# data[].attributes.body
Name: Betti Numbers
Distances: · Euclidean distance between Betti curves
Short Explanation: attempt to construct simulated models; TDA project computes persistent homology of all the models to see what looks most realistic
Reference to first use: Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function.  Michael W Reimann, et al
Pros: Euclidean distance easy to measure
Cons: ???

# data[].attributes.body
Name: Betti Numbers
Distances: ·Euclidean distance between lines in crocker plot
Short Explanation: Motivated by classification
Reference to first use: ??
Pros: K-mediods (R Package) easy to use, Euclidean distance is very easy to compute
Cons: Trying to find another way to measure distance: interleaving distance

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Name: Carlsson and Carlsson et al. summaries
Distances: ·any distance on $\mathbb{R}^n$
Short Explanation: Short set of statistics that are easily computed from the birth and death coordinates of persistence points.
Reference to first use: Carlsson and Carlsson et al. (need to look up reference!)
Pros: This is a simple summary of persistence diagrams, since there are several summaries of these points, it is
Cons: You might lose information in this simplification.

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Name: Persistence Landscapes
Distances: ·L_\infty distance
Short Explanation: Turn PD 45 degrees and create an arrangement of lines to summarize the size of features
Reference to first use: Bubenik 2013
Pros: Based on barcode in vector space
Cons: sensitive to outliers

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Name: Euler Curve
Distances: ·distances between curves
Short Explanation: compute EC at every point in a filtered space
Reference to first use: ???
Pros: fast
Cons: not descriptive

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Name: PHT Transform
Distances between descriptors: sum of d_B between matching of diagrams
Short explanation: Set of persistence diagrams related to simplicial complex/mesh embedded in R^k
Reference to first use: Kate Turner's PHT
Pros: straight foward to describe geometrically, captures geometry, stable
Cons: sensitive to rotation, can involve a lot of computation depending on the shape,
information loss can be large if not done correctly

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Name: CROCKER plot
Distances: ·The matrix can be vectorized, so any distance that works in Euclidean space is applicable, i.e. L_1, L_2, L_\infty, angle between subspaces, etc.
Short Explanation: Simple representation of two-parameter persistence, where we compute the number of topological features at each of the two parameters. This gives a betti count (i.e. dimension of the homology vector space) for every input and can be represented as a matrix and visualized as a contour plot
Reference to first use: Topological Data Analysis of Biological Aggregation Models – PLOS One, 2015, Topaz, Ziegelmeier, Halverson
Pros: can concatenate different features, vector representation, can perform machine learning tasks, can observe how homology changes
Cons: not stable, same features are not tracked from one set of parameters to the next.

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Name: Persistence Images
Distances: any metric on Rn
Short explanation: Vectorize each persistence diagram by centering a weighted kernel at each point in the persistence diagram
Reference to first use:  Adams et al. Persistence Images, JMLR 2017
Pros: Opens the door to many machine learning algorithms that require more than just the distance between points. There is lots of flexibility. PIs are stable with respect to 1-Wasserstein metric. You have the option of using lots of different metrics.
Cons: can't go backwards from this vector representation to the persistence diagram again. There are some parameters to choose (like resolution, the weighting function). There is not a way to recover the persistence diagram from the image.

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Name: Metric Graph
Distances between descriptors: persistence-distortion distance
Short explanation: The Hausdorff distance between two sets of persistent diagrams obtained from the graphs.
Reference to first use: Dey, Shi, Wang; Comparing graphs via Persistence Distortion
Pros: Stable under perturbations
Cons: Computationally not easy, not a metric, non-isomorphic graphs can have distance zero.

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Name: Mapper/Reeb Cosheaf
Distances between descriptors: Mapper can be seen as a cosheaf construction, and there is a natural interleaving metric defined between cosheaves.
Short explanation: Measures the amount of smoothing necessary to be able to map one Mapper into the other.
Reference to first use: Elizabeth Munch and Bei Wang, Convergence between Categorical Representations of Reeb Space and Mapper
Pros: Easy to visualize and explain, computation of Mapper is efficient.
Cons: Computing distances is difficult

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Name: Persistence Diagrams
Distances between descriptors: Focusing on bottleneck distance between two persistence diagrams
Short explanation: Measures the largest distance between the matching of two points (or one point and the diagonal) in the "best" matching between two diagrams (and their subsequent diagonals)
Reference to first use: Edelsbrunner and Harer, Computational Topology, an Introduction
Pros: Fast, simple to visualize and describe, stable relative to the distance between the real-valued functions on the topological space
Cons: Sensitive to outliers