Interesting Paths in the Mapper
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The Mapper produces a compact summary of high dimensional data as a simplicial complex. We study the problem of quantifying the interestingness of subpopulations in a Mapper, which appear as long paths, flares, or loops. First, we create a weighted directed graph G using the 1-skeleton of the Mapper. We use the average values at the vertices of a target function to direct edges (from low to high). The difference between the average values at vertices (high-low) is set as its weight. Covariation of the remaining h functions (independent variables) is captured by a h-bit binary signature assigned to the edge. An interesting path in G is a directed path whose edges all have the same signature. We define the interestingness score of such a path as a sum of its edge weights multiplied by a nonlinear function of their ranks in the path. Second, we study three optimization problems on this graph G. In the problem Max-IP, we seek an interesting path in G with the maximum interestingness score. We show that Max-IP is NP-complete. For the special case when G is a directed acyclic graph (DAG), we show that Max-IP can be solved in polynomial time - in O(mnd_i) where d_i is the maximum indegree of a vertex in G. In the more general problem IP, the goal is to find a collection of interesting paths such that these paths form an exact cover of E (hence they are edge-disjoint) and the overall sum of interestingness scores of all paths is maximized. We also study a variant of IP termed k-IP, where the goal is to identify a collection of edge-disjoint interesting paths each with k edges, and the total interestingness score of all paths is maximized. While k-IP can be solved in polynomial time for k <= 2, we show k-IP is NP-complete for k >= 3. Further, we show that k-IP remains NP-complete for k >= 3 even for the case when G is a DAG. We develop polynomial time heuristics for IP and k-IP on DAGs.
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