Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Comparison beten multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link beten the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological information is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties.   introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations  take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the above filtrations can induce different matchings beten the associated persistent diagrams. In our paper  prove that the coherent 2D matching distance is ll-defined and stable.
['Andrea Cerri', 'Marc Ethier', 'Patrizio Frosini']
I am writing a research paper on a idea The Coherent Matching Distance in 2D Persistent Homology
[{"from": "human", "value": "I am writing a research paper on a idea The Coherent Matching Distance in 2D Persistent Homology"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Comparison beten multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link beten the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological information is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties.   introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations  take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the above filtrations can induce different matchings beten the associated persistent diagrams. In our paper  prove that the coherent 2D matching distance is ll-defined and stable."}]