Nonnegative Matrix Factorization: Theory with an application to translations invariant image processing

Nonnegative Matrix Factorization(NMF) is a common used technique in machine learning to extract features out of data such as text documents and images thanks to its natural clustering properties and the easy interpretation of the output data. We review the original NMF problem, its common variants, and the main solving algorithms used nowadays. We'll also see how its particular framework makes it suitable for a lot of applications like clustering and text mining. One of the main applications of NMF is the analysis and decomposition of images, but it can't recognize the objects if they're located in different places on multiple images, so the input data must always be pre-calibrated and adjusted. We present a way to fix this problem, that keeps the interpretability property of the output to represent the wanted parts of images, doesn't change the original input data, and bounds the computational cost by the number of effective features we want to find. We'll describe a new domain for the variables in the matrices, and we devise a method to solve the new problem, with experiments on handmade data

Analyse und Berechnung niedrigdimensionaler Darstellungen von Lösungsmengen zur nichtnegativen Matrixfaktorisierung

In der Habilitationsschrift werden niedrigdimensionale Darstellungen von Lösungsmengen zur nichtnegativen Matrixfaktorisierung untersucht. Im Fokus stehen die sogenannten Mengen zulässiger Lösungen. Die dabei genutzte niedrigdimensionale Darstellung basiert auf der Perron-Frobenius-Theorie nichtnegativer Matrizen. Die Mengen zulässiger Lösungen werden analysiert, und ausgehend von den zentralen Eigenschaften werden verschiedene Methoden zur geometrischen Konstruktion beziehungsweise zur numerischen Approximation entwickelt und untersucht