Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

Abstract The combination of persistent homology and discrete Morse theory has proven very effective in visualizing analyzing big heterogeneous data. Indeed, topology provides computable coarse summaries data independently from specific coordinate systems does so robustly to noise. Moreover, the geometric content a gradient vector field is useful for visualization purposes. case multivariate still demands further investigations, on one hand, computational reasons, it important reduce necessary amount be processed. On other analysis requires detection interpretation possible interdepedance among components. To this end, paper we introduce study notion perfectness fields with respect multi-parameter homology, called relative-perfectness. As natural generalization usual relative-perfectness entails having least number critical cells relevant persistence. first contribution, support our definition by generalizing inequalities filtration structure where groups involved are relative subsequent sublevel sets. In order allow an 2-parameter persistence, second contribution consists two bounding Betti tables persistence modules above below, via cells. Our last result proof that existing algorithms based local homotopy expansions efficient computability over simplicial complexes up dimension 2.