Deep Neural Networks," arXiv:1802.09210 [stat.ML].

© 2018 Springer. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Springer.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.