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Determinantal consensus clustering

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Abstract

Random restart of a given algorithm produces many partitions that can be aggregated to yield a consensus clustering. Ensemble methods have been recognized as more robust approaches for data clustering than single clustering algorithms. We propose the use of determinantal point processes or DPPs for the random restart of clustering algorithms based on initial sets of center points, such as k-medoids or k-means. The relation between DPPs and kernel-based methods makes DPPs suitable to describe and quantify similarity between objects. DPPs favor diversity of the center points in initial sets, so that sets with similar points have less chance of being generated than sets with very distinct points. Most current inital sets are generated with center points sampled uniformly at random. We show through extensive simulations that, contrary to DPPs, this technique fails both to ensure diversity, and to obtain a good coverage of all data facets. The latter are two key properties that make DPPs achieve good performance. Simulations with artificial datasets and applications to real datasets show that determinantal consensus clustering outperforms consensus clusterings which are based on uniform random sampling of center points.

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Acknowledgements

This work was partially funded by a Discovery grant 2019-05444 from the Natural Sciences and Engineering Research Council of Canada, and by a Fundamental Research Projects grant PRF-2017-20 from the Institute for Data Valorization (IVADO) of Quebec, Canada.

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  1. Département de Mathématiques et de Statistique, Université de Montréal, Montreal, Canada

    Serge Vicente & Alejandro Murua-Sazo

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Appendices

Appendix: Proof that an increasing sequence of thresholds gives rise to hierarchical cluster partitions

Consider s given sequence of increasing thresholds \(\theta _1< \theta _2< \cdots < \theta _m\). As mentioned in Sect. 4, the associated consolidated matrices \(C^{(1)}, \ldots , C^{(m)}\) are nested in the sense that when \(s < t\), \(C^{(s)}_{ij} \ge C^{(t)}_{ij}\) for all pairs (i, j) \(s, t=1, \ldots ,m\). Let \(\mathcal {P}_s\) and \(\mathcal {P}_t\) be the connected component partitions associated with the consolidated matrices \(C^{(s)}\) and \(C^{(t)}\), respectively. Let points \(x_i\) and \(x_j\) be in the same connected component of \(\mathcal {P}_t\). That is, there is a path \(x_i \mapsto x_{l_1} \mapsto \cdots \mapsto x_{l_{g-1}} \mapsto x_j\) of points satisfying \(C^{(t)}_{l_h, l_{h+1}} = 1\), \(h=0, \ldots , g-1\), with \(l_0 = i,\) and \(l_{g} = j\). Since \(C^{(s)}_{ij} \ge C^{(t)}_{ij}\) for all pairs (i, j), necessarily points \(x_i\) and \(x_j\) are also in the same connected component of \(\mathcal {P}_s\). The converse is not true. So suppose that points \(x_i\) and \(x_j\) are in the same connected component, say \(\mathcal {C}_\ell \) of \(\mathcal {P}_s\). That is, there is a path \(x_i \mapsto x_{l_1} \mapsto \cdots \mapsto x_{l_{g-1}} \mapsto x_j\) of points satisfying \(C^{(s)}_{l_h, l_{h+1}} = 1\), \(h=0, \ldots , g-1\), with \(l_0 = i,\) and \(l_{g} = j\). Suppose as well that for some \(h\in \{0, \ldots , g-1\}\), \(C^{(t)}_{l_h, l_{h+1}} = 0\). Then the proportion of times the pair \((x_i, x_j)\) appear together is at least \(\theta _s\), but less than \(\theta _t\). Points \(x_i\) and \(x_j\) are necessarily in different connected components of \(\mathcal {P}_t\). This means that the component \(\mathcal {C}_\ell \in \mathcal {P}_s\) has split in several connected components isolated from all other components of \(\mathcal {P}_s\). In this sense the sequence of thresholds \(\theta _1< \theta _2< \cdots < \theta _m\), leads to a sequence of partitions satisfying \(\mathcal {P}_1 \supseteq \mathcal {P}_2 \supseteq \cdots \supseteq \mathcal {P}_m\), where \(\mathcal {P}_s \supseteq \mathcal {P}_t\) means that each connected component in \(\mathcal {P}_t\) arises from a connected component of \(\mathcal {P}_s\), and the number of connected components of \(\mathcal {P}_t\) is larger or equal to the number of connected components of \(\mathcal {P}_s\).

Appendix: Scattering formulas in the transformed space: using the kernel trick

Consider the RBF kernel \(\kappa (x_i,x_j)\) defined by (3). The mean scattering induced by the kernel on the data (Vert et al. 2004; Fan et al. 2010) is defined as

$$\begin{aligned} V_{\mathcal {S}} = \frac{1}{n}\sum _{i=1}^{n}\left\{ \kappa (x_i,x_i)-\sum _{j=1}^{n}\frac{2\, \kappa (x_i,x_j)}{n}+\sum _{j, \ell =1}^{n}\frac{\kappa (x_j,x_\ell )}{n^2} \right\} ^{\frac{1}{2}}. \end{aligned}$$

This formula is derived using the kernel trick which links kernel evaluations with inner products between transformed points in a high-dimensional space. It can be shown that this quantity corresponds to the average distance between the transformed points and the mean of the transformed points. Let \(\mathcal {V}= \{\mathcal {V}_1, \ldots , \mathcal {V}_K\}\) be a clustering configuration with K clusters. Also, let \(n_k\) be the size of cluster \(\mathcal {V}_k\), and let \(J_k = \{ i : x_i \in \mathcal {V}_k\}\), \(k=1,\ldots ,K.\) As with the mean scattering, Fan et al. (2010) define similarly, the kernel induced scattering within each cluster \(\mathcal {V}_k\),

$$\begin{aligned} W_{\mathcal {V}_k} = \frac{1}{n_k}\sum _{i\in J_k} \left\{ \kappa (x_i,x_i)-\sum _{j\in J_k}\frac{2\, \kappa (x_i,x_j)}{n_k}+\sum _{j, \ell \in J_k} \frac{\kappa (x_j,x_\ell )}{n_k^2} \right\} ^{\frac{1}{2}} \end{aligned}$$

which can be seen as the average distance between the mean of the clusters \(\mathcal {V}_k\) in the transformed space and the transformed points of cluster \(\mathcal {V}_k\). Finally, to measure the total scattering between clusters, one considers the distance between clusters in the transformed space,

$$\begin{aligned} B^2(\mathcal {V}_i,\mathcal {V}_j) = \bigl [ B(\mathcal {V}_i,\mathcal {V}_j) \bigr ]^2&= \sum _{i, j \in J_i} \frac{\kappa (x_i,x_j)}{n_i^2} - \sum _{i\in J_i}\sum _{j\in J_j}\frac{2 \kappa (x_i,x_j)}{n_in_j} +\sum _{i, j\in J_j} \frac{\kappa (x_i,x_j)}{n_j^2}. \end{aligned}$$

Appendix: Scheme to sample \(\varvec{Y}\sim DPP_{\mathcal {S}}(L)\)

Below \(e_i\) represents the ith standard basis vector, a n-dimensional vector that is all zeros except for a one in the ith position. The span of a given finite collection of vectors V, denoted by \({{\,\mathrm{span}\,}}(V)\), is the set containing of all linear combinations of the vectors in V, i.e., \({{\,\mathrm{span}\,}}(V)=\left\{ \sum _{v\in V}a_v v \mid a_v\in \mathbb {R}\right\} \).

figure b

Appendix: Sensitivity analysis on kernel choice

As mentioned in Sect. 5.3, we considered three kernels: the linear kernel, \(\kappa (x_i,x_j)=x_i^Tx_j\), and two variants of the q-stable kernel (Bilodeau and Nangue 2017), also known as the q-Gaussian kernel (Verleysen and François 2005), which is given by \(\kappa (x_i, x_j)=\exp \left( -\left\Vert x_i-x_j\right\Vert ^{q}/\sigma ^q\right) \), \(q\in (0,2]\), where \(\sigma >0\) is the kernel bandwidth. The variants considered are the Laplace kernel (\(q=1\)), and the Gaussian kernel (\(q=2\)).

Fig. 5
figure 5

Mean ARIs over five runs associated with DPP, CC-PAM and CC-k-means\(++\) for different scenarios with datasets with diverse shape clusters. The horizontal axis is divided by the different powers \(\lambda \) involved in the inverse Box–Cox transformations. The results are aggregated over the three types of data sizes (\(n\in \{150, 500, 1500\}\)). The segments on top of the bars represent \(\pm 1\) Standard Error

Full size image

Two estimates of the bandwidth associated with each one of the two q-stable kernels studied were considered. For the Laplace kernel, the bandwidth estimates were the mean \(\hat{\sigma }_2\) and the median \(\tilde{\sigma }_2\) of all pairwise Euclidean distances. For the Gaussian kernel, the estimates were the mean \(\hat{\sigma }^2\) and the median \(\tilde{\sigma }^2\) of all pairwise squared Euclidean distances. Note that the Gaussian kernel with bandwidth estimate \(\hat{\sigma }^2\) corresponds to our proposed DPP kernel.

We further investigated the behavior of the kernels on data whose clusters are non-Gaussian. For this simulation, we followed the setup of Sect. 5.2. Recall that the data were generated from 24 experimental conditions of a \(3^3\)-factorial design. For this study, we generated five replicas from each experimental condition. The non-ellipsoidal (non-Gaussian) clusters were obtained by applying the inverse Box–Cox transformation to the datasets, As before, we selected \(\lambda =1/3\), \(\lambda =1/2\), and \(\lambda = 2\) for the power parameter \(\lambda \) of the Box–Cox transformations. We run the consensus DPP algorithm described in Sect. 4 on each dataset with the five different kernels. The values of the consensus DPP parameters were set to \(a=0.5\), \(R=200\), and \(\tau =0.6\). To select the optimal clustering configuration, we used the kernel-based validation index KVI\(_\mathcal {V}\) given by (6). To measure its quality, we use the adjusted Rand index (see Sect. 4.2). The results are displayed in Fig. 5 in the form of barplots. Each bar represents a different kernel or consensus method. The segments on top of the bars represents ± one standard error. We also compare the results from the different kernels with those yielded by CC-PAM and CC-k-means\(++\).

For both q-stable kernels, estimating the bandwidth parameter with the median or the mean of all the pairwise distances do not produce any major difference in the ARI results.

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Vicente, S., Murua-Sazo, A. Determinantal consensus clustering. Adv Data Anal Classif 17, 829–858 (2023). https://doi.org/10.1007/s11634-022-00514-6

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