On different witness complex filtrations and their landmark choices

The main aim of this thesis is to study different Witness Complex filtrations. By the notion of Witness Complex we mean filtrations of simplicial complexes characterized by the fact that the data set they are based on is not automatically included in the complex but rather landmark points are chosen and the rest of the points is used for additional information on the higher dimensional simplices between them. We consider two constructions that are commonly referred to in literature simply by the term Witness Complex, two constructions based on the Cech and Vietoris-Rips Complexes as well as a possible construction for a Witness Complex bifiltration. First, we compare them to one another and to other known filtrations, look at methods of choosing the landmark points and search for approximations to the Vietoris-Rips Complex before introducing multiparameter filtrations. One of the main parts of this work is after that to construct a different method of acquiring landmark points that is robust with respect to outliers in the data set. This method is derived from the DBSCAN algorithm and with it we can compare one of the Witness Complex constructions to the Degree-Rips filtration which is a density-sensitive bifiltration of the classical Vietoris-Rips Complex. Another goal of this work is then to expand on some existing interleaving results for Witness Complexes to all of the different introduced constructions and especially to expand them before finally stating some additional stability results for all of the constructions which allow both the landmark points as well as the data points to shift to a certain extent.