St. Olaf Pages - MapperMapper in Applications Nielson, et al. “Topological data analysis for...
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Mapper An algorithm for extracting shape characteristics of high-dimensional data sets. Image credit: Ayasdi
Transcript of St. Olaf Pages - MapperMapper in Applications Nielson, et al. “Topological data analysis for...
MapperAn algorithm for extracting shape
characteristics of high-dimensional data sets.
Image credit: Ayasdi
Mapper
Mapper was introduced in 2007 by Gurjeet Singh, Facundo Mémoli, and Gunnar Carlsson.
Mapper involves partial clustering of the data guided by a filter function.
Example: Continuous Setting
𝑋
𝒰
𝑓
ℝ
Image credit: Steve Oudot
Example: Continuous Setting
𝑋
𝑓
ℝ
𝒱
nerve 𝒱
Image credit: Steve Oudot
Example: Continuous SettingInput• Topological space 𝑋• Continuous function 𝑓: 𝑋 → ℝ• A cover 𝒞 of im𝑓 by open intervals: im𝑓 ⊆ 𝐼∈𝒞 𝐼
Method• Compute the pullback cover 𝒰 = 𝑓−1 𝐼 𝐼∈𝒞 of 𝑋.• Separate each 𝑈 ∈ 𝒰 into components, producing a refined cover 𝒱 of 𝑋.• Compute the nerve of 𝒱, a simplicial complex containing:
o A vertex for each 𝑉 ∈ 𝒱,o An edge for each intersection 𝑉1 ∩ 𝑉2 ≠ ∅, with 𝑉1, 𝑉2 ∈ 𝒱,
o A 𝑘-simplex for each (𝑘 + 1)-fold intersection 𝑖=0𝑘 𝑉𝑖 ≠ ∅, with
𝑉1, … , 𝑉𝑘 ∈ 𝒱.
Example: In Practice
In practice, we start with a discrete metric space (a point cloud).
𝑃
𝒰
𝑓
ℝImage credit: Steve Oudot
Example: In Practice
In practice, we start with a discrete metric space (a point cloud).
𝑃
𝒱
𝑓
ℝnerve 𝒱
Image credit: Steve Oudot
Parameters
Filter Function• May be motivated by the data• Or use trial-and-error • Common choices: distance, density• May use multiple functions (or a function into ℝ𝑑)
Other parameters• Size of intervals in cover of ℝ• Size of overlap of intervals in cover of ℝ• Clustering method used to refine the cover 𝒱 to obtain 𝒰.
Mapper in Applications
Monica Nicolau, Arnold J. Levine, Gunnar Carlsson. “Topology-Based Data Analysis Identifies a Subgroup of Breast Cancers With a Unique Mutational Profile and Excellent Survival.” Proceedings of the National Academy of Sciences. Vol. 108, No. 17, 2011, p. 7265 – 7270.
Classification of cancers
Mapper in Applications
Nielson, et al. “Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury.” Nature Communications. Vol. 6, 2015.
Recovery from spinal cord and brain injuries
Mapper in Applications
Mapper in Applications
