ROOTCLUS: Searching for “ROOT CLUSters” in Three-Way Proximity Data

Abstract

In the context of three-way proximity data, an INDCLUS-type model is presented to address the issue of subject heterogeneity regarding the perception of object pairwise similarity. A model, termed ROOTCLUS, is presented that allows for the detection of a subset of objects whose similarities are described in terms of non-overlapping clusters (ROOT CLUSters) common across all subjects. For the other objects, Individual partitions, which are subject specific, are allowed where clusters are linked one-to-one to the Root clusters. A sound ALS-type algorithm to fit the model to data is presented. The novel method is evaluated in an extensive simulation study and illustrated with empirical data sets.

Appendix

In order to solve the constrained problem (14) in those cases when the diagonal entries of the H similarity matrices $\mathbf {S}_{h}$ are not of interest, the steps 1 to 4 of the ALS-type algorithm presented in Sect. 4 can be modified straightforwardly as follows.

Since only the off-diagonal elements of matrices $\mathbf {S}_{h}$ ($h=1,\ldots ,H$) need to be considered, the loss function (14) becomes

$$\begin{aligned} f_{\text {off}}(\mathbf {P}, \mathbf {W}, \mathbf {M}_h, \mathbf {V}_h, c_h)=\; F_{\text {off}}(\mathbf {P}, \mathbf {W}, \mathbf {M}_h, \mathbf {V}_h, c_h) +\lambda G, \end{aligned}$$

(23)

where

$$\begin{aligned} \begin{aligned}&F_{\text {off}}(\mathbf {P}, \mathbf {W}, \mathbf {M}_h, \mathbf {V}_h, c_h) = \\&\quad \frac{\sum _{h=1}^{H} \left\| \mathbf {S}_h - \mathbf {P} \mathbf {W} \mathbf {P}^\prime - \bigl (\mathbf {P}+\mathbf {M}_{h} \bigr ) \mathbf {V}_h \bigl (\mathbf {P}+\mathbf {M}_{h} \bigr )^\prime - c_{h} \mathbf {1}_N \mathbf {1}_{N}^{\prime } \right\| _{\text {off}}^2}{\sum _{h=1}^{H} \left\| \mathbf {S}_h \right\| _{\text {off}}^2} \end{aligned} \end{aligned}$$

(24)

and $\left\| \mathbf {Z} \right\| _{\text {off}}^2 = \sum _{x=1}^{X}\sum _{\begin{array}{c} y=1\; ;\; \end{array}{y\ne x}}^{Y} z_{xy}^2$.

In step 1, the loss function (23), instead of (14), is minimized over $\mathbf {P}$ and $\mathbf {M}_h$ ($h=1,\ldots ,H$).

In steps 2 to 4, all the rows of $\mathbf {s}_{h}$, $\mathbf {T}$, $\mathbf {Q}_{h}$ and $\mathbf {1}_{N^2}$ in model (15), corresponding to the diagonal entries of the matrices in (14), need to be left out. Such reduced structures are obtained as follows:

$$\begin{aligned} {\tilde{\mathbf {s}}}_h&= \mathbf {s}_h \odot \mathbf {d} \qquad (h=1,\ldots ,H) \,, \end{aligned}$$

(25)

$$\begin{aligned} {\tilde{\mathbf {T}}}&= \mathbf {T} \odot \mathbf {D}\,, \end{aligned}$$

(26)

$$\begin{aligned} {\tilde{\mathbf {Q}}}_h&= \mathbf {Q}_h \odot \mathbf {D} \qquad (h=1,\ldots ,H)\,, \end{aligned}$$

(27)

$$\begin{aligned} {\tilde{\mathbf {1}}}_{N^2}&= \mathbf {1}_{N^2} \odot \mathbf {d} \,, \end{aligned}$$

(28)

where $\odot$ denotes the Hadamard product, $\mathbf {d}$ is the column vector of size $N^2$ of the vectorized matrix $\big (\mathbf {1}_N \mathbf {1}_N^\prime - \mathbf {I}_N\big )$, being $\mathbf {I}_N$ the identity matrix of size N, and $\mathbf {D}$ is the $N^2 \times J$ matrix having all its columns equal to $\mathbf {d}$.

Therefore, model (15) is rewritten in terms of (25)–(28) and Steps 2, 3 and 4 modified accordingly.