This work on information dimension means a lot to me, and I’d like to let you know how it all started. I had come into contact with Renyi’s information dimension in 2012, when I worked on the information lossin principal components analysis. Back then I was considering only random variables, but in the summer of 2013, when I was investigating the information loss in anti-aliasing filters, I had the feeling that there should be a similar definition also for stochastic processes. I somehow managed to get the results published at the Int. Zurich Seminar in 2014, even though I have to admit that some of the more interesting results were obtained only with the help of two Assumptions, which would have been theorems had the results on information dimension already existed. I was certain at this point that a proper definition of the information dimension of a stochastic process must be connected with the bandwidth of the process.
After my PhD, I spent some time at TU Munich as a postdoc. My Schroedinger grant from the Austria Science Fund allowed me to work on information-theoretic reduction of Markov models. The first year though, which was funded directly by my boss Prof. Gerhard Kramer, I set out to work on “Two Little (?) Problems” - at least that was the title under which I presented these two problems to my colleagues during our winter retreat in the beautiful village of Stilfs in early 2015. The two problems were the fractal properties of polar codes (which are an entirely different story) and the information dimension of stochastic processes. While my work on polar codes progressed quickly, I was stuck on the other problem. In the meantime I have found out that researchers published a possible definition of information dimension for stochastic processes. But I was not able to pursue this further until I met Stefan Moser and Tobi Koch at the Int. Zurich Seminar in 2016 (where I presented my work on polar codes). I’ve met both of them before and I read some of their works – I just knew that they were able to help me.
Long story short, Tobi was strongly interested and invited me to stay with him at Universidad Carlos III de Madrid later this year. The stay was amazing: Late breakfasts, dinners at 9 pm, a dry 40 degress weather, and ice cold beer at night to cool down. Most importantly, we got a good way towards proving the connection between information dimension – defined differently than previously suggested – and bandwidth, at least for Gaussian processes. Over the coming months our results were refined and extended, and now it feels as if there is not much important left to do. It seems as if both of my “Two Little (?) Problems” are solved now.