Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the Vietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size becomes prohibitively large for reasonable computations. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structure.  investigate a complex called the graph induced complex that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the por of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably.  show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a lean subsample, (ii) reconstruct a surface in three dimensions from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample.  provide experimental evidences in support of our theory.
['Tamal K. Dey', 'Fengtao Fan', 'Yusu Wang']
I am writing a research paper on a idea Graph induced complex on point data
[{"from": "human", "value": "I am writing a research paper on a idea Graph induced complex on point data"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the Vietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size becomes prohibitively large for reasonable computations. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structure.  investigate a complex called the graph induced complex that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the por of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably.  show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a lean subsample, (ii) reconstruct a surface in three dimensions from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample.  provide experimental evidences in support of our theory."}]