Computational Topology - Mapper Jiaqi Ni Eindhoven University of Technology June 14, 2018
Outline Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary
Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary
Introduction ◮ Mapper is a computational method for extracting simple descriptions of high dimensional data sets in the form of simplicial complexes.
Recap about Reeb Graph Definition: The Reeb graph of f is the set of contours R(f) .
Recap about Reeb Graph We can get similar result as Reeb Graph with Mapper .
Recap about Reeb Graph We can also get the more different results from Reeb Graph with Mapper .
Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary
Cover of space If the set X is a topological space, then a cover C of X is a collection of subsets U of X whose union is the whole space X . In this case we say that C covers X , or that the sets U cover X .
Cover of space If the set X is a topological space, then a cover C of X is a collection of subsets U of X whose union is the whole space X . In this case we say that C covers X , or that the sets U cover X . Topological Space X Cover of Space X
Cover of space If Y is a subset of X , then a cover of Y is a collection of subsets of X whose union contains Y, � i.e., C is a cover of Y if Y ⊆ U α α ∈ C
Cover of space If Y is a subset of X , then a cover of Y is a collection of subsets of X whose union contains Y, � i.e., C is a cover of Y if Y ⊆ U α α ∈ C
Cover refinement ◮ A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained in some set in C.
Cover refinement ◮ A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained in some set in C. ◮ Formally: D = { V β ∈ B } is a refinement of C = { U α ∈ A } when ∀ β ∃ α V β ⊆ U α
Cover refinement ◮ A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained in some set in C. ◮ Formally: D = { V β ∈ B } is a refinement of C = { U α ∈ A } when ∀ β ∃ α V β ⊆ U α Space X Cover of Space X Refinement of Cover
Mapper in the continuous setting Input: ◮ Continuous function( filter ) f : X → R � ◮ Cover C of im(f) by open intervals: im ( f ) ⊆ c c ∈ C Method: ◮ Compute pullback cover U of X : U = f − 1 ( c ) c ∈ C ◮ Refine U by separating each of its elements into its various connected components → connected cover V ◮ The Mapper is the nerve of V : ◮ 1 vertex per element V ∈ V ◮ 1 edge per intersection V ∪ V ′ � = ø, V , V ′ ∈ V ◮ 1 k-simplex per (k + 1)-fold intersection , � k i =0 V i � = ø , V 0 , V 1 ... V k ∈ V
Example of Mapper in the continuous setting
Example of Mapper in the continuous setting
Example of Mapper in the continuous setting
Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary
Mapper in practice Input: ◮ Point cloud P with distance matrix ◮ Continuous function( filter ) f : P → R � ◮ Cover C of im(f) by open intervals: im ( f ) ⊆ c c ∈ C Method: ◮ Compute pullback cover U of X : U = f − 1 ( c ) c ∈ C ◮ Refine U by applying clustering algorithm(with distance threshold δ ) → connected cover V ◮ The Mapper is the nerve of V : ◮ 1 vertex per element V ∈ V ◮ 1 edge per intersection V ∪ V ′ � = ø, V , V ′ ∈ V ◮ 1 k-simplex per (k + 1)-fold intersection , � k i =0 V i � = ø , V 0 , V 1 ... V k ∈ V
Example of Mapper in practice
Example of Mapper in practice
Introduction Mapper in the continuous setting Mapper in practice Parameters of Mapper in practice Applications Summary
Parameters of Mapper in practice ◮ Filter f : P → R
Parameters of Mapper in practice ◮ Filter f : P → R ◮ Cover C of im(f) by open intervals:
Parameters of Mapper in practice ◮ Filter f : P → R ◮ Cover C of im(f) by open intervals: ◮ Clustering algorithm
Parameters of Mapper in practice - Filter functions ◮ The outcome of Mapper is highly dependent on the function chosen to partition (filter) the data set and the choice of functions depends mostly on the dataset. ◮ Possible functions: ◮ Density ◮ Eccentricity ◮ Graph Laplacians ◮ sum/average/max/min ◮ x/y- axis projection
Filter function examples
Filter function examples
Filter function examples
Filter function examples
Parameters of Mapper in practice - Cover ◮ Uniform cover I ◮ resolution / granularity: r (diameter of intervals) ◮ gain: g (percentage of overlap)
Parameters of Mapper in practice - Cover ◮ Uniform cover I ◮ resolution / granularity: r (diameter of intervals) ◮ gain: g (percentage of overlap) ◮ Example:
Parameters of Mapper in practice - Cover ◮ Uniform cover I ◮ resolution / granularity: r (diameter of intervals) ◮ gain: g (percentage of overlap) ◮ Example: ◮ Modification of r and g can highly effect the result.
Cover examples
Cover examples
Cover examples
Cover examples
Mapper for Y-shape point cloud data
Mapper for Y-shape point cloud data
Parameters of uniform Cover Parameter r: ◮ Small r : fine cover, Mapper close to Reeb Graph , but sensitive to δ . ◮ Large r : rough cover, less sensitive to δ , but Mapper far from Reeb Graph . Parameter g: ◮ Large g(close to 1): more points inside intersections, less sensitive to δ but far from Reeb Graph . ◮ Small g(close to 0): controlled Mapper dimension, close to Reeb Graph .
Parameters of Mapper in practice - Clustering algorithm Single-linkage clustering is one of several methods of hierarchical clustering. ◮ Based on grouping clusters in bottom-up fashion (agglomerative clustering). ◮ At each step combining two clusters that contain the closest pair of elements not yet belonging to the same cluster as each other.
Example of Single-linkage clustering
Example of Single-linkage clustering
Example of Single-linkage clustering
Example of Single-linkage clustering
Example of Single-linkage clustering
Example of Single-linkage clustering
Example of Single-linkage clustering
Example of Clustering algorithm with different parameters
Example of Clustering algorithm with different parameters
Example of Clustering algorithm with different parameters
Example of Clustering algorithm with different parameters
Parameters of graph neighborhood size Parameter δ : ◮ Large δ : fewer nodes, clean Mapper but far from Reeb Graph (more straight lines). ◮ Small δ : presence of topological structure but lots of nodes (noisy).
Higher Dimensional Parameter Spaces ◮ We use 1 function and let R to be our 1-dimensional parameter space.
Higher Dimensional Parameter Spaces ◮ We use 1 function and let R to be our 1-dimensional parameter space. ◮ We can use M functions and let R M to be our M-dimensional parameter space, remain to find a covering of an M-dimensional hypercube which is defined by the ranges of the M functions.
Example of parameter space R 2 ◮ Assume we have a point could dataset P (2-Dim) as following. ◮ Assume we have two filter functions f : P → R , g : P → R , and f = f − 1 and g = g − 1 .
Example of parameter space R 2 ◮ Moreover, assume we have the following cover C , which is also the cover of P since f = f − 1 and g = g − 1 .
Example of parameter space R 2 ◮ Moreover, assume we have the following cover C , which is also the cover of P since f = f − 1 and g = g − 1 . ◮ Assume the clustering algorithm group every points in each rectangle as one cluster.
Example of parameter space R 2 ◮ Moreover, assume we have the following cover C , which is also the cover of P since f = f − 1 and g = g − 1 . ◮ Assume the clustering algorithm group every points in each rectangle as one cluster.
Example of parameter space R 2 ◮ Moreover, assume we have the following cover C , which is also the cover of P since f = f − 1 and g = g − 1 . ◮ Assume the clustering algorithm group every points in each rectangle as one cluster. ◮ Whenever clusters corresponding to any n vertices have non empty intersection, add a corresponding n-1 simplex.
Example of parameter space R 2 ◮ Two clusters intersection = 1 edge.
Example of parameter space R 2 ◮ Three clusters intersection = 1 triangle.
Example of parameter space R 2 ◮ Four clusters intersection = 1 tetrahedron.
Example of parameter space R 2 ◮ Final simplical complex.
Higher Dimensional Parameter Spaces Mapper to the parameter space R M can be extended in a similar fashion (by finding a covering of an M-dimensional hypercube which is defined by the ranges of the M functions).
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