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An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

Frédéric Chazal, Bertrand Michel

Frontiers in Artificial Intelligence · 2021 · Review · CC BY

doi:10.3389/frai.2021.667963 (opens in a new tab)

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Abstract

With the recent explosion in the amount, the variety, and the dimensionality of available data, identifying, extracting, and exploiting their underlying structure has become a problem of fundamental importance for data analysis and statistical learning. Topological data analysis (tda) is a recent and fast-growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. It proposes new well-founded mathematical theories and computational tools that can be used independently or in combination with other data analysis and statistical learning techniques. This article is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for nonexperts.

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Figures14

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FIGURE 1
Left: the Hausdorff distance between two subsets A and B of the plane. In this example, d H (A, B) is the distance between the point a in A which is the farthest from B and its nearest neighbor b on B. Right: the Gromov–Hausdorff distance between A and B. A can be rotated—this is an isometric embedding of A in the plane—to reduce its Hausdorff distance to B. As a consequence, d GH (A, B) ≤ d H (A, B).
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Left: the Hausdorff distance between two subsets A and B of the plane. In this example, d H (A, B) is the distance between the point a in A which is the farthest from B and its nearest neighbor b on B. Right: the Gromov–Hausdorff distance between A and B. A can be rotated—this is an isometric embedding of A in the plane—to reduce its Hausdorff distance to B. As a consequence, d GH (A, B) ≤ d H (A, B).
Open FIGURE 1 full size (opens in a new tab)

FIGURE 2
Čech complex Cechα(X) (left) and the Vietoris–Rips Rips2α(X) (right) of a finite point cloud in the plane R2. The bottom part of Cechα(X) is the union of two adjacent triangles, while the bottom part of Rips2α(X) is the tetrahedron spanned by the four vertices and all its faces. The dimension of the Čech complex is 2. The dimension of the Vietoris–Rips complex is 3. Notice that this latter is thus not embedded in R2.
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Čech complex Cechα(X) (left) and the Vietoris–Rips Rips2α(X) (right) of a finite point cloud in the plane R2. The bottom part of Cechα(X) is the union of two adjacent triangles, while the bottom part of Rips2α(X) is the tetrahedron spanned by the four vertices and all its faces. The dimension of the Čech complex is 2. The dimension of the Vietoris–Rips complex is 3. Notice that this latter is thus not embedded in R2.
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FIGURE 3
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Point cloud sampled in the plane and a cover of open sets for this point cloud (left). The nerve of this cover is a triangle (right). Edges correspond to a set of the cover whereas a vertex corresponds to a non-empty intersection between two sets of the cover.
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FIGURE 4
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(A) Refined pull-back cover of the height function on a surface in R3 and its nerve. (B) Mapper algorithm on a point cloud sampled around a circle and the height function. First, the pull-back cover of the height function defined on the point cloud is computed and refined (left). Second, the nerve of the refined pull-back is visualized as a graph (right).
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FIGURE 5
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Example of a point cloud Xn sampled on the surface of a torus in R3 (top left) and its offsets for different values of radii r 1 < r 2 < r 3. For well-chosen values of the radius (e.g., r 1 and r 2), the offsets are clearly homotopy equivalent to a torus.
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FIGURE 6
Some examples of chains, cycles, and boundaries on a two-dimensional complex K: c 1, c 2, and c 4 are one-cycles; c 3 is a one-chain but not a one-cycle; c 4 is the one-boundary, namely, the boundary of the two-chain obtained as the sum of the two triangles surrounded by c 4. The cycles c 1 and c 2 span the same element in H 1(K) as their difference is the two-chain represented by the union of the triangles surrounded by the union of c 1 and c 2.
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Some examples of chains, cycles, and boundaries on a two-dimensional complex K: c 1, c 2, and c 4 are one-cycles; c 3 is a one-chain but not a one-cycle; c 4 is the one-boundary, namely, the boundary of the two-chain obtained as the sum of the two triangles surrounded by c 4. The cycles c 1 and c 2 span the same element in H 1(K) as their difference is the two-chain represented by the union of the triangles surrounded by the union of c 1 and c 2.
Open FIGURE 6 full size (opens in a new tab)

Show all 14 figures

FIGURE 7
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Betti numbers of the circle (top left), the two-dimensional sphere (top right), and the two-dimensional torus (bottom). The blue curves on the torus represent two independent cycles whose homology class is a basis of its one-dimensional homology group.
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FIGURE 8
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Effect of outliers on the sublevel sets of distance functions. Adding just a few outliers to a point cloud may dramatically change its distance function and the topology of its offsets.
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FIGURE 9
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(A) Example 1: the persistence barcode and the persistence diagram of a function f:[0,1]→R. (B) Example 2: the persistence barcode and the persistence diagram of the height function (projection on the z-axis) defined on a surface in R3.
Open FIGURE 9 full size (opens in a new tab)

FIGURE 10
The sublevel set filtration of the distance function to a point cloud and the construction of its persistence barcode as the radius of balls increases. The blue curves in the unions of balls represent one-cycles associated with the blue bars in the barcodes. The persistence diagram is finally defined from the persistence barcodes. (A) For the radius r = 0, the union of balls is reduced to the initial finite set of points, each of them corresponding to a zero-dimensional feature, that is, a connected component; an interval is created for the birth for each of these features at r = 0. (B) Some of the balls started to overlap, resulting in the death of some connected components that get merged together; the persistence diagram keeps track of these deaths, putting an end point to the corresponding intervals as they disappear. (C) New components have merged, giving rise to a single connected component and, so, all the intervals associated with a zero-dimensional feature have been ended, except the one corresponding to the remaining components; two new one-dimensional features have appeared, resulting in two new intervals (in blue) starting on their birth scale. (D) One of the two one-dimensional cycles has been filled, resulting in its death in the filtration and the end of the corresponding blue interval. (E) All the one-dimensional features have died; only the long (and never dying) red interval remains. As in the previous examples, the final barcode can also be equivalently represented as a persistence diagram where every interval (a, b) is represented by the point of coordinates (a, b) in R2. Intuitively, the longer an interval in the barcode or, equivalently, the farther from the diagonal the corresponding point in the diagram, the more persistent, and thus relevant, the corresponding homological feature across the filtration. Notice also that for a given radius r, the kth Betti number of the corresponding union of balls is equal to the number of persistence intervals corresponding to k-dimensional homological features and containing r. So, the persistence diagram can be seen as a multiscale topological signature encoding the homology of the union of balls for all radii as well as its evolution across the values of r.
Full legend
The sublevel set filtration of the distance function to a point cloud and the construction of its persistence barcode as the radius of balls increases. The blue curves in the unions of balls represent one-cycles associated with the blue bars in the barcodes. The persistence diagram is finally defined from the persistence barcodes. (A) For the radius r = 0, the union of balls is reduced to the initial finite set of points, each of them corresponding to a zero-dimensional feature, that is, a connected component; an interval is created for the birth for each of these features at r = 0. (B) Some of the balls started to overlap, resulting in the death of some connected components that get merged together; the persistence diagram keeps track of these deaths, putting an end point to the corresponding intervals as they disappear. (C) New components have merged, giving rise to a single connected component and, so, all the intervals associated with a zero-dimensional feature have been ended, except the one corresponding to the remaining components; two new one-dimensional features have appeared, resulting in two new intervals (in blue) starting on their birth scale. (D) One of the two one-dimensional cycles has been filled, resulting in its death in the filtration and the end of the corresponding blue interval. (E) All the one-dimensional features have died; only the long (and never dying) red interval remains. As in the previous examples, the final barcode can also be equivalently represented as a persistence diagram where every interval (a, b) is represented by the point of coordinates (a, b) in R2. Intuitively, the longer an interval in the barcode or, equivalently, the farther from the diagonal the corresponding point in the diagram, the more persistent, and thus relevant, the corresponding homological feature across the filtration. Notice also that for a given radius r, the kth Betti number of the corresponding union of balls is equal to the number of persistence intervals corresponding to k-dimensional homological features and containing r. So, the persistence diagram can be seen as a multiscale topological signature encoding the homology of the union of balls for all radii as well as its evolution across the values of r.
Open FIGURE 10 full size (opens in a new tab)

FIGURE 11
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Example of a persistence landscape (right) associated with a persistence diagram (left). The first landscape is in blue, the second one in red, and the last one in orange. All the other landscapes are zero.
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FIGURE 12
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Perfect matching and the bottleneck distance between a blue and a red diagram. Notice that some points of both diagrams are matched to points of the diagonal.
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FIGURE 13
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(A,B) Two persistence diagrams for two configurations of MBP. (C) MDS configuration for the matrix of bottleneck distances. (D) Persistence diagram and confidence region for the persistence diagram of an MBP.
Open FIGURE 13 full size (opens in a new tab)

FIGURE 14
(A) First three landscapes for zero-homology of the alpha shape filtration defined for a time series of acceleration of Walker A. (B) Variable importances of the landscape coefficients for the classification of walkers. The first 3,000 coefficients correspond to the three landscapes of dimension 0 and the last 3,000 coefficients to the three landscapes of dimension 1. There are 1,000 coefficients per landscape. Note that the first landscape of dimension 0 is always the same using the Rips complex (a trivial landscape), and consequently, the corresponding coefficients have a zero-importance value.
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(A) First three landscapes for zero-homology of the alpha shape filtration defined for a time series of acceleration of Walker A. (B) Variable importances of the landscape coefficients for the classification of walkers. The first 3,000 coefficients correspond to the three landscapes of dimension 0 and the last 3,000 coefficients to the three landscapes of dimension 1. There are 1,000 coefficients per landscape. Note that the first landscape of dimension 0 is always the same using the Rips complex (a trivial landscape), and consequently, the corresponding coefficients have a zero-importance value.
Open FIGURE 14 full size (opens in a new tab)

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