Stefan Klus, Tim Sullivan
Ilja Klebanov (ZIB)
01.01.2019 – 31.12.2020
The application domain of this project is to model customer demand and control item prices in an e-commerce setting using both reproducing kernel Hilbert space (transfer) operator approaches and models inspired by recurrent neural networks. The collaboration partner Zalando will use the developed prototypical methods to improve supply planning and pricing, taking into account real-world constraints.
The underlying mathematical challenges involve the statistical analysis and optimal control of time series in high-dimensional non-linear spaces. The embedding of these objects into appropriate reproducing kernel Hilbert feature spaces offers a way to faithfully linearise these problems and make them amenable to computation.
The first publication stemming from this project is “A rigorous theory of conditional mean embeddings”. Conditional mean embeddings (CMEs) have proven themselves to be a powerful tool in many machine learning applications. They allow the efficient conditioning of probability distributions within the corresponding reproducing kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the kernel mean embeddings of the respective joint and conditional probability distributions. Both centred and uncentred covariance operators have been used to define CMEs in the existing literature. In this paper, we develop a mathematically rigorous theory for both variants, discuss the merits and problems of each, and significantly weaken the conditions for applicability of CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian conditioning in Hilbert spaces.
Follow-on work in the context of this project will apply the CME theory developed in the first period of the project to the analysis of time series data from our industrial partners and the optimal control of design variables (commercial decisions).
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