Attraction-repulsion clustering: a way of promoting diversity linked to demographic parity in fair clustering

Abstract

We consider the problem of diversity enhancing clustering, i.e, developing clustering methods which produce clusters that favour diversity with respect to a set of protected attributes such as race, sex, age, etc. In the context of fair clustering, diversity plays a major role when fairness is understood as demographic parity. To promote diversity, we introduce perturbations to the distance in the unprotected attributes that account for protected attributes in a way that resembles attraction-repulsion of charged particles in Physics. These perturbations are defined through dissimilarities with a tractable interpretation. Cluster analysis based on attraction-repulsion dissimilarities penalizes homogeneity of the clusters with respect to the protected attributes and leads to an improvement in diversity. An advantage of our approach, which falls into a pre-processing set-up, is its compatibility with a wide variety of clustering methods and whit non-Euclidean data. We illustrate the use of our procedures with both synthetic and real data and provide discussion about the relation between diversity, fairness, and cluster structure.

1 Introduction

Artificial Intelligence (AI) in its different branches is pervasive in today’s world. It is almost impossible to ignore its effects on individuals, groups, and even entire populations. Although the AI revolution has brought a variety of major positive breakthroughs, as it is common in any revolution, time has evidenced some of the limitations and harms involved in heavily relying on algorithms and data. An account of the history of AI development and some of the causes of its limitations are given in Cristianini (2019). The author argues that the negative effects of AI that we are facing are due to an ‘ethical debt’, a kind of externality of our technological infrastructure. This ‘ethical debt’ is formed by, among others, the cost of fixing the lack of fairness and transparency of machine decisions, the cost of regulating AI and the cost of making services secure against surveillance and manipulation.

An entire research field, known as fair learning, is currently dedicated to ‘repaying’ part of the ‘ethical debt’ by studying causes and mitigating or eliminating effects of bias and unfairness in machine learning. It is widely agreed that supervised and unsupervised classification procedures are increasingly more influential in people’s life since they are used in credit scoring, article recommendation, risk assessment, spam filtering or sentencing recommendations in courts of law, among others. Hence, controlling the outcome of such procedures, in particular ensuring that some variables which should not be considered due to moral or legal issues are not playing a role in the classification of the observations, has become an important field of research. The main concern is to detect whether decision rules, learnt from variables X, are biased with respect to a subcategory of the population driven by some variables called protected or sensitive. Such variables induce a bias in the observations and are correlated to other observations. Hence avoiding this effect cannot be achieved by the naive solution of ignoring such protected attributes. Indeed, if the data at hand reflects a real-world bias, machine learning algorithms can pick on this behaviour and emulate it. An extensive overview on types of biases and discrimination and on fair machine learning strategies can be found in Mehrabi et al. (2019). Additional overviews of such legal issues and mathematical solutions to address them can be found in (Lum and Johndrow 2016; Chouldechova 2017; Besse et al. 2018) or Friedler et al. (2018).

Any fair machine learning procedure starts with a particular definition of fairness, i.e., with a statement of an abstract mathematical proxy for some fairness notion that must be considered. A non-exhaustive list of definitions can be found in Mehrabi et al. (2019). It is worth mentioning that some popular definitions of mathematical fairness are incompatible between each other, and that different situations may require different definitions of fairness. Once a definition for fairness is provided, a way of imposing or promoting it can be established, for example, by transforming the data to avoid correlation between the set of sensitive attributes and the rest of the data (Feldman et al. 2015; del Barrio et al. 2018) or by modifying the objective functions of the algorithms in a way that favours fairness (Zafar et al. 2017; Kehrenberg et al. 2018). Notice that this scheme has sources of bias, one is the subjective choice of a definition of fairness, and another one is the choice of a mathematical proxy for that definition. Therefore, a problem specific approach may be a good way of avoiding pitfalls.

The problem we will be interested in is intimately related to the problem of fair clustering. Classical (or standard) cluster analysis or clustering is the task of dividing a set of objects in such a way that elements in the same group or cluster are more similar, according to some dissimilarity measure, than elements in different groups. Such methods have been extensively investigated in the literature. We refer to Hennig et al. (2015) and references therein for an in-depth overview. However, classical clustering is insensitive to notions of mathematical fairness. Suppose we observe data that includes information about attributes that we know, or suspect are biased with respect to the protected class. If the biased variables are dominant enough, a standard clustering on the unprotected data will result in some biased clusters, that is, clusters with possibly large differences in distribution of the protected attributes. Therefore, if we classify new instances based on this partition, we will incur in biased decisions. In essence, this will be a violation of the notion of demographic parity (Feldman et al. (2015)), a popular proxy for fairness, adapted to the clustering setting.

A natural mathematical definition for fairness in the setting of clustering, and the one that we adopt, is based on the notion of demographic parity, namely, on looking for balanced clusters, as introduced in Chierichetti et al. (2017). Quoting from Abbasi et al. (2021) ‘What does it mean for a clustering to be fair? One popular approach seeks to ensure that each cluster contains groups in (roughly) the same proportion in which they exist in the population. The normative principle at play is balance: any cluster might act as a representative of the data, and thus should reflect its diversity’. Hence, a fair clustering would be the situation in which, in the partition of the data, the proportions of the protected attributes are the same in each cluster (hence, the same as the proportions in the whole dataset). We might argue that balance (demographic parity) is not necessarily equivalent to fairness, and proper consideration must be given to the problem of when it is appropriate and useful to identify fairness with balance. Nonetheless, diversity in itself may be a desirable property. Even more, diversity may be easier to formalize and impose, since it is less charged with ethical connotations. Hence, we will focus on clustering procedures that favour diversity in the protected attributes, in what we call diversity enhancing clustering.

In our setting, diversity enhancing clustering is a relaxation of diversity preserving clustering (which is usually considered as a fair clustering method). The latter can be described as follows. Let us have a dataset $\mathcal{D}\mathcal{S}=\{(X_1,S_1),\dots ,(X_n,S_n)\}$, with $S_i\in S$, and for simplicity assume that the protected attributes S are discrete. Notice that the unprotected attributes $X_i$ and the protected ones $S_i$ should be correlated for diversity preserving clustering to be relevant. A diversity preserving clustering or a diversity preserving partition, ${\mathcal {C}}_f=\{{\mathcal {C}}_i\}_{i=1}^k$, where ${\mathcal {C}}_i\subset \mathcal{D}\mathcal{S}$, fulfils that

$$\begin{aligned} \frac{|\{(x,s)\in {\mathcal {C}}_i: s=j\}|}{|{\mathcal {C}}_i|}= & {} \frac{|\{(x,s)\in \mathcal{D}\mathcal{S}: s=j\}|}{|\mathcal{D}\mathcal{S}|} \nonumber \\{} & {} \quad \text {for every} \quad j\in S \quad \text {and}\quad i=1,\dots , k. \end{aligned}$$

(1)

In this case the proportion of individuals in any cluster for any protected class is the same as the respective proportion in the total data, hence we could say that proportions are independent of the cluster. This means that any decision taken with respect to a particular cluster will affect individuals in the same proportion as if it were taken for the entire population. Therefore, disparate impact for some sub-population would be avoided. Diversity enhancing clustering seeks to improve diversity with respect to a classical clustering procedure, without imposing the diversity preserving condition (1). Here, improving diversity means that clusters have more diverse composition with respect to the protected attributes compared with the clusters obtained by a classical clustering procedure.

Our approach to diversity enhancing clustering falls into the category of pre-processing. This means that we want to transform the unprotected data to obtain more diverse partitions. Since we are in an unsupervised setting, which means that there is no ground truth, looking for an adequate transformation of the data becomes very challenging. To address this problem, we propose a transformation based on a heuristic that incorporates some perturbations in the original dissimilarities of the data to favour heterogeneity with respect to the protected variable. For this, we present a new methodology inspired by electromagnetism and based on attraction-repulsion dissimilarities. These new dissimilarities aim at increasing the separation between points with the same values of the protected class and/or decreasing the separation between points with different values of the protected class. Hence they favour the formation of clusters that are more heterogeneous, leading to a possible gain in diversity. The lack of ground truth also leads us to consider a trade-off between diversity and some other quality measures of the clusterings on the data at hand, in the spirit of Empirical Risk Minimization.

The main contributions of our work are the following:

• The introduction of novel dissimilarities that perturb the distance in the unprotected attributes considering the protected ones.

• A new methodology based on a heuristic that promotes diversity and is applicable to almost any clustering algorithm with no or minimal modifications to the original objective functions.

• A call for diversity and, more broadly, fairness in clustering to be considered jointly with other relevant aspects that make a partition sensible.

• A methodology that is easily adaptable to non-Euclidean data and clusters with non-convex boundaries.

• An R package to perform Attraction-repulsion clustering freely available at https://github.com/HristoInouzhe/AttractionRepulsionClustering

The structure of this work is the following. Section 2 presents a brief overview of related works. In Sect. 3 we introduce attraction-repulsion dissimilarities. Clustering methods are developed in Sect. 4 while Sect. 5 is devoted to their extension to non-Euclidean spaces through a kernel transformation. We study the tuning of our methods in Sect. 6. Experiments for our technique are given in Sect. 7. We provide a discussion on synthetic datasets in 7.1. In Sect. 7.2, we provide comparison between our methods and the ones proposed in Chierichetti et al. (2017), on synthetic and real data, showing the different behaviour of both procedures. In Sect. 7.3 we give a full example of attraction-repulsion clustering on a real dataset. Section 8 is devoted to some final remarks and some future work proposals.

2 Related work

Related works belong to the field of fair clustering. One of the first and most popular approaches to fair clustering is to impose constraints on the objective functions of classical clustering procedures, and corresponds to what we have called diversity preserving clustering. In Chierichetti et al. (2017) constrained k-center and k-median clustering was introduced. The authors proposed a model where data are partitioned into two groups, codified by red and blue, where disparate impact is avoided by maintaining balance between the proportion of points in the two categories. Their goal is to achieve a partition of the data where the respective objective function is minimized while balance in each cluster is maintained over a given threshold. Their approach initially designed for data with two different protected classes (two-state) has led to various extensions (multi-state). Contributions to the k-center problem can be found in (Rösner and Schmidt 2018; Bercea et al. 2018), to the k-median problem in Backurs et al. (2019) and to the k-means problem in (Bercea et al. 2018; Bera et al. 2019; Schmidt et al. 2018). These works discuss a broader range of clustering problems while imposing constraints related to some minimum and/or maximum value for the proportions of the protected classes in each cluster. A multi-state deep learning fair clustering procedure was developed in Wang and Davidson (2019). A multi-state non-disjoint extension was presented in Huang et al. (2019). In Mazud Ziko et al. (2019) a variational framework of fair clustering compatible with several popular clustering objective functions was proposed.

Somewhat different approaches, that do not fall into diversity preserving clustering, can be found in the following works. In Ahmadian et al. (2019) the goal is to cluster the points to minimize the k-center cost but with the additional constraint that no colour is over-represented in any cluster. Chen et al. (2019) cluster n points with k centers, with fairness meaning that any n/k points are entitled to form their own cluster if there is a center that is closer in distance for all n/k points. In (Abbasi et al. 2021; Ghadiri et al. 2021) fair center based clustering is understood to be fair if the centers ‘represent’ different groups equally well and they provide algorithms for this purpose.

Fair clustering is a vibrant field with a growing number of new contributions, so we do not claim this to be an exhaustive overview of relevant works. For more information we refer to the previously mentioned works and sources therein.

3 Charged clustering via multidimensional scaling

Clustering relies on the choice of dissimilarities that control the part of information conveyed by the data that will be used to gather points into the same cluster, expressing how such points share some similar features. To obtain a diversity enhancing clustering we aim at obtaining clusters which are not governed by the protected variables but are rather mixed with respect to these variables. For this, we introduce interpretable dissimilarities in the space $(X,S)\in {\mathbb {R}}^{d}\times {\mathbb {R}}^p$ that separate points with the same value of the protected classes. Using an analogy with electromagnetism, the labels S play the role of an electric charge and similar charges tend to have a repulsive effect while dissimilar charges tend to attract themselves. We stress that $S\in {\mathbb {R}}^p$, hence it can take non-integer values, and all our procedures are well suited for such values. However, our examples concentrate on integer values for interpretability and simplicity.

Our guidelines for choosing these dissimilarities are that we would like the dissimilarities to

i):

induce diversity into subsequent clustering techniques (decrease, or, at best, eliminate, identifiability between clusters and protected attributes),

ii):

keep the essential geometry of the data (with respect to non-protected attributes) and

iii):

be easy to use and interpret.

To fulfil i), we introduce new dissimilarities that consider the geometry of the unprotected attributes while penalizing homogeneity with respect to the protected ones. For ii), we will combine these dissimilarities with multidimensional scaling, to retain most of the original geometry. With respect to iii), we refer to the discussion below. Hence, we propose the following dissimilarities.

Definition 1

(Attraction-Repulsion Dissimilarities)

$$\begin{aligned} \delta _1\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) = 1'U1 + S_1'VS_2 + \Vert X_1-X_2\Vert ^2 \end{aligned}$$

(2)

with U, V symmetric matrices in ${\mathbb {R}}^{p\times p}$;

$$\begin{aligned} \delta _2\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) = \left( 1 + ue^{-v\Vert S_1-S_2\Vert ^2}\right) \Vert X_1-X_2\Vert ^2 \end{aligned}$$

(3)

with $u,v\ge 0$;

$$\begin{aligned} \delta _3\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) = \Vert X_1-X_2\Vert ^2 - u\Vert S_1-S_2\Vert ^2 \end{aligned}$$

(4)

with $u\ge 0$.

Let $0\le u\le 1$ and $v,w\ge 0$,

$$\begin{aligned} \delta _4\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right)= & {} \left( 1 + \textrm{sign}(S_1'VS_2)u\left( 1-e^{-v(S_1'VS_2)^2}\right) e^{-w\Vert X_1-X_2\Vert }\right) \nonumber \\{} & {} \Vert X_1-X_2\Vert . \end{aligned}$$

(5)

Remark 1

$\delta _1((X,S),(X,S))\ne 0$ and therefore it is not strictly a dissimilarity in usual sense, see, for example, Chapter 3 in Everitt et al. (2011). Yet, for all practical purposes discussed in this work this does not affect the proposed procedures.

Remark 2

Both $\delta _1$ and $\delta _3$ could produce negative values. For this reason, we add a step that ensures positiveness in the algorithms at the end of this and the next sections.

Remark 3

All our dissimilarities can handle categorical variables as protected attributes S, once they are appropriately codified in numerical values. Furthermore, $\delta _2$ and $\delta _3$ are applicable to more general situations where the protected class is not discrete, as, for example, age or income. This is due to the treatment of S as a quantitative variable in these dissimilarities.

To the best of our knowledge this is the first time that such dissimilarities have been proposed and used in the context of clustering (in Ferraro and Giordani (2013) repulsion was introduced modifying the objective function, considering only distances between points, to maintain centers of clusters separated). Dissimilarities (2) to (5) are natural in the context of diversity enhancing clustering because they penalize the Euclidean distance considering the protected class of the points involved. Hence, some gains in diversity could be obtained.

The dissimilarities we consider are interpretable, providing the practitioner with the ability to understand and control the degree of perturbation introduced. Dissimilarity (2) is an additive perturbation of the squared Euclidean distance where the intensity of the penalization is controlled by matrices U and V, with V controlling the interactions between elements of the same and of different classes S. Dissimilarity (4) presents another additive perturbation but the penalization is proportional to the difference between the classes $S_1$ and $S_2$, and the intensity is controlled by the parameter u.

Dissimilarity (3) is a multiplicative perturbation of the squared Euclidean distance. With u we control the amount of maximum perturbation achievable, while with v we modulate how fast we diverge from this maximum perturbation when $S_1$ is different to $S_2$.

Dissimilarity (5) is also a multiplicative perturbation of the Euclidean distance. However, it has a very different behaviour with respect to (2)-(4). It is local, i.e., it affects less points that are further apart. Through w we control locality. With bigger w the perturbation is meaningful only for points that are closer together. With matrix V we control interactions between classes as in (2), while with u we control the amount of maximum perturbation as in (3). Again, v is a parameter controlling how fast we diverge from the maximum perturbation.

Next, we present a simple example for the case of a single binary protected attribute, coded as $-1$ or 1. This is an archetypal situation in which there is a population with an (often disadvantaged) minority, that we code as $S=-1$, and the new clustering has to be independent (or not too dependent) on S.

Example 1

Let us take $S_1,S_2\in \{-1,1\}$. For dissimilarity (2) we fix $U=V=c\ge 0$, therefore

$$\begin{aligned}\delta _1\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) = c(1 + S_1S_2) + \Vert X_1-X_2\Vert ^2.\end{aligned}$$

If $S_1\ne S_2$, we have the usual squared distance $\Vert X_1-X_2\Vert ^2$, while when $S_1=S_2$ we have $2c+\Vert X_1-X_2\Vert ^2$, effectively we have introduced a repulsion between elements with the same class. For dissimilarity (3) let us fix $u=0.1$ and $v = 100$,

$$\begin{aligned}\delta _2\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) = \left( 1 + 0.1e^{-100\Vert S_1-S_2\Vert ^2}\right) \Vert X_1-X_2\Vert ^2.\end{aligned}$$

When $S_1\ne S_2$, $\delta _2\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) \approx \Vert X_1-X_2\Vert ^2$, while when $S_1=S_2$, $\delta _2\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) \approx 1.1\Vert X_1-X_2\Vert ^2$, again introducing a repulsion between elements of the same class. For dissimilarity (4), when $S_1 = S_2$, $\delta _3\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) =\Vert X_1-X_2\Vert ^2$ and when $S_1\ne S_2$ we get $\delta _3\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) =\Vert X_1-X_2\Vert ^2-2u$, therefore we have introduced an attraction between different members of the sensitive class. When using dissimilarity (5), fixing $V = c > 0$, $u = 0.1$, $v = 100$, $w = 1$, we get

$$\begin{aligned} \delta _4\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right)= & {} \left( 1 + 0.1\textrm{sign}(cS_1'S_2)\left( 1-e^{-100(cS_1'S_2)^2}\right) e^{-\Vert X_1-X_2\Vert }\right) \\{} & {} \Vert X_1-X_2\Vert .\end{aligned}$$

If $S_1=S_2$, $\delta _4\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) \approx \left( 1+0.1e^{-\Vert X_1-X_2\Vert }\right) \Vert X_1-X_2\Vert$, therefore we have a repulsion. If $S_1\ne S_2$, $\delta _4\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) \approx \left( 1-0.1e^{-\Vert X_1-X_2\Vert }\right) \Vert X_1-X_2\Vert$, which can be seen as an attraction. $\square$

Our proposals are flexible thanks to the freedom in choosing the class labels. If we encode S as $\{-1,1\}$, as in the previous example, we can only produce attraction between different classes and repulsion between the same classes (or exactly the opposite if $V<0$) in (2) and (5). On the other hand, if we encode S as $\{(1,0),(0,1)\}$, we have a wider range of possible interactions induced by V. For example, taking $V = ((1,-1)'|(-1,0)')$ we produce attraction between different classes, no interaction between elements labelled as (0, 1) and repulsion between elements labelled as (1, 0). If we had three classes we could use $\{(1,0,0),(0,1,0),(0,0,1)\}$ as labels and induce a tailor-made interaction between the different elements via a $3\times 3$ matrix V. For example, $V=((0,-1,-1)'|(-1,0,-1)'|(-1,-1,0)')$ provides attraction between different classes and no interaction between elements of the same class. Extensions to more than three classes are straightforward. More details on parameter and class codification selection will be given in Sects. 6 and 7.

These dissimilarities can be used directly in some agglomerative hierarchical clustering methods, as described in Sect. 4. However, there is a way to extend our methodology to a broader set of clustering methods. This is done using our dissimilarities to produce some embedding of the data into a suitable Euclidean space which allows the use of optimization clustering techniques (in the sense described in Chapter 5 in Everitt et al. (2011)) on the embedded data. For example, the dissimilarities $\delta _l$ can be combined with a common optimization clustering technique, such as k-means, via some embedding of the data. We note that our dissimilarities aim at increasing the separation of points with equal values in the protected attributes while respecting otherwise the geometry of the data. Using multidimensional scaling (MDS) we can embed the original points in the space ${\mathbb {R}}^{d'}$ with $d'\le d$ and use any clustering technique on the embedded data. The approach of using MDS to embed dissimilarities in an Euclidean space and then perform clustering has been pursued in Hausdorf and Hennig (2003) and Oh and Raftery (2007). Quoting Cox and Cox (2000), multidimensional scaling ‘is the search for a low dimensional space, usually Euclidean, in which points in the space represent the objects, one point representing one object, and such that the distances between the points in the space, match, as well as possible, the original dissimilarities’. Thus, applied to dissimilarities $\delta _l$, MDS will lead to a representation of the original data that approximates the original geometry of the data in the unprotected attributes and, at the same time, favours clusters with diverse values in the protected attributes. We stress that we are using MDS as a tool for embedding our dissimilarity relations in a Euclidean space, not necessarily as a dimension reduction technique. However, it is worth mentioning that MDS embeddings produce some loss of the original dissimilarity structure. Therefore, we advise to use embeddings in a space with similar dimension as the original data to minimize this adverse effect. In fact, in our experiments, we use the same dimension as that of the original data’s unprotected attributes.

Here is an outline of how to use the dissimilarities $\delta _l$ coupled with MDS for a sample $(X_1,S_1),\dots ,(X_n,S_n)$.

Attraction-Repulsion MDS For any $l \in \{1,2,3,4\}$

• Compute the dissimilarity matrix $[\varDelta _{i,j}]=[\delta _l((X_i,S_i),(X_j,S_j))]$ with a particular choice of the free parameters.

• If $\textrm{min} \varDelta _{i,j} \le 0$, transform the original dissimilarity to have positive entries: $\varDelta _{i,j} = \varDelta _{i,j} + |\textrm{min}\varDelta | + \epsilon$, where $\epsilon$ is small.

• For $\delta _1,\delta _2,\delta _3$: $\varDelta _{i,j} = \sqrt{\varDelta _{i,j}}$.

• Use MDS to transform $(X_1,S_1),\dots ,(X_n,S_n)$ into $X'_1, \dots , X'_n\in {\mathbb {R}}^{d'}$, where $D_{i,j} = \Vert X'_i-X'_j\Vert$ is similar to $\varDelta _{i,j}$.

• Apply a clustering procedure on the transformed data $X'_1, \dots , X'_n$.

This procedure will be studied in Sect. 7 for some synthetic and real datasets.

4 Charged hierarchical clustering

Agglomerative hierarchical clustering methods (bottom-top clustering) encompass many of the most widely used methods in unsupervised learning. Rather than a fixed number of clusters, these methods produce a hierarchy of clusterings starting from the bottom level, at which each sample point constitutes a group, to the top of the hierarchy, where all the sample points are grouped into a single unit. We refer to Murtagh and Contreras (2011) for an overview. The main idea is simple. At each level, the two groups with the lowest dissimilarity are merged to form a single group. The starting point is typically a matrix of dissimilarities between pairs of data points. Hence, the core of a particular agglomerative hierarchical clustering lies at the way in which dissimilarities between groups are measured. Classical choices include single linkage, complete linkage, average linkage or McQuitt’s method. Additionally, some other very popular methods are readily available for using dissimilarities, as, for example, PAM (Partitioning Around Medoids, also known as k-medoids) introduced in Kaufman and Rousseeuw (1987) and DBSCAN (Density-Based Spatial Clustering of Applications with Noise) introduced in Ester et al. (1996).

When a full data matrix (rather than a dissimilarity matrix) is available it is possible to use a kind of agglomerative hierarchical clustering in which every cluster has an associated prototype (a center or centroid) and dissimilarity between clusters is measured through dissimilarity between the prototypes. A popular choice (see Everitt et al. (2011)) is Ward’s minimum variance clustering: dissimilarities between clusters are measured through a weighted squared Euclidean distance between mean vectors within each cluster. More precisely, if clusters i and j have $n_i$ and $n_j$ elements and mean vectors $g_i$ and $g_j$ then Ward’s dissimilarity between clusters i and j is

$$\begin{aligned} \delta _{W}(i,j)={\textstyle \frac{n_i n_j}{n_i+n_j}}\Vert g_i -g_j\Vert ^2, \end{aligned}$$

where $\Vert \cdot \Vert$ denotes the usual Euclidean norm. Other methods based on prototypes are the centroid method or Gower’s median method (see Murtagh and Contreras (2011)). However, these last two methods may present some undesirable features (the related dendrograms may present reversals that make the interpretation harder, see, e.g., Everitt et al. (2011)) and Ward’s method is the most frequently used within this prototype-based class of agglomerative hierarchical clustering methods.

Hence, in our approach to diversity enhancing clustering we will focus on Ward’s method. Given two clusters i and j consisting of points $\{(X_{i'},S_{i'})\}_{i'=1}^{n_i}$ and $\{(Y_{j'},T_{j'})\}_{j'=1}^{n_j}$, respectively, we define the charged dissimilarity between them as

$$\begin{aligned} \delta _{W,l}(i,j) = \frac{n_i n_j}{n_i+n_j}\delta _l(({\bar{X}}_i,{\bar{S}}_i), ({\bar{Y}}_j,{\bar{T}}_j)) \end{aligned}$$

(6)

where $\delta _l$, $l=1,\ldots ,4$ is any of the point dissimilarities defined by (2) to (5) and the bar notation refers to the standard sample mean over the points in the respective cluster.

The practical implementation of agglomerative hierarchical methods depends on the availability of efficient methods for the computation of dissimilarities between merged clusters. This is the case of the family of Lance-Williams methods (see Lance and Williams (1967), Murtagh and Contreras (2011) or Everitt et al. (2011)) for which a recursive formula allows to update the dissimilarities when clusters i and j are merged into cluster $i\cup j$ in terms of the dissimilarities of the initial clusters. This allows to implement the related methods using computer time of order $O(n^2\log n)$. We show next that a recursive formula like the Lance-Williams class holds for the dissimilarities $\delta _{W,l}$ and, consequently, the related agglomerative hierarchical method can be efficiently implemented. The fact that we are dealing differently with genuine and protected attributes results in the need for some additional notation (and storage). Given clusters i and j consisting of points $\{(X_{i'},S_{i'})\}_{i'=1}^{n_i}$ and $\{(Y_{j'},T_{j'})\}_{j'=1}^{n_j}$, respectively, we denote

$$\begin{aligned} d_x(i,j) = \big \Vert {\bar{X}}_i-{\bar{Y}}_j\Vert . \end{aligned}$$

(7)

Note that $d_x(i,j)$ is simply the Euclidean distance between the means of the X-attributes in clusters i and j. Similarly, we set

$$\begin{aligned} d_s(i,j) = \big \Vert {\bar{S}}_i-{\bar{T}}_j\Vert . \end{aligned}$$

(8)

Proposition 1

For $\delta _{W,l}$ as in (6), $d_x(i,j)$ as in (7) and $d_s(i,j)$ as in (8) and assuming that clusters i, j and k have sizes $n_i,n_j$ and $n_k$, respectively, we have the following recursive formulas:

• i) $\delta _{W,1}(i\cup j,k) = \frac{n_i + n_k}{n_i+n_j + n_k}\delta _{W,1}(i,k) + \frac{n_j + n_k}{n_i+n_j+n_k}\delta _{W,1}(j,k)-\frac{n_k}{n_i+n_j+n_k}d^2_{W,x}(i,j);$

• ii)
  $$\begin{aligned}{} & {} \delta _{W,2}(i\cup j,k) \\{} & {} \quad = \Big (1+ue^{-v (\frac{n_i}{n_i+n_j}d^2_s(i,k)+\frac{n_j}{n_i+n_j}d^2_s(j,k) -\frac{n_in_j}{(n_i+n_j)^2}d^2_s(i,j))}\Big )\\{} & {} \qquad \times \Big ({\textstyle \frac{n_i + n_k}{n_i+n_j + n_k}}d^2_{W,x}(i,k) +{\textstyle \frac{n_j + n_k}{n_i+n_j+n_k}}d^2_{W,x}(j,k)-{\textstyle \frac{n_k}{n_i+n_j+n_k}}d^2_{W,x}(i,j)\Big );\\ \end{aligned}$$

• iii) $\delta _{W,3}(i\cup j,k) = \frac{n_i+n_k}{n_i+n_j+n_k}\delta _{W,3}(i,k) + \frac{n_j+n_k}{n_i+n_j+n_k}\delta _{W,3}(j,k) - \frac{n_k}{n_i+n_j + n_k}\delta _{W,3}(i,j)$,

where $d^2_{W,x}(i,j)=\frac{n_i n_j}{n_i+n_j}d_x^2(i,j)$.

Proof

For i) we just denote by $R_s$, $S_t$ and $T_r$ the protected attributes in clusters i, j and k, respectively and note that

$$\begin{aligned} \delta _{W,1}(i\cup j,k)&= {\textstyle \frac{(n_i+n_j)n_k}{n_i+n_j+n_k}}\Big (1'U1 + {\textstyle \frac{1}{n_i+n_j}\big (\sum _{s=1}^{n_i}R_s+\sum _{t=1}^{n_j}S_t\big )'V {\bar{T}}_k}+d^2_{x}(i\cup j, k)\Big )\\&= {\textstyle \frac{(n_i+n_j)n_k}{n_i+n_j+n_k}\frac{n_i}{n_i+n_j}\Big (1'U1+{\bar{R}}_i'V{\bar{T}}_k\Big )}\\&\quad + {\textstyle \frac{(n_i+n_j)n_k}{n_i+n_j+n_k}\frac{n_j}{n_i+n_j}\Big (1'U1+{\bar{S}}_j'V{\bar{T}}_k\Big )+d^2_{W,x}(i\cup j, k)}\\&={\textstyle \frac{n_i+n_k}{n_i+n_j+n_k}\frac{n_in_k}{n_i+n_k}\Big (1'U1+{\bar{R}}_i'V{\bar{T}}_k + d_{x}^2(i,k)\Big )}\\&\quad + {\textstyle \frac{n_j+n_k}{n_i+n_j+n_k}\frac{n_jn_k}{n_j+n_k}\Big (1'U1+{\bar{S}}_j'V{\bar{T}}_k + d_x^2(j,k)\Big )}\\&\quad -{\textstyle \frac{n_k}{n_i+n_j+n_k}d^2_{W,x}(i,j)}\\&={\textstyle \frac{n_i+n_k}{n_i+n_j+n_k}\delta _{W,1}(i,k)+\frac{n_j+n_k}{n_i+n_j+n_k}\delta _{W,1}(j,k)}-{\textstyle \frac{n_k}{n_i+n_j+n_k}d^2_{W,x}(i,j)}. \end{aligned}$$

Observe that we have used the well-known recursion for Ward’s dissimilarities, namely,

$$\begin{aligned} d_{W,x}^2(i\cup j, k)={\textstyle \frac{n_i+n_k}{n_i+n_j+n_k}d_{W,x}^2(i,k)+\frac{n_j+n_k}{n_i+n_j+n_k}d_{W,x}^2(j,k)-{\textstyle \frac{n_k}{n_i+n_j+n_k}d^2_{W,x}(i,j)}}\nonumber \\ \end{aligned}$$

(9)

(see, e.g., Everitt et al. (2011)). The update formulas ii) and iii) are obtained similarly. We omit details. $\square$

From Proposition 1 we see that a practical implementation of agglomerative hierarchical clustering based on $\delta _{W,l}$, $l=1,2$ would require the computation of $d_{W,x}^2(i,j)$, which can be done using the Lance-Williams formula (9). In the case of $\delta _{W,2}$ we also need $d_s^2(i,j)$, which again can be obtained through a Lance-Williams recursion. This implies that agglomerative hierarchical clustering based on $\delta _{W,l}$, $l=1,2$ or 3 can be implemented using computer time of order $O(n^2\log n)$ (at most twice the required time for the implementation of a ‘standard’ Lance-Williams method).

We end this section with an outline of the implementation details for our proposal for diversity enhancing agglomerative hierarchical clustering based on dissimilarities $\delta _{W,l}$.

Iterative Attraction-Repulsion Clustering For $l \in \{1,2,3\}$

• Compute the dissimilarity matrix $[\varDelta _{i,j}]=[\delta _l((X_i,S_i),(X_j,S_j))]$ with a particular choice of the free parameters.

• If $\textrm{min}\, \varDelta _{i,j}\le 0$, transform the original dissimilarity to have positive entries: $\varDelta _{i,j} = \varDelta _{i,j} + |\textrm{min}\varDelta | + \epsilon$, where $\epsilon$ is arbitrarily small.

• Use the Lance-Williams type recursion to determine the clusters i and j to be merged; iterate until there is a single cluster

5 Diversity enhancing clustering with kernels

Clustering techniques based on the minimization of a criterion function typically result in clusters with a particular geometrical shape. For instance, given a collection of points $x_1,\dots x_n \in {\mathbb {R}}^d$, the classical k-means algorithm looks for a grouping of the data into $K\le n$ clusters $C = \{C_1,\dots , C_K\}$ with corresponding means $\{\mu _1,\dots , \mu _K\}$ such that the objective function

$$\begin{aligned} \sum _{k=1}^{K}\sum _{x\in C_i}\Vert x-\mu _i\Vert ^2 \end{aligned}$$

is minimized. The clusters are then defined by assigning each point to the closest center (one of the minimizing $C_i$’s). This results in convex clusters with linear boundaries. It is often the case that this kind of shape constraint does not adapt well to the geometry of the data. A non-linear transformation of the data could map some clustered structure to make it more adapted to convex linear boundaries (or some other pattern). In some cases, this transformation can be implicitly handled via kernel methods. This approach is commonly called the ‘kernel trick’. The use of kernels in statistical learning is well documented, see, for instance, Cristianini (2004) and Chapter 16 in Shalev-Shwartz and Ben-David (2014). For completeness, let us state that in this work a kernel is a symmetric and non-negative function $\kappa : {\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\rightarrow {\mathbb {R}}$. Additionally, a Mercer (positive semidefinite) kernel, is a kernel for which there is a transformation $\phi :{\mathbb {R}}^d \rightarrow \varOmega$, with $\varOmega$ a Hilbert space, such that $\kappa (x,y)=<\phi (x),\phi (y)>_{\varOmega }$. In this section we explore how the attraction-repulsion dissimilarities that we have introduced can be adapted to the kernel clustering setup, focusing on the particular choice of kernel k-means.

Kernel k-means is a non-linear extension of k-means that allows to find arbitrary shaped clusters introducing a suitable kernel similarity function, where the role of the squared Euclidean distance between two points x, y in the classical k-means is taken by

$$\begin{aligned} d_\kappa ^2(x,y) = \kappa (x,x)+\kappa (y,y)-2\kappa (x,y). \end{aligned}$$

(10)

Details of this algorithm can be found in Schölkopf et al. (1998).

In a first approach, we could try to introduce a kernel function for vectors $(X_1,S_1), (X_2,S_2)\in {\mathbb {R}}^{d+p}$ such that $d_\kappa ^2$ takes into account the squared Euclidean distance between $X_1$ and $X_2$ but also tries to separate points of the same class and/or tries to bring closer points of different classes, i.e., makes use of $S_1,S_2$. Some simple calculations show that this is not an easy task, if possible at all in general. If we try, for instance, a joint kernel of type $\kappa ((X_1,S_1),(X_2,S_2)) = \tau (S_1,S_2) + k(X_1, X_2)$, $S_1,S_2\in \{-1,1\}$ with $\tau , k$ Mercer (positive semi-definite) kernels (this covers the case $k(X_1,X_2)=X_1\cdot X_2$, the usual scalar product in ${\mathbb {R}}^d$), our goal can be written as

$$\begin{aligned} d_\kappa ^2((X_1,S_1),(X_2,S_1)) > d_\kappa ^2((X_1,S_1),(X_2,S_2)), \end{aligned}$$

(11)

for any $X_1,X_2$, with $S_1\ne S_2$. However, (11) implies that

$$\begin{aligned} 2\tau (S_1,S_2)> \tau (S_1,S_1)+\tau (S_2,S_2) \end{aligned}$$

which contradicts that $\tau$ is positive-semi-definite. Therefore, there is no kernel on the sensitive variables that we can add to the usual scalar product. Another possibility is to consider a multiplicative kernel, $\kappa ((X_1,S_1),(X_2,S_2)) = \tau (S_1,S_2)k(X_1, X_2)$, $S_1,S_2\in \{-1,1\}$ with $\tau , k$ Mercer kernels. From (11) we get

$$\begin{aligned}2\left( \tau (S_1,S_1)-\tau (S_1,S_2)\right) k(X_1, X_2)<\left( \tau (S_1,S_1)-\tau (S_2,S_2)\right) k(X_2,X_2)\end{aligned}$$

which depends on $k(X_1,X_2)$ and makes it challenging to eliminate the dependence of the combinations $X_1,X_2$.

The previous observations show that it is difficult to think of a simple and interpretable kernel $\kappa$ that can be a simple combination of a kernel in the space of unprotected attributes and a kernel in the space of sensitive attributes. This seems to be caused by our desire to separate vectors that are similar in the sensitive space, which goes against our aim to use norms induced by scalar products. In other words, a naive extension of the kernel trick to our approach to diversity enhancing clustering seems to be inappropriate.

Nonetheless, the difficulty comes from a naive desire to carry out the (implicit) transformation of the attributes and the penalization of homogeneity in the protected attributes in the clusters in a single step. We still may obtain gains in diversity, while improving the separation of the clusters in the unprotected attributes if we embed the X data into a more suitable space by virtue of some sensible kernel $\kappa$ and consider the corresponding kernel version of $\delta _l$, with $\delta _l$ as in (2) to (5). Instead of using the Euclidean norm $\Vert X_1-X_2\Vert$ we should use $d_\kappa (X_1,X_2)$. In the case of $\delta _1$, for instance, this would amount to consider the dissimilarity

$$\begin{aligned} \delta _{\kappa ,1}\left( \left( X_1,S_1\right) ,\left( X_2,S_2\right) \right) = 1'U1 + S_1'VS_2 + d_\kappa ^2(X_1,X_2), \end{aligned}$$

(12)

with similar changes for the other dissimilarities. Then we can use an embedding (MDS the simplest choice) as in Sect. 3 and apply a clustering procedure to the embedded data. This would keep the improvement in cluster separation induced (hopefully) by the kernel trick and apply, at the same time, a diversity correction. An example of this adaptation of the kernel trick to our setting is given in Sect. 7.1.2. We recall that our procedure inherits the possible benefits of the kernel trick, but also the difficulties related to it, for example the appropriate selection of kernels.

6 Parameter selection and tunning

There are two main steps required for tuning our methods. First, the user must select some criteria for diversity and some criteria for the quality of a clustering which are suitable to the problem at hand. Second, the practitioner should choose values in an informed way or just define a set of possible values. Then, an optimal selection of parameters and clustering methods can be done. A complete example of how to select the best parameters, the best dissimilarity, and the best clustering method (among some reasonable selection of methods) is provided in Sect. 7.3. We do not advise trying to select the best method among all available clustering methods, since this seems rather infeasible.

6.1 Diversity and quality of a clustering

There are different measures for diversity, which, in a context where diversity is a good proxy for fairness, can also measure fairness. We will be mainly interested in two of them. As introduced in (Chierichetti et al. 2017), the balance of a set of points X with protected attributes $S=\{red,black\}$ is defined as

$$\begin{aligned} \textrm{balance}(X)=\min \left( \frac{\# Black}{\# Red},\frac{\# Red}{\# Black}\right) \end{aligned}$$

and the balance of ${\mathcal {C}}$, a clustering of the data in X, is given by

$$\begin{aligned} \textrm{balance}({\mathcal {C}})=\min _{C\in {\mathcal {C}}}\textrm{balance}(C). \end{aligned}$$

(13)

Let ${\mathcal {C}}$ be a clustering of a dataset into K clusters. For each cluster k, $1\le k\le K$, there is an associated proportions vector $p_k$, where $p_k$ is formed by the proportion of each value of the protected attributes in the cluster k. A simple measure for the diversity of a partition is

$$\begin{aligned} \textrm{imbalance}({\mathcal {C}}) = \frac{1}{K}\sum _{k=1}^K\Vert p_k-p_t\Vert , \end{aligned}$$

(14)

where $\Vert \cdot \Vert$ denotes the usual Euclidean norm and $p_t$ the vector of proportions of each value of the protected attributes in the whole dataset. Notice that $\textrm{imbalance}({\mathcal {C}}) = 0$ means that the partition fulfils the diversity preserving criteria (1).

Due to the lack of ground truth in unsupervised learning and due to the characteristics of clustering, many notions of what a good clustering is and how to compare different partitions exist (see Section VI in Hennig et al. (2015)). One popular approach is the Silhouette Index which measures how similar objects are in the same cluster with respect to objects in different clusters. Hence, a good partition has clusters that are cohesive and well separated. We recall that the silhouette index of an observation $X_i$ is given by

$$\begin{aligned} s(i) = \frac{b(i)-a(i)}{\max (a(i),b(i))} \end{aligned}$$

(15)

where a(i) is the average distance to $X_i$ of the observations in the same group as $X_i$, and b(i) is the average distance to $X_i$ of the observations in the closest group different from the one of $X_i$ (see Rousseeuw (1987)). The average silhouette index of a group is the average of the silhouette indexes of the members of the group and the average silhouette index is the average of the silhouette indexes of all points.

On the other hand, in diversity enhancing clustering, a good benchmark for comparisons could be the partition obtained by a ‘standard’ clustering procedure. A well-established measure for comparing partitions is the Adjusted Rand Index (ARI) (see Hubert and Arabie (1985)). Hence, there is a simple quantity that measures the difference between a diversity enhancing clustering and a standard one that we can interpret as the effect that a diversity correction has on the clustering structure.

6.2 Parameter proposal

The attraction-repulsion dissimilarities (2) - (5) introduced in Sect. 3 depend on two main sets of parameters. The first set consist of free parameters used to balance the influence of the variables X and the protected variable S. The second set is formed by the labels for the different classes, where different encoding of the labels allows for different interactions between the groups. Below, we propose some guidelines on the choice of these parameters focusing on how to understand them and what their effect on diversity is.

The dissimilarities we consider can be divided into two groups: (3) and (4) do not depend on the codification of the class variable, while (2) and (5) do depend on such a choice. In our method, the level of perturbation, which influences the level of diversity, is imposed through the choice of the parameters in the dissimilarities. Choosing the parameters enables us to balance diversity and the original structure of the data which may convey information that should not be erased by diversity constraints.

Consider first dissimilarities (3) and (4). They rely on two parameters u and v. In the multiplicative dissimilarity (3), v is a parameter that measures how sudden the change in the distance is when switching from elements with a different protected class to elements with the same protected class. For v large enough, $e^{-v\Vert S_1-S_2\Vert ^2}$ is small when $S_1\ne S_2$, which implies that the diversity dissimilarity only modifies the distance between points inside the same protected class, increasing heterogeneity of the clusters.

Once v has been fixed, the main parameter u controls the intensity of the perturbation in a similar way for both dissimilarities (3) and (4). To illustrate the effect of this parameter, we focus on (3) and perform a diversity enhancing clustering, with MDS or hierarchically, for different values of the intensity parameter u and measure the diversity of the clusters obtained. Such an example is depicted in the left middle row of Fig. 1. We can see that, as expected, increasing the values for u puts more weight on the part of the dissimilarity that enforces heterogeneity of the clusters. $u=0$ leads to the usual clustering. Indeed, varying u from 0 to 4.5 in steps of 0.5 increases the diversity achieved for both clusters, with a saturation effect from 4.5 to 5 where we do not appreciate an improvement in diversity. Hence, maximum diversity is achieved for $u = 4.5$ and gives the lowest perturbation that achieves the highest level of diversity. If one aims at preserving more of the structure of the original information at the expense of a lower level of diversity, some smaller value of u can be selected. For example, in the right middle row of Fig. 1 we provide the result of choosing $u=1$. Hence u balances both effects of closeness to the usual dissimilarities and the amount of heterogeneity reached in the clustering.

Next, dissimilarities (2) and (5), as described in Sect. 3, depend on the values chosen for the protected variable S, and a matrix V, which plays the role of the matrix of interactions for different classes. When dealing with a two-class discrimination problem where the protected class has only two values, labelling the classes as $\{-1,1\}$ or $\{(1,0),(0,1)\}$ can lead to the same results for appropriate choices of V. However, for more than two protected classes we will use only the following vectorial codification: for q different values of the protected class, we will codify the values as the q unitary vectors $\{a_1,\dots ,a_q\}$ where $a_{i,j} =1$ if $i = j$ and $a_{i,j} =0$ if $i \ne j$ for $1\le i,j\le q$.

To build the interaction matrix we proceed as follows. First, consider a matrix ${\tilde{V}}_{i,j}$ with $1\le i,j\le q$. We fix ${\tilde{V}}_{i,j} = 0$ if we want no interaction between classes i and j, in particular, if $i=j$ this means that there is no interaction between elements with the same class. We take ${\tilde{V}}_{i,j} = 1$ if we want repulsion (relative increase in the distance) between classes i and j. We fix ${\tilde{V}}_{i,j}=-1$ if we want attraction (relative decrease in distance) between classes i and j. Hence, if the practitioner wants to promote diversity for a class represented by $a_{i^*}$, it is recommendable to set values of ${\tilde{V}}_{i^*,j}=-1$ for $j\ne i^*$. As an example, in Sect. 7.1.2, we have chosen the interaction matrix $V = ((1,-1)'|(-1,0)')$, to model repulsion between elements of the same class (1, 0), attraction between elements of the classes (1, 0) and (0, 1), and no interaction between the elements of the same class (0, 1). Then intensity of the interaction is modelled using a constant $v_0 > 0$, and we set $V=v_0{\tilde{V}}$. In the previous example $v_0=1$. The parameter v for dissimilarity (5) has the same meaning as the corresponding parameter for (3) and can be selected in the same way.

Finally, matrix U for dissimilarity (2) represents an extra additive shift. In many cases it can be set to $U = 0$ (the zero matrix).

We provide an example to explain how to select the intensity $v_0$ for dissimilarity (2) in the top left image of Fig. 1. Notice that using $V>0$ and $S\in \{-1,1\}$ is the same as using $V=v_0{\tilde{V}}$ with ${\tilde{V}} = ((1,-1)'|(-1,1)')$ and $S\in \{(1,0),(0,1)\}$. We plot the variation of diversity in each cluster when we vary the intensity of the interaction between 0 and 4.4 with steps of size 0.44. There is a steady improvement in diversity in both clusters until the intensity reaches $v_0 = 3.52$, but from this level, as previously, there is no more improvement in diversity. Therefore, if a practitioner wants to achieve the highest level of diversity, $v_0 = 3.52$ should be the proper intensity, since it corresponds to the smallest perturbation to the geometry that achieves the best observed diversity. However, a smaller correction in diversity may be of interest, we have a representation of that top right in Fig. 1 for $v_0 = 1.32$.

For dissimilarity (5), after choosing the interaction matrix ${\tilde{V}}$, we can try to find a maximum in diversity, fixing a grid formed by different combinations for the vector of parameters $(v_0,u,w)$. In the second and third column of Table 2 we see the diversity of the respective clusters when we look at the grid (1, u, 0.05) with $u \in \{0,0.098,\dots , 0.98\}$. What we notice is an improvement in diversity for all values of u, therefore a practitioner would be advised to select $u = 0.98$ where we obtain the best diversity values.

7 Experiments

In this section we provide examples of attraction-repulsion clustering and some insights into the complex relation between diversity and other properties of clustering.

In the first two subsections we deal with simulated examples since we mainly want to describe how attraction-repulsion clustering works, how it compares to some diversity preserving clustering procedures and how diversity relates to cluster structure. To the best of our knowledge, this last point seems to be overlooked in the fair clustering literature, whenever diversity is the proxy used for fairness, in favour of obtaining high fairness with respect to some fairness measure.

The last subsection is a full example on a non-trivial real data set where full tuning of our methods is provided and where preservation of the geometrical structure is of crucial importance. We use a range of different clustering procedures to illustrate that our methods are, as claimed, easily adaptable to a wide range of clustering algorithms. We do not advocate for any particular clustering method, but we do try to use some of the extensions we have developed in Sect. 4.

7.1 Synthetic data

7.1.1 Diversity and cluster structure

Our first example shows how our procedures behave and gives an intuition of how diversity can affect cluster structure. We generate 50 points from four distributions,

$$\begin{aligned}{} & {} \mu _1 \sim N((-1,0.5),\textrm{diag}(0.25,0.25)),\mu _2 \sim N((-1,-0.5),\textrm{diag}(0.25,0.25)); \\{} & {} \mu _3 \sim N((1,0.5),\textrm{diag}(0.25,0.25)),\mu _4 \sim N((1,-0.5),\textrm{diag}(0.25,0.25)), \end{aligned}$$

and label the samples from $\mu _1$ and $\mu _2$ as $S = 1$ (squares) and the samples from $\mu _3$ and $\mu _4$ as $S = -1$ (circles). A representation of the data in the original space is given in the third column of Fig. 1. We can think of the x direction of the data as a source of a lack of diversity, therefore any sensible clustering procedure is going to have clusters that are highly homogeneous in the class S when the original coordinates are used. This is exemplified in Table 1, as we look for different number of clusters: with k-means we are detecting almost pure groups (1st row); the same happens with a complete linkage hierarchical clustering with the Euclidean distance (5th row) and with Ward’s method with the Euclidean distance (9th row).

Therefore, it may be useful to apply our procedures to the data to gain diversity in S. In the first column of Fig. 1 we study the gain in diversity from the increase in intensity of the corrections we apply, the relation between the gains in diversity and the disruption of the cluster structure of the original classes after MDS, and the relation between stronger correction and the change with respect to the original k-means partition measured by ARI. In the first row we use dissimilarity (2), where we fix $U = 0$, and we vary $V= 0,0.44,0.88,\dots ,4.4$. In the second row we work with dissimilarity (3), where we fix $v = 20$ and set $u = 0,0.5,1,\dots ,5$. In the last row we work with dissimilarity (5) fixing $V = 1,\,v = 20,\, w = 1$ and we vary $u = 0, 0.099,0.198,\dots ,0.99$. We do not show results for dissimilarity (4), since in this example it gives results very similar to dissimilarity (2). With some abuse of notation, throughout Sect. 7 we will use S as the name of the protected variable. Solid and dashed black lines represent, respectively, the proportion of class $S = 1$ in the two clusters found by k-means after the MDS transformation. Solid and dashed red lines represent, respectively, the average silhouette index of class $S=1$ and class $S=-1$. The green line is the Adjusted Rand Index between the k-means partition obtained in the original data and the different corrected partitions.

What we see top-left and middle-left in Fig. 1 is that higher intensity (higher V an u, respectively) relates to greater diversity and to significantly different partitions compared to the original k-means clustering, but also relates to lower silhouette index. This can be interpreted as the fact that greater intensity in dissimilarities (2) and (3) has a greater impact in the geometry of the original problem. In essence, the greater the intensity, the more indistinguishable $S = 1$ and $S= -1$ become after MDS, therefore, any partition with k-means will result in very diverse clusters in S. This is equivalent to saying that diversity is essentially destroying the original cluster structure of the data, since it is mainly due to the variable x. Hence, here a diversity enhancing partition and a good partition are competing against each other. By construction this is not what happens with dissimilarity (5). The strong locality penalty ($w = 1$) allows to conserve the geometry, shown by the little reduction in silhouette index (row 3 column 1), but results in smaller corrections in the proportions and in small differences with the original k-means clustering.

Fig. 1

Top row: dissimilarity (2). Middle row: dissimilarity (3). Bottom row: dissimilarity (5). Left column: proportions of $S = 1$ in each cluster (black lines), average silhouette indexes for $S = 1$ and $S = -1$ in the transformed space (red lines) and ARI with respect to the k-means in the original space (green line), for input parameters that vary. Middle column: two clusters in the transformed space for a particular choice of parameters. Right column: same two clusters in the original space (colour figure online)

Fig. 2

Top row: dissimilarity (2). Middle row: dissimilarity (3). Bottom row: dissimilarity (5). Left column: proportions of $S = 1$ in the clusters. Middle column: average silhouette indexes for $S = 1$ in the transformed space. Right column: ARI index with respect to the unperturbed k-means partition. Curves represent averages over 20 samples. Colours correspond to an increase in perturbation on the label space, ranging from red (lowest amount) to magenta (highest amount) (colour figure online)

In the rest of Fig. 1, we show the actual clusters in the MDS embedding obtained with k-means (column 2) and the same clusters in the original space (column 3), for some moderate intensities. For dissimilarity (2) we take $V = 1.32$, for (3) $u = 1$ and for (5) we use $u = 0.99$. A short remark is that a rotation of a MDS is a MDS, and that is the cause of the rotations that we see in column 2. Indeed, after MDS the geometry of the groups is not heavily modified, but at the same time some corrections to the proportions are achieved when clustering. These corrections appear very natural once we return to the original space.

Table 1 Proportion of class S = 1 in every group in different clustering procedures

For the same values as the previous paragraph, we present Table 1, where we look for 2, 3 and 4 clusters with MDS and k-means, but also using the approximation-free complete linkage hierarchical clustering and our Ward’s-like method. Since we are applying a small perturbation, we see some, but not a drastic, improvement in the heterogeneity of the groups. We also see that the more clusters we want the smaller the improvement. We stress that we can produce some improvements in diversity while modifying slightly the geometry of the data. This is a desirable situation when a lot of relevant information is codified in the geometry of the data. We also notice that we produce a partition that is almost diversity preserving with a stronger perturbation as shown in the last row of Table 3.

In Fig. 2 we study the behaviour of the same set-up as in Fig. 1 but with perturbation on the sensitive attribute labels. We attempt to empirically estimate a kind of break point behaviour with respect to the sensitive class, i.e., the maximal amount of perturbation in the original labels that our methods can sustain before returning no gains in diversity. The results can be interpreted as a measure of robustness against perturbations in the sensitive class, but also as a further study of the behaviour of attraction-repulsion clustering. Specifically, we select uniformly without replacement the same number of points belonging originally to class $S=1$ and $S=-1$ and we switch their labels. We take 20 samples of this kind, perform our methods, and average the resulting proportions, silhouette indexes and ARIs. The amount of perturbation goes from changing labels of $1,2,\dots ,10,15$ to 20 points in each sensitive class. Red represents the smallest perturbation in the labels and the perturbation increases as we get closer to magenta, following the rainbow colour convention. We observe that diversity gains diminish as we increase the number of swapped labels, becoming almost non-existent when we change 20 percent of the labels in each sensitive class. At that level, essentially all dissimilarities return a clustering very similar to the k-means partition in the original space represented in Fig. 3. This clustering does not fulfil the diversity preserving condition (1), however it can be easily argued that clusters capture spatial information well and are diverse.

These observations about diversity are similar to the familiar notion that fairness definitions can differ sharply, together with the fact that adequate cluster structure can depend on the particular goals of the analysis.

Fig. 3

A partition obtained with k-means when randomly swapping the sensitive labels of 20 points in each original class

7.1.2 Diversity in a non-linear setting

We explore the methods introduced in Sect. 5. With this example we want to stress that our methodology is also well-suited for non-linear clustering structure. Additionally, we want to emphasize that good partition properties and diversity may not be enough to capture some other relevant information in the data, which in this example is represented by the geometric structure.

We consider the data in the top-left image of Fig. 4. These data have a particular geometrical shape and are split into two groups. There is an inside ring of squares, a middle ring of circles, and then an outer ring of squares. There are 981 observations, and the proportions of the classes are approximately 3 to 1 (circles are 0.246 of the total data).

It is natural to apply to the original data some clustering procedure as k-means or a robust extension as tclust (deals with groups with different proportions and shapes and with outliers (García-Escudero et al. 2008)). Looking for two clusters, we are far from capturing the geometry of the groups, but the clusters have proportions of the classes that are like the total proportion, hence, the diversity preserving condition (1) is satisfied, and there is a nice cluster structure as measured by average silhouette index. Indeed, this is what we see in Fig. 4 middle-left when we apply k-means to the original data.

On the other hand, the kernel trick is convenient in this situation. In this toy example it is easy to select an appropriate kernel function, for instance, $\kappa (x,y) = x_1^2y_1^2 + x_2^2y_2^2$, which corresponds to a transformation $\phi ((x_1,x_2)) = (x_1^2,x_2^2)$. Indeed, this kernel produces linear separation between the groups. The data in the transformed space is depicted in the top-right of Fig. 4. Our adaptation to the kernel trick uses $d_\kappa$ as defined in (10) and dissimilarity (5) in the form

$$\begin{aligned} \delta _{\kappa ,4}((X_1,S_1),(X_2,S_2))= & {} \big (1 + \textrm{sign}(S_1'VS_2)u(1-e^{-v(S_1'VS_2)^2})e^{-wd_\kappa (X_1,X_2)}\big )\nonumber \\{} & {} d_\kappa (X_1,X_2), \end{aligned}$$

(16)

for $X_1,X_2$ in the original two-dimensional space, as described in Sect. 5.

Considering the discussion at the end of Sect. 3 and Sect. 6 we use dissimilarity (16) with $S_1,S_2\in \{(1,0),(0,1)\}$. In our setting circles are labelled as (1, 0) and squares as (0, 1). Now if we fix $u = 0$, use (16) to calculate the dissimilarity matrix $\varDelta$ and use MDS, essentially, we will be in the space depicted top-right on Fig. 4. Looking for two clusters with tclust, allowing groups with different sizes, we get the result depicted middle-right in Fig. 4. We have captured the geometry of the clusters but the proportions of the class S are not the best, as seen in row 1 columns 2 and 3 of Table 2 (ideally they should be close to 0.754).

Fig. 4

Top row: data in the original space (left) and after transformation $\phi$ (right). Middle row: k-means in the original space (left) and tclust applied in the transformed space and plotted in the original one (right). Bottom row: tclust after diversity enhancing corrections, corresponding to the case shown in the last row of Table 2, applied in the transformed space (left) and represented in the original space (right)

To gain diversity in what is referred to as cluster 2 in Table 2 (in red in the plots), we vary the intensity u of our local dissimilarity, with the other parameters set as indicated in Table 2. We see that as we increase the intensity of the interactions we gain heterogeneity in cluster 2, and both proportions come closer to the total proportion 0.754 (columns 2-3). Again, this is achieved without destroying the geometry of the original classes after the MDS, as seen in the small variation of the average silhouette index in columns 4-5.

We plot the best performance, given by $u = 0.98$, after MDS in bottom-left and in the original space in bottom-right of Fig. 4. Clearly, we have been able to capture the geometry of the groups and to produce relatively divers clusters.

Table 2 Effect of varying the intensity u of the local dissimilarity (16), for fixed $V = ((1,-1)'|(-1,0)')$, $v = 20$ and $w = 0.05$. First two columns contain the proportion of points with $S = (0,1)$ in the clusters found with tclust in the transformed space. Last two columns show the silhouette of the original classes in the MDS

7.2 Comparison with fair clustering through fairlets

In this section we present a comparison of the results of our diversity enhancing clustering methods with results obtained by implementing, in Python and R, the fair clustering procedure introduced in Chierichetti et al. (2017) based on fairlets decomposition. We recall that this clustering method is an example of what we call diversity preserving clustering. Since our examples are concerned with two values for the protected class it is justified to use Chierichetti et al. (2017) for comparison since it is well suited for this situation. Our implementation of the case when the size of both protected classes is the same, which reduces to an assignment problem, is implemented using the function max_bipartite_matching of the package igraph in R. In the case of different sizes, we must solve a min cost flow problem as stated in Chierichetti et al. (2017), which can be done in Python with the function min_cost_flow of the package networkx (also it can be solved with a min_cost_flow solver in ortools).

When implemented as a min cost flow problem, Chierichietti’s et al. methodology has two free parameters, $t'$ and $\tau$. First, for a partition ${\mathcal {C}}$ of data X, we have that $1/t'\le \textrm{balance}({\mathcal {C}})\le \textrm{balance}(X)$ and hence $t'$ controls the lower bound of the diversity of the partition. Recall the definition of balance in (13). Second, $\tau$ is a free parameter related to the distance between points, and has a defined lower limit given by the maximum distance taken from the set formed by the distance between each point in one class and its respective closest point in the other class. For more information we refer to Chierichetti et al. (2017).

We start with the data used for the example studied in Fig. 1. We will address k-median clustering, for which Chierichetti et al. (2017) has a fair implementation. Since the data has two groups of the same size, we can solve an assignment problem or use a min cost flow problem. For the min cost flow problem, we have the set of parameters $\{(t',\tau )\}$ with $t' = 2,3,4$ and $\tau = \infty , 2.42$. Values for $\tau$ represent no locality and maximum locality. As comparison we will use k-median clustering after perturbing the data with $\delta _1$ with parameters $(U = 0, V = 4.4)$ and doing a MDS embedding. Results are shown in Table 3. Since the data is in one-to-one correspondence between the two classes, both the alignment solution and the different min cost flow solutions give balance $=1$, hence the diversity preserving condition (1) is fulfilled. Our method gives balance close to 1 but does not fulfil (1). However, the average silhouette index of our method is higher, which means that clusters are more identifiable and compact, and the k-median objective function is also lower, and hence better. A plot of some of the different clusterings can be seen in Fig. 5.

Table 3 Rows 1-7 are implementations of fair clustering with fairlets while the last row is k-median clustering after using $\delta _1$, with $(U = 0, V = 4.4)$, and MDS embedding

Fig. 5

Left: clusters obtained by fair k-median as an assignment problem (Chierichetti et al. (2017)). Middle: clusters obtained by fair k-median as min cost flow problem with $t' = 4,\tau = 2.42$ (Chierichetti et al. (2017)). Right: clusters obtained by attraction-repulsion clustering with dissimilarity (2) and MDS, using k-median

Our next comparison is for the data used in Fig. 4. We stress that for fairlets we are working with the data after the implicit kernel transformation, i.e., the data shown top-right in Fig. 4, and therefore results are comparable with attraction-repulsion clustering in the kernel trick setting. Results are shown in Table 4. We see that consistently the fair k-median implementation gives balance values very close to $241/740 \approx 0.3257$, showing a high degree of diversity in the clusters. Our method gives a lower balance value; hence groups are less diverse, but as we see from the silhouette and k-median objective function values, the groups are more identifiable and more compact. Even more, comparing the middle-right of Fig. 4 and left of Fig. 6, we see that our procedure is even able to capture the underlying geometry of the data.

Fig. 6

Clustering of the data in top-left of Fig. 4 obtained using fair k-median as min cost flow problem with $t' = 4,\tau = 19.62$ (Chierichetti et al. (2017))

Table 4 Rows 1-6 are implementations of fair (diversity preserving) clustering with fairlets as a min cost flow problem with different parameters. Last row is a tclust clustering after using $\delta _4$, with $(u = 0.98, v = 20, w = 0.05, V = ((1,-1)'|(-1,0)')$, and MDS embedding

Table 5 Rows 1-8 are implementations of fair clustering with fairlets as a min cost flow problem with different parameters. Rows 9-10 are the best results we have obtained with attraction-repulsion clustering. The last row represents the values for a k-means clustering in the original data

Fig. 7

Diversity enhancing, diversity preserving and standard clustering for the Ricci dataset, where circles represent not white individuals and squares represent white individuals. First row: k-means for 2 and 4 clusters in the unperturbed (original) data. Second row: k-means for 2 and 4 clusters in the MDS setting with $\delta _1$. Third row: Ward’s method for 2 and 4 clusters with $\delta _2$. Fourth row: fair k-median as in Chierichetti et al. (2017) with $t'= 5,\tau = 18.62$ for 2 and 4 clusters

Our last comparison is on a real data set known as the Ricci dataset, consisting of scores in an oral and a written exam of 118 firefighters, where the sensitive attribute is the race of the individual. This dataset was a part of the case Ricci v. DeStefano presented before the Supreme Court of the United States.

For applying our attraction-repulsion clustering we codify white individuals as $S = 1$ and black and hispanic individuals as $S = -1$. The appropriate parameters for the dissimilarities are chosen to give a good performance (after a grid search as suggested in Sect. 6). The best results are obtained with our adaptation of Ward’s method with $\delta _2$ and parameters $(u = 3.125, v = 10)$ and k-means after applying $\delta _1$, with parameters $(U = 0, V= 500)$, and a MDS embedding. Results are given in Table 5. We see the balance given by using fairlets is higher than the one obtained with our procedures. However, we see that our procedures produce more identifiable and compact clusters. As a remark, we see that both procedures achieve a nice improvement in diversity compared to the k-means solution in its standard version.

A plot of some of the clusterings can be seen in Fig. 7. Visually it is quite clear why the average silhouette index is higher in the attraction-repulsion clustering than in the fair k-median. It also clarifies what we mean by more identifiable and compact clusters.

7.3 Civil rights data collection

In this section we are going to apply our procedure to the Schools Civil Rights Data Collection (CRDC) for the year 2015-2016 available for download in the link https://ocrdata.ed.gov/DownloadDataFile. We are going to work with the data for entry, middle level, and high schools in the state of New Jersey.

From the CRDC data we can collect the number of students in each school that belong to six distinct categories: Hispanic, American Indian/Alaska Native, Asian, Native Hawaiian/Pacific Islander, Black, White. An entry of our dataset looks like this.

Additionally, using geolocate of the package ggmap in R we can extract latitude and longitude coordinates of the schools in New Jersey. Hence we have a precise location for the schools and can calculate distances in a straight line between them, i.e., geodesic distance on Earth between schools. A plot of the locations can be seen in Fig. 8.

Fig. 8

Left: location of schools in the state of New Jersey. Right: mds-embedding of the straight-line distances between the schools

A relevant problem in public management is how to group some smaller entities (such as individual residences, schools, buildings, etc...) into some bigger entities (such as districts) to improve decision making, efficiency or some other relevant aspect. This is a typical problem in governance, with a famous example being electoral districts (constituencies) in the USA.

If we were faced with the problem of making diverse school districts, how should we proceed? Usually, districts incorporate meaningful spatial information and the diversity part in this case is related to the overall ethnic composition of the district. A possible approach to the problem is to perform diversity enhancing clustering, hence, identifying clusters of schools with districts. Next, we must consider what is the purpose of a district, since this is highly relevant for what are the ethical ramifications of diversity and for what are considered good clusters. For example, if all districts will be affected by some centralized decision, enforcing demographic parity through (1) may be a good proxy for fairness. Hence, diversity and fairness may be understood as interchangeable. However, if we were interested in a similar problem, which is making fair voting districts in a winner-takes-all system, promoting diversity may be highly inadequate (see for example Abbasi et al. (2021)). If we decide that we want to use diversity enhancing clustering, we should decide what types of clusters are adequate for our purposes? A simple approach is to look for spatially compact clusters because vicinity is usually a reasonable proxy for other shared features. At this point, we should be aware that diversity and vicinity requirements may be conflicting, and therefore some kind of trade-off between them is advisable. This setting is the one that we adopt and present attraction-repulsion clustering results below.

We will measure the degree of diversity of the partition by comparing the ethnic composition of each of its clusters to the ethnic composition of the overall dataset which is $p_t = (0.2423, 0.0018, 0.0988,0.0028, 0.1604,0.4741).$ This is done by using the imbalance index defined in (14).

Recall that the average silhouette index is a measure of the compactness of the clusters in a partition. Hence our methodology is to use attraction-repulsion clustering where tuning is done over a mesh of distinct parameters, and the best parameters are those that give the lowest $\textrm{imbalance}$ while keeping the average silhouette index over some threshold $\tau$. In this way we impose the maximum improvement in diversity while keeping part of the geographical information codified by the distance. This procedure can be seen in Algorithm 1.

Codification of the protected attributes in this case is straightforward, it is just the number of students at the school in each category. Hence, for instance Catham High School has $S_{C.H.S.} = (46,0,106,2,19,1021)$. We want to stress that another easy possibility is to use proportions with respect to the total number of students, i.e., (0.0382, 0, 0.0880, 0.0158, 0.0017, 0.8480). However, we choose to use the number of students since we think it should be considered, and that information is lost when using proportions.

Selecting the grid or mesh of parameters for the different dissimilarities is an important task. For some of them, $\delta _2$ and $\delta _3$, it is mainly an analytical task of selecting the best values. However, in proposing candidates for V in $\delta _1$ and $\delta _4$ we can use available information as we will see below. In this example we will concentrate on $\delta _1,\delta _2$ and $\delta _4$. To explore the effects of attraction-repulsion dissimilarities on different clustering methods we select a tiny sample. As hierarchical clustering methods, since we only have distances between schools, we will use complete, average, and single linkage. Through a MDS embedding we will use k-means. Hence, cluster.methods = (complete, average, single, k-means) in Algorithm 1.

The grid we use for $\delta _2$ in Algorithm 1 is formed by $parameters=\{(u_i,v_j)\}_{i,j=1}^{i=19,j=10}$ with $\{u_i\}_{i=1}^{19} = \{0.1, 0.2, 0.3, 0.4, 0.5,0.6, 0.7, 0.8, 0.9, 1, 2, 3, 4, 5, 6,7, 8, 9, 10\}$ and $\{v_j\}_{j=1}^{10} = \{0.1, 0.3, 0.5, 0.7, 0.9, 1, 3.25,5.5, 7.75, 10\}$.

As we stated previously, for proposing values for $V = v_0{\tilde{V}}$ in $\delta _1$ and $\delta _4$ we are going to use some a priori information that can be corroborated by the data. The most numerous minorities are the Hispanic, Asian, and Black communities. Even more, it is well known that poor neighbourhoods have higher concentrations of minorities, and therefore schools in those areas should be representative of that. Hence, this is a major source for a lack of diversity in a mainly geographical clustering of the schools. Since white students are the majority, values of the proportion for the previously mentioned minorities and white students will affect our imbalance index the most. Hence we should attempt to achieve mixing in precisely these groups. Our first three proposals are variations of schemes that should improve mixing in the above-mentioned communities.

$$\begin{aligned} {\tilde{V}}= & {} \left\{ \left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} 1 &{} -1 &{} -1 &{} -1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}1 \\ \end{array} } \right) , \left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}1 \\ \end{array} } \right) , \left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}1 \\ \end{array} } \right) \right. ,\\{} & {} \left. \left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ -1 &{} 1 &{} -1 &{} -1 &{} -1 &{} -1 \\ -1 &{} -1 &{} 1 &{} -1 &{} -1 &{} -1 \\ -1 &{} -1 &{} -1 &{} 1 &{} -1 &{} -1 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} 1 \\ \end{array} } \right) , \left( {\begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} 1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} -1 &{} 1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{array} } \right) , \left( {\begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} 1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} -1 &{} 1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array} } \right) \right\} . \end{aligned}$$

The fourth proposal is the obvious one, which tries to produce mixing in all communities. The fifth and sixth proposals try to impose mixing mainly for the smallest minorities American Indians/Alaska Natives and Native Hawaiians/Pacific Islanders.

Fig. 9

Top row is a comparison between unperturbed clustering methods, bottom row is a comparison of clustering methods after using $\delta _1$. The clustering methods are complete linkage in black, k-means in red, single linkage in green and average linkage in blue. We stress that single linkage with $\delta _1$, green line in bottom row, does not achieve $\tau >0$ for $k\ge 9$ (colour figure online)

Fig. 10

Top: a comparison between effects of unperturbed and perturbed situations for k-means. Bottom: same comparison as in top but for complete linkage clustering. In black we have the unperturbed situation, in red we have the best $\delta _1$ perturbation, in green the best $\delta _2$ and in blue the best $\delta _4$ perturbation (colour figure online)

Now we can define the values we will use for the input parameters in Algorithm 1. For $\delta _1$ we have $parameters = \{(U,v_0,V')\}$ where $U = 0_{6\times 6}$, $v_0 \in \{0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009,0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08,0.09,0.1,0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1\}$ and $V'\in {\tilde{V}}$. For $\delta _4$ another parameter that can incorporate a priori information is w, which in this case tells us how strong the influence between schools that are further apart should be. For the local dissimilarity $\delta _4$ we propose to use $parameters = \{(u,v,w,V)\}$ such that $u \in \{0.5,2,8\}$, $v \in \{0.1,1,10\}$, $w \in \{0.1,0.5,0.9,1,5.5,10\}$ and $V\in {\tilde{V}}$.

Before showing results for the attraction-repulsion procedures, we will cluster the data without considering race information, hence only geographical distance is of importance. From the top row of Fig. 9, we can say that k-means (red) is giving a good performance since it has low imbalance index and a reasonably high average silhouette index. We stress that the k-means procedure is done in the MDS embedding shown on the right in Fig. 8. In the left column of Fig. 11 we can see k-means clustering for 5 and 11 clusters. We clearly see that spatial proximity is the driving force of the clustering. The values of imbalance and average silhouette index are respectively (0.13, 0.42) and (0.21, 0.42).

To apply our attraction-repulsion clustering as shown in Algorithm 1 we need to fix the silhouette bound. To do this we take $\tau = 0$, and hence we are not imposing very strong compactness criteria, recall that the silhouette index varies between -1 and 1, but we still want clusters to be relatively compact. In this way we are making a trade-off between reduction in imbalance and spatial coherence. In Fig. 10 we see the effects of the different best parameters for dissimilarities $\delta _1,\delta _2$ and $\delta _4$ for k-means and complete linkage clustering. Generally, we see that $\delta _1$ (in red) is the dissimilarity that produces the strongest reduction in imbalance (solid line) and of course this is on behalf of a reduction in average silhouette index (dashed line). Also as expected a local dissimilarity as $\delta _4$ (blue) brings only a modest reduction of imbalance but maintains a high spatial coherence.

Hence, if we want to slightly gerrymander districts to improve diversity while maintaining a very high geographical coherence we should use $\delta _4$. On the other hand, if we want the maximum imbalance reduction achievable with our procedure we should use $\delta _1$.

To decide which clustering method is the best, from our small selection of methods, we use the bottom row of Fig. 9. There we see that for attraction-repulsion clustering using $\delta _1$, k-means (red) and complete linkage (black) produce similar reduction in imbalance while k-means keeps a relatively higher average silhouette index. Therefore, in this case k-means with perturbation $\delta _1$ and a MDS embedding seems to be the best procedure.

In Fig. 11 we can see a visual comparison between k-means clustering in the different situations. The best parameters for the dissimilarities in the cases we have shown are the following

$$\begin{aligned} params({\mathcal {C}}^{\delta _4}_5)= & {} \left( 2,1,10,\left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} 1 &{} -1 &{} -1 &{} -1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}1 \\ \end{array} } \right) \right) ,\\ params({\mathcal {C}}^{\delta _4}_{11})= & {} \left( 2,0.1,1,\left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}1 \\ \end{array} } \right) \right) , \\ params({\mathcal {C}}^{\delta _1}_5)= & {} \left( 0_{6\times 6},0.03,\left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ -1 &{} 1 &{} -1 &{} -1 &{} -1 &{} -1 \\ -1 &{} -1 &{} 1 &{} -1 &{} -1 &{} -1 \\ -1 &{} -1 &{} -1 &{} 1 &{} -1 &{} -1 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} 1 \\ \end{array} } \right) \right) ,\\ params({\mathcal {C}}^{\delta _1}_{11})= & {} \left( 0_{6\times 6},0.07,\left( {\begin{array}{cccccc} 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \\ -1 &{} 1 &{} -1 &{} -1 &{} -1 &{} -1 \\ -1 &{} -1 &{} 1 &{} -1 &{} -1 &{} -1 \\ -1 &{} -1 &{} -1 &{} 1 &{} -1 &{} -1 \\ -1 &{} -1 &{} -1 &{} -1 &{} 1 &{} -1\\ -1 &{} -1 &{} -1 &{} -1 &{} -1 &{} 1 \\ \end{array} } \right) \right) . \end{aligned}$$

What we see is that the matrices V that we introduced trying to codify available a priori information work very well. Hence real-world intuitions are compatible with our model. In the case of the parameters for $\delta _1$ again what we expected from intuition is seen, i.e., to reduce imbalance for a bigger number of clusters it is necessary to use a stronger perturbation (a higher value for $v_0$). In the plots we clearly see the behaviour we have previously mentioned. The local dissimilarity $\delta _4$ does a sort of positive gerrymandering while $\delta _1$ imposes a stronger reduction in imbalance and hence significantly alters the clustering results.

Fig. 11

Results of k-means clustering, top row looking for 5 clusters and bottom row looking for 11. Left: unperturbed situation. Middle: $\delta _4$ perturbation and MDS embedding. Right: $\delta _1$ perturbation and MDS embedding

8 Conclusions

We have introduced a pre-processing heuristic based on novel dissimilarities that is applicable virtually to any clustering method and can produce gains in diversity. Hence, when diversity is a good proxy for fairness, for example when demographic parity is required, our methods may fall into a fair clustering setting. Our heuristic approach allows for tractable and simple computations at the cost of tuning requirements and no theoretical guarantees for enforcing the diversity preserving condition (1).

The experiments described in Sect. 7, while simple, allow us to reach some meaningful conclusions. To begin with, pre-processing transformations looking to impose diversity through independence of the marginal distributions with respect to the protected attributes, but ignoring the geometry of the data, can eliminate any meaningful cluster structure as seen in Sect. 7.1.1. This can also happen when using ad hoc fair (diversity preserving) clustering objective functions as shown in Sect. 7.2. Hence, it is advisable to reach a trade-off between the desire to find some cluster structure present in the geometry of the data and the desire to impose diversity preserving conditions. Furthermore, the justification of diversity, and more broadly, fairness concerns in clustering rely heavily on the purpose of the practitioner and the data itself, something that is implicit in Sect. 7.3. The previous considerations make us urge practitioners to be clearly aware of:

• What is the purpose of the partitions that they are looking for

• The definitions of diversity, with its relations to fairness, and of cluster structure that are suitable to their purpose

• The conflict that may exist between diversity and the groupings present in the data

• The level of trade-off between diversity and sensible cluster structure that is reasonable for the users purpose

As for future directions of research, there are two clear paths. One is to try to use a transformation of the data that can give some theoretical guarantees for the demographic parity condition (1), most likely restricting clustering procedures to a certain type. Another is to consider the trade-off between diversity, cluster structure and maybe some other relevant criteria for the practitioner through a multi-objective optimization formulation. An additional alternative is to try to explore model-based clustering in the setting of attraction-repulsion dissimilarities.