Neighborhood Co-regularized Multi-view Spectral Clustering of Microbiome Data

Abstract

In many unsupervised learning problems data can be available in different representations, often referred to as views. By leveraging information from multiple views we can obtain clustering that is more robust and accurate compared to the one obtained via the individual views. We propose a novel algorithm that is based on neighborhood co-regularization of the clustering hypotheses and that searches for the solution which is consistent across different views. In our empirical evaluation on publicly available datasets, the proposed method outperforms several state-of-the-art clustering algorithms. Furthermore, application of our method to recently collected biomedical data leads to new insights, critical for future research on determinants of the cervicovaginal microbiome and the cervicovaginal microbiome as a risk factor for the transmission of HIV. These insights could have an influence on the interpretation of clinical presentation of women with bacterial vaginosis and treatment decisions.

E. Tsivtsivadze and H. Borgdorff contributed equally to this work.

Appendix

Given the matrix formulation of our optimization problem, we can find the following closed form for the solution. Taking the partial derivative of $J(W_i)$ with respect to $\mathbf {w}^{(v)}_i$ we get

$$\begin{aligned} \frac{\partial }{\partial \mathbf {w}^{(v)}_i} J(W_i) =&-2 X^{(v)}_i (\mathbf {q}^{(v)}_i - X^{(v)T}_i \mathbf {w}^{(v)}_i ) + 2 \lambda \mathbf {w}^{(v)}_i\\&- 4\nu \sum _{u,v=1, u \ne v }^M X^{(v)}_i (X^{(v)T}_i \mathbf {w}^{(v)}_i - X^{(u)T}_i\mathbf {w}^{(u)}_i ). \end{aligned}$$

By defining $G^\nu = 2\nu (M -1) X^{(v)}_i X^{(v)T}_i$, $G^\lambda = \lambda X^{(v)T}_i$ and $G = X^{(v)}_i X^{(v)T}_i$, we can rewrite the above term as

$$\begin{aligned} \frac{\partial }{\partial \mathbf {w}^{(v)}_i}J(W_i) =&2(G + G^\nu + G^\lambda ) \mathbf {w}^{(v)}_i -2 X^{(v)T}_i \mathbf {q}^{(v)}_i\\&\,-4\nu \sum _{u,v=1, u \ne v }^M X^{(v)}_i X^{(u)T}_i \mathbf {q}^{(u)}_i. \end{aligned}$$

At the optimum we have $\frac{\partial }{\partial \mathbf {w}^{(v)}_i}J(W_i)=0$ for all views, thus we get the exact solution by solving

$$\begin{aligned} \left( \begin{array}{ccc} G_1 &{} -2\nu X^{(1)}_i X^{(2)T}_i &{} \ldots \\ \\ -2\nu X^{(2)}_i X^{(1)T}_i &{} G_2 &{} \ldots \\ \\ \vdots &{} \vdots &{} \ddots \end{array} \right) \left( \begin{array}{cc} \mathbf {q}^{(1)}_i \\ \\ \mathbf {q}^{(2)}_i \\ \\ \vdots \end{array} \right) = \left( \begin{array}{cc} X^{(1)T}_i \mathbf {q}^{(1)}_i \\ \\ X^{(2)T}_i \mathbf {q}^{(2)}_i\\ \\ \vdots \end{array} \right) \end{aligned}$$

with respect to $\mathbf {w}^{(1)}_i,\ldots ,\mathbf {w}^{(M)}_i$. Note that the left-hand side matrix is positive definite and therefore invertible.