CASUS Institute Seminar, rehearsal talks of two master theses

Ashish Yashwanth Kangen, Student assistant at the Center for Advanced Systems Understanding CASUS, Helmholtz-Zentrum Dresden-Rossendorf (HZDR)

Title: Architectures and applications of polynomial neural nets — Master Thesis.

Abstract of the talk//Since their inception in 1957 conventional neural network architectures appear as the pivotal choice for machine learning tasks till today.
Seeking to approximate multi-dimensional functions, the architectures rest on the composition of linear and analytic non-linear activation functions. In this master thesis alternate architectures are explored and analyzed, resting on products of linear operators in order to realize polynomial networks. Based on the recent work of Grigorios G. Chrysos on P-nets: Deep polynomial neural networks this thesis extends and analyzes alternate architectures.

Two modifications are the center of investigations: Firstly, specific Chebyshev and Legendre polynomial networks are introduced by modifying the recursive relationship of the initial standard polynomial network. Secondly, in an effort of interpretability, the underlying polynomial basis of the polynomial network is identified due to an extraction scheme. This insight serves as a guidance for several sparsity strategies simplifying the polynomial network without losing approximation power.

The thesis further provides two implementations of applications: Firstly, by utilizing the basis extraction scheme a quadrature rule is provided, enabling exact integration of the polynomial network. Especially in high dimensions, demonstrations show high efficiency of this approach and an increase in performance due to the novel networks. Secondly, sparse polynomial network are used to realize a variational autoencoder, which shows an increase in efficiency compared to the standard polynomial network and an increase in reconstruction quality compared to a standard multilayer perceptron variational autoencoder.

Title: Implementation and complexity analysis of bases transformations of multivariate Lagrange- and Newton-polynomials — Bachelor Thesis.

Phil-Alexander Hofmann, Student assistant at the Center for Advanced Systems Understanding CASUS, Helmholtz-Zentrum Dresden-Rossendorf (HZDR)

Abstract of the talk//Polynomial interpolation goes back to Newton, Lagrange, and others, and its fundamental importance in mathematics and computing is undisputed.
For downward closed polynomial spaces these classics have been extend to the multivariate case. The scope of this thesis is to analyze and implement novel algorithms realizing the underlying basis transformations of the Lagrange and Newton polynomials.

While the theoretical analysis shows a complexity reduction from quadratic runtime up to linearithmic time, empirical demonstrations validate this increase in efficiency. In this talk, the impact of the contribution in comparison to applicability and implementations of Fast Fourier Transformations is discussed.

The seminar will take about one hour in total. The two presentations will be held in presence in the CASUS lecture room. However, as the event is organized in a hybrid format that includes a videoconferencing tool by Zoom Inc., people interested in the topic have the chance to also join the talk remotely. Please ask for the login details via