Flattening Multiparameter Hierarchical Clustering Functors

One of the most useful ways to process a dataset represented as a finite metric space is to cluster the dataset, or break it into groups. An important step is choosing the desired granularity of the clustering, and multiparameter hierarchical clustering algorithms accept a hyperparameter vector to control this.

Since applying a multiparameter hierarchical clustering algorithm with different hyperparameter values may produce different partitions, we can view such algorithms as mapping a finite metric space $(X,d_{X})$ to a function from the hyperparameter space to the space of partitions of $X$. A flattening procedure then maps this function to a single partition.

Many popular clustering algorithms, such as HDBSCAN [7] and ToMATo [3], include a flattening step that operates on the same intuition that drives the analysis of persistence diagrams in TDA. That is, the most important clusters (homological structures) of a dataset are those which exist at multiple scales (have large differences between their birth and death times).

In this paper we will characterize and study clustering algorithms as functors, similarly to [1, 4, 12]. We will particularly focus on multiparameter hierarchical clustering algorithms with partially ordered hyperparameter spaces. This perspective allows us to guarantee that the clustering algorithms we study preserve both non-expansive maps between metric spaces and the ordering of the hyperparameter space. Our contributions are:

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  We describe an algorithm for flattening multiparameter hierarchical clusterings, which we demonstrate is a functorial map from a category of multiparameter hierarchical partitions to a category of binary integer programs.

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  We introduce a Bayesian update algorithm for learning a distribution over clustering hyperparameters from data.

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  We prove that the composition of the Bayesian update algorithm and the flattening procedure is consistent.

In this work we will define flat clustering algorithms to map a finite metric space $(X,d_{X})$ to a partition of $X$. We will primarily work with the following categories:

We will also work in the subcategories $\mathcal{\mathbf{Met}}_{bij},\mathbf{Part}_{bij}$ of $\mathcal{\mathbf{Met}},\mathbf{Part}$ respectively in which morphisms are further restricted to be bijective.

Now recall that the set of connected components of the $\delta$-Vietoris-Rips complex of $(X,d_{X})$ is the partioning of $X$ into subsets with maximum pairwise distance no greater than $\delta$. Given $a\in(0,1]$, an example of a flat clustering functor on $\mathcal{\mathbf{Met}}$ is the $a$-single linkage functor $\mathcal{SL}(a)$, which maps a metric space to the connected components of its $-log(a)$-Vietoris-Rips complex [12, 4, 1]. Given $a_{1},a_{2}\in(0,1]$, an example of a flat clustering functor on $\mathcal{\mathbf{Met}}_{bij}$ is the robust single linkage functor $\mathcal{SL}_{\mathcal{R}}(a_{1},a_{2})$ which maps a metric space $(X,d_{X})$ to the connected components of the $-log(a_{2})$-Vietoris-Rips complex of $(X,d^{a_{1}}_{X})$ where:

and $\mu_{X_{a_{1}}}(x_{1})$ is the distance from $x_{1}$ to its $\left\lfloor{a_{1}*|X|}\right\rfloor$th nearest neighbor [2]. Intuitively, robust single linkage reduces the impact of dataset noise by increasing distances in sparse regions of the space. Note that robust single linkage is not a flat clustering functor on $\mathcal{\mathbf{Met}}$ because it includes a $k$-nearest neighbor computation that is sensitive to $|X|$.

Like single linkage and robust single linkage, many flat clustering algorithms are configured by a hyperparameter vector that governs their behavior. In the case that this hyperparameter vector is an element of a partial order $O$ we can represent the output of such an algorithm with a functor $O\rightarrow\mathbf{Part}$.

Recall that $\mathbf{Part}^{O}$ is the category of functors from $O$ to $\mathbf{Part}$ and natural transformations between them. We will write $\mathbf{Part}^{\overline{O}}$ to represent the subcategory of such functors that commute with the forgetful functor $U:\mathbf{Part}\rightarrow\mathbf{Set}$. Given $F:O\rightarrow\mathbf{Part}$ in $\mathbf{Part}^{\overline{O}}$ we will call the image of $U\circ F$ the underlying set of $F$. Note that there also exists a forgetful functor $\mathbf{Part}^{\overline{O}}\rightarrow\mathbf{Set}$ that maps $F:O\rightarrow\mathbf{Part}$ to its underlying set and that any natural transformation in $\mathbf{Part}^{\overline{O}}$ between the functors $F_{X}:O\rightarrow\mathbf{Part}$ and $F_{Y}:O\rightarrow\mathbf{Part}$ with underlying sets $X$ and $Y$ is fully specified by a function $f:X\rightarrow Y$.

For example, single linkage $\mathcal{SL}:\mathcal{\mathbf{Met}}\rightarrow\mathbf{Part}^{\overline{(0,1]^{op}}}$ is a $(0,1]^{op}$-clustering functor on $\mathcal{\mathbf{Met}}$ and robust single linkage $\mathcal{SL}_{\mathcal{R}}:\mathcal{\mathbf{Met}}_{bij}\rightarrow\mathbf{Part}_{bij}^{\overline{(0,1]^{op}\times(0,1]^{op}}}$ is a $(0,1]^{op}\times(0,1]^{op}$-clustering functor on $\mathcal{\mathbf{Met}}_{bij}$.

We will write the elements of $S_{X}$ (subsets of $X$) with the notation $S_{X}=\{s_{{1}_{X}},s_{{2}_{X}},\cdots,s_{{n}_{X}}\}$.

In practice it is often convenient to “flatten” a functor $F_{X}\in\mathbf{Part}^{\overline{O}}$ to a single partition of $X$ by selecting a non-overlapping collection of sets from its partition collection $S_{X}$. Since we will express this selection problem as a binary integer program we will work in the following category:

When we construct a binary integer program from an object $F_{X}$ in $\mathbf{Part}^{\overline{O}}$ with underlying set $X$, we weight the elements of its partition collection $S_{X}$ according to a model of the importance of different regions of $O$. In this work we will only consider $O$ that are Borel measurable, so we can represent this model with a probability measure $\mu$ over $O$. This probabilistic interpretation will be useful in Section 2.2 when we update this model with labeled data. We can then view the flattening algorithm as choosing a non-overlapping subset $\mathcal{P}_{X}\subseteq S_{X}$ (the elements of $\mathcal{P}_{X}$ are subsets of $X$ where no element of $X$ belongs to more than a single set in $\mathcal{P}_{X}$) that maximizes the expectation of the function that maps $a$ to the number of $s_{{i}_{X}}\in\mathcal{P}_{X}$ that are also in $F_{X}(a)$. Formally, the algorithm maximizes $\int_{a\in O}\left|\{s_{{i}_{X}}\ |\ s_{{i}_{X}}\in\mathcal{P}_{X},s_{{i}_{X}}\in F_{X}(a)\}\right|d\mu$ . If $\mu$ is uniform this is similar to the Topologically Motivated HDBSCAN described in [7].

For example, if $\mu$ is uniform then the connected components of the Vietoris-Rips filtration of $(X,d_{X})$ that have the largest differences between their birth and death times will be a solution to $(Flatten_{\mu}\circ\mathcal{SL})(X,d_{X})$. Note also that $Flatten_{\mu}$ is only functorial over $\mathbf{Part}_{bij}^{\overline{O}}$, and not all of $\mathbf{Part}^{\overline{O}}$. Intuitively, this is because $Flatten_{\mu}$ maps natural transformations between elements of $\mathbf{Part}_{bij}^{\overline{O}}$ to linear maps that only exist when the functions underlying these natural transformations are bijective.