Sensitivity analysis of discrete preference functions using Koszul simplicial complexes

We use a monomial ideal I to model a discrete preference function on a set of n factors. We can measure the sensitivity of each point represented by a monomial m by calculating its formal partial derivatives with respect to each variable. These derivatives can be used to define the Koszul simplicial complex of the ideal I at m. We refer to points at which the homology of their Koszul complex is not null as sensitive corners. In the context of preference analysis, the ranks of the homology groups are not precise enough to distinguish between sensitive corners that have the same homology but correspond to different sensitivity behaviors. To address this issue, we propose using a filtration on the Koszul complexes of the sensitive corners based on the lcm-lattice of the ideal I. This filtration induces a persistent homology at each corner m. We then use unsupervised Machine Learning methods to classify the corners based on the distance between their persistence diagrams.