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Subspace clustering of high-dimensional data: a predictive approach

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Abstract

In several application domains, high-dimensional observations are collected and then analysed in search for naturally occurring data clusters which might provide further insights about the nature of the problem. In this paper we describe a new approach for partitioning such high-dimensional data. Our assumption is that, within each cluster, the data can be approximated well by a linear subspace estimated by means of a principal component analysis (PCA). The proposed algorithm, Predictive Subspace Clustering (PSC) partitions the data into clusters while simultaneously estimating cluster-wise PCA parameters. The algorithm minimises an objective function that depends upon a new measure of influence for PCA models. A penalised version of the algorithm is also described for carrying our simultaneous subspace clustering and variable selection. The convergence of PSC is discussed in detail, and extensive simulation results and comparisons to competing methods are presented. The comparative performance of PSC has been assessed on six real gene expression data sets for which PSC often provides state-of-art results.

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Notes

  1. Software available at http://www2.imperial.ac.uk/~gmontana/psc.htm

  2. http://www.broadinstitute.org/cgi-bin/cancer/datasets.cgi

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Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments and the EPSRC (Engineering and Physical Science Research Council) for funding this project.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, UK

    Giovanni Montana

  2. Department of Informatics, ETH, Zürich, Switzerland

    Brian McWilliams

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  1. Brian McWilliams
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Correspondence to Giovanni Montana.

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Responsible editor: Ian Davidson.

Appendices

Appendix 1: Derivation of predictive influence

Using the chain rule, the gradient of the PRESS for a single latent factor is

$$\begin{aligned} \frac{\partial J^{(1)}}{\partial \varvec{x}_i} = \frac{1}{2} \frac{\partial }{\partial \varvec{x}_i}\left\| \varvec{e}^{(1)}_{-i}\right\| ^2 = \frac{1}{2}\varvec{e}^{(1)}_{-i} \frac{\partial }{\partial \varvec{x}_i}\varvec{e}^{(1)}_{-i}. \end{aligned}$$

For notational convenience we drop the superscript in the following. Using the quotient rule, the partial derivative of the \(i{\text{ th }}\) leave-one-out error has the following form

$$\begin{aligned} \frac{\partial }{ \partial \varvec{x}_i} \varvec{e}_{-i} = \frac{\frac{\partial }{ \partial \varvec{x}_i} \varvec{e}_i (1-h_i) + \varvec{e}_i\frac{\partial h_i }{ \partial \varvec{x}_i}}{(1-h_i)^2} \end{aligned}$$

which depends on the partial derivatives of the \(i{\text{ th }}\) reconstruction error and the \(h_i\) quantities with respect to the observation \(\varvec{x}_i\). The computation of these two partial derivatives are straightforward and are, respectively

$$\begin{aligned} \frac{\partial }{ \partial \varvec{x}_i} \varvec{e}_i = \frac{\partial }{\partial \varvec{x}_i} \varvec{x}_i \left( \varvec{I}_P - \varvec{v}{\varvec{v}}^{\top }\right) = \left( \varvec{I}_P - \varvec{v}{\varvec{v}}^{\top }\right) , \end{aligned}$$

and

$$\begin{aligned} \frac{\partial }{ \partial \varvec{x}_i} h_i = \frac{\partial }{\partial \varvec{x}_i} \varvec{x}_i\varvec{v} D \varvec{v}^{\top }\varvec{x}_i^{\top }= 2\varvec{v} D d_i . \end{aligned}$$

The derivative of the PRESS, \(J\) with respect to \(\varvec{x}_i\) is then

$$\begin{aligned} \frac{1}{2}\frac{\partial }{ \partial \varvec{x}_i}\left\| \varvec{e}_{-i}\right\| ^2 = \varvec{e}_{-i} \frac{\partial }{ \partial \varvec{x}_i} \varvec{e}_{-i}= \varvec{e}_{-i} \frac{ \left( \varvec{I}_P - \varvec{v}{\varvec{v}}^{\top }\right) (1-h_i) + 2\varvec{e}_i \varvec{v} D d_i }{(1-h_i)^2}. \end{aligned}$$
(28)

However, examining the second term in the sum, \(\varvec{e}_i \varvec{v} D d_i \), we notice

$$\begin{aligned} \varvec{e}_i\varvec{v}Dd_i = (\varvec{x}_i-\varvec{x}_i\varvec{vv}^{\top })\varvec{v}Dd_i = \varvec{x}_i\varvec{v}Dd_i - \varvec{x}_i\varvec{vv}^{\top }\varvec{v}Dd_i = 0 . \end{aligned}$$

Substituting this result back in Eq. (28), the gradient of the PRESS for a single PCA component with respect to \(\varvec{x}_i\) is given by

$$\begin{aligned} \frac{1}{2}\frac{\partial }{ \partial \varvec{x}_i} \left\| \varvec{e}_{-i}\right\| ^2 = \varvec{e}_{-i} \frac{ \left( \varvec{I}_P - \varvec{v}{\varvec{v}}^{\top }\right) (1-h_i)}{(1-h_i)^2} = \varvec{e}_{-i} \frac{ \left( \varvec{I}_P - \varvec{v}{\varvec{v}}^{\top }\right) }{(1-h_i)} . \end{aligned}$$

In the general case for \(R>1\), the final expression for the predictive influence \(\varvec{\pi }(\varvec{x}_i)\in \mathbb{R }^{P\times 1}\) of a point \(\varvec{x}_i\) under a PCA model then has the following form:

$$\begin{aligned} \varvec{\pi }(\varvec{x}_i;\varvec{V}) = \varvec{e}^{(R)}_{-i} \left( \sum _{r=1}^{R} \frac{ \left( \varvec{I}_p - \varvec{v}^{(r)}{\varvec{v}^{(r)}}^{\top }\right) }{\left( 1-h^{(r)}_i\right) } - (R-1) \right) . \end{aligned}$$

Appendix 2: Proof of Lemma 1

From Appendix 1, for \(R=1\), the predictive influence of a point \(\varvec{\pi }({\varvec{x}_i};\varvec{v})\) is

$$\begin{aligned} \varvec{\pi }(\varvec{x}_i;\varvec{v}) =\frac{\varvec{e}_{i}}{(1-h_i)^2} \end{aligned}$$
(29)

This is simply the \(i{\text{ th }}\) leave-one-out error scaled by \(1-h_i\). If we define a diagonal matrix \(\varvec{\varXi }\in \mathbb{R }^{N\times N}\) with diagonal entries \({\varXi }_{i} = (1-h_i)^2\), we can define a matrix \(\varvec{\Pi }\in \mathbb{R }^{N\times P}\) whose rows are the predictive influences, \(\varvec{\Pi }=[\varvec{\pi }(\varvec{x}_1;\varvec{v}) ^{\top },\ldots , \varvec{\pi }(\varvec{x}_N;\varvec{v}) ^{\top }]^{\top }\). This matrix has the form

$$\begin{aligned} \varvec{\Pi } = \varvec{\varXi }^{-1}\left( \varvec{X} - \varvec{X}\varvec{vv}^{\top }\right) . \end{aligned}$$

Now, solving (21) is equivalent to minimising the squared Frobenius norm,

$$\begin{aligned}&\min _{\varvec{v}} \text{ Tr } \left( \left( \varvec{X} - \varvec{X}\varvec{vv}^{\top }\right) ^{\top }\varvec{\varXi }^{-2} \left( \varvec{X} - \varvec{X}\varvec{vv}^{\top }\right) \right) \nonumber \\&\text{ subject } \text{ to } \left\| \varvec{v} \right\| =1 . \end{aligned}$$
(30)

Expanding the terms within the trace we obtain

$$\begin{aligned} \text{ Tr } \left( \left( \varvec{X} - \varvec{X}\varvec{vv}^{\top }\right) ^{\top }\varvec{\varXi }^{-2} \left( \varvec{X} - \varvec{X}\varvec{vv}^{\top }\right) \right)&= \text{ Tr } \left( \varvec{X}^{\top }\varvec{\varXi }^{-2} \varvec{X} \right) - 2\text{ Tr }\left( \varvec{vv}^{\top }\varvec{X} ^{\top }\varvec{\varXi }^{-2} \varvec{X} \right) \nonumber \\&+ \text{ Tr }\left( \varvec{vv}^{\top }\varvec{X} ^{\top }\varvec{\varXi }^{-2} \varvec{X}\varvec{vv}^{\top }\right) . \end{aligned}$$

By the properties of the trace, the following equalities hold

$$\begin{aligned} \text{ Tr }\left( \varvec{vv}^{\top }\varvec{X} ^{\top }\varvec{\varXi }^{-2} \varvec{X} \right) = \varvec{v}^{\top }\varvec{X}^{\top }\varvec{\varXi }^{-2} \varvec{X} \varvec{v}, \end{aligned}$$

and

$$\begin{aligned} \text{ Tr } \left( \varvec{vv}^{\top }\varvec{X} ^{\top }\varvec{\varXi }^{-2} \varvec{X}\varvec{vv}^{\top }\right)&= \text{ Tr }\left( \varvec{\varXi }^{-1}\varvec{X}\varvec{vv}^{\top }\varvec{vv}^{\top }\varvec{X}^{\top }\varvec{\varXi }^{-1}\right) \\&= \varvec{v}^{\top }\varvec{X}^{\top }\varvec{\varXi }^{-2} \varvec{X} \varvec{v}, \end{aligned}$$

since \(\varvec{\varXi }\) is diagonal and \(\varvec{v}^{\top }\varvec{v}=1\). Therefore, (30) is equivalent to

$$\begin{aligned}&\min _{\varvec{v}} \text{ Tr } \varvec{X}^{\top }\varvec{\varXi }^{-2} \varvec{X} - \varvec{v}^{\top }\varvec{X}^{\top }\varvec{\varXi }^{-2}\varvec{Xv} , \nonumber \\&\text{ subject } \text{ to } ~ \left\| \varvec{v} \right\| =1 . \end{aligned}$$
(31)

It can be seen that under this constraint, (31) is minimised when \(\varvec{v}^{\top }\varvec{X}^{\top }\varvec{\varXi }^{-2}\varvec{Xv}\) is maximised which, for a fixed \(\varvec{\varXi }\) is achieved when \(\varvec{v}\) is the eigenvector corresponding to the largest eigenvalue of \(\varvec{X}^{\top }\varvec{\varXi }^{-2} \varvec{X}\).

Appendix 3: Proof of Lemma 2

In this section we provide a proof of Lemma 2 As an additional consequence of this proof, we develop an upper bound for the approximation error which can be shown to depend on the leverage terms. We derive this result for a single cluster, \(\mathcal{C }^{(\tau )}\) however it holds for all clusters.

We represent the assignment of points \(i=1,\ldots ,N\) to a cluster, \(\mathcal{C }^{(\tau )}\) using a binary valued diagonal matrix \(\varvec{A}\) whose diagonal entries are given by

$$\begin{aligned} A_{i}= \left\{ \begin{array}{ll} 1, &{} \text{ if } i\in \mathcal{C }^{(\tau )} \\ 0,&{} \text{ otherwise }, \end{array} \right. \end{aligned}$$
(32)

where \(\text{ Tr }(\varvec{A})=N_k\). We have shown in Lemma 1 that for a given cluster assignment, the parameters which optimise the objective function can be estimated by computing the SVD of the matrix

$$\begin{aligned} \sum _{i\in \mathcal{C }_k^{(\tau )}} \varvec{x}_i^{\top }{\varXi }_{i}^{-2} \varvec{x}_i = \varvec{X}^{\top }\varvec{\varXi }^{-2}\varvec{A} \varvec{X} , \end{aligned}$$
(33)

within each cluster where the \(i{\text{ th }}\) diagonal element of \({\varvec{\varXi }}\) is \(\varXi _{i}=(1-h_i)^2\le 1\), so that \(\varXi _{i}^{-2}\ge 1\). We can then represent \({\varvec{\varXi }}^{-2} = \varvec{I}_N + \varvec{\varPhi }\) where \(\varvec{\varPhi }\in \mathbb{R }^{n\times n}\) is a diagonal matrix with entries \(\varPhi _{i}=\phi _i\ge 0\). Now, we can represent Eq. (33) at the next iteration as

$$\begin{aligned} \varvec{M} = \varvec{X}^{\top }\varvec{A}(\varvec{I}_N + \varvec{\varPhi })\varvec{X} . \end{aligned}$$
(34)

We can quantify the difference between the optimal parameter, \(\varvec{v}^{*}\) obtained by solving (22) using \( \varvec{M}\) and the new PCA parameter estimated at iteration \(\tau +1\), \(\varvec{v}^{(\tau )}\) as,

$$\begin{aligned} E(\mathcal{S }^*,\mathcal{S }^{(\tau )})= {\varvec{v}^{*}}^{\top }\varvec{M}^{(\tau )} \varvec{v}^{*} - {\varvec{v}^{(\tau )}}^{\top }\varvec{X}^{\top }\varvec{A} \varvec{X}\varvec{v}^{(\tau )}, \end{aligned}$$

where \(\varvec{v}^{(\tau )}\) is obtained through the SVD of \( \varvec{X}^{\top }\varvec{A}\varvec{X} \). We can express \(E(\mathcal{S }^*,\mathcal{S }^{(\tau )})\) in terms of the spectral norm of \(\varvec{M}\). Since the spectral norm of a matrix is equivalent to its largest singular value, we have \({\varvec{v}^{(\tau )}}^{\top }\varvec{X}^{\top }\varvec{A} \varvec{X}\varvec{v}^{(\tau )} =\left\| \varvec{X}^{\top }\varvec{A}\varvec{X} \right\| \) Since \(\varvec{\varPhi }\) is a diagonal matrix, its spectral norm, \(\left\| \varvec{\varPhi } \right\| = \max (\varvec{\varPhi })\). Similarly, \(\varvec{A}\) is a diagonal matrix with binary valued entries, so \(\left\| \varvec{A} \right\| = 1\).

$$\begin{aligned} E(\mathcal{S }^*,\mathcal{S }^{(\tau )})&\le \left\| \varvec{M} - \varvec{X}^{\top }\varvec{A}\varvec{X} \right\| \nonumber \\&= \left\| \varvec{X}^{\top }\varvec{A}\varvec{\varPhi }\varvec{X} \right\| \nonumber \\&\le \max (\varvec{\varPhi }) \left\| \varvec{X}^{\top }\varvec{X} \right\| . \end{aligned}$$
(35)

Where the triangle and Cauchy-Schwarz inequalities have been used. In a similar way, we now quantify the difference between the optimal parameter and the old PCA parameter \(\varvec{v}^{(\tau -1)}\),

$$\begin{aligned} E(\mathcal{S }^*,\mathcal{S }^{(\tau -1)}) = {\varvec{v}^{*}}^{\top }\varvec{M} \varvec{v}^{*} - {\varvec{v}^{(\tau -1)}}^{\top }\varvec{X}^{\top }\varvec{A} \varvec{X}\varvec{v}^{(\tau -1)}. \end{aligned}$$

Since \(\varvec{v}^{(\tau )}\) is the principal eigenvector of \(\varvec{X}^{\top }\varvec{A}\varvec{X}\), by definition, \({\varvec{v}^{(\tau )}}^{\top }\varvec{X}^{\top }\varvec{A}\varvec{Xv}^{(\tau )}\) is maximised, therefore we can represent the difference between the new parameters and the old parameters as

$$\begin{aligned} E(\mathcal{S }^{(\tau )},\mathcal{S }^{(\tau -1)})={\varvec{v}^{(\tau )}}^{\top }\varvec{X}^{\top }\varvec{A} \varvec{Xv}^{(\tau )} - {\varvec{v}^{(\tau -1)}}^{\top }\varvec{X}^{\top }\varvec{A} \varvec{Xv}^{(\tau -1)}\ge 0. \end{aligned}$$

Using this quantity, we can express \(E(\mathcal{S }^*,\mathcal{S }^{(\tau -1)})\) as

$$\begin{aligned} E(\mathcal{S }^*,\mathcal{S }^{(\tau -1)})&\le \left\| \varvec{M} \right\| - {\varvec{v}^{(\tau -1)}}^{\top }\varvec{X}^{\top }\varvec{A} \varvec{X}\varvec{v}^{(\tau -1)} \nonumber \\&\le \left\| \varvec{X}^{\top }\varvec{\varPhi } \varvec{A} \varvec{X} \right\| + \left\| \varvec{X}^{\top }\varvec{A} \varvec{X} \right\| - {\varvec{v}^{(\tau -1)}}^{\top }\varvec{X}^{\top }\varvec{A} \varvec{X}\varvec{v}^{(\tau -1)} \nonumber \\&\le \max (\varvec{\varPhi } )\left\| \varvec{X}^{\top }\varvec{X} \right\| + E(\mathcal{S }^{(\tau )},\mathcal{S }^{(\tau -1)}), \end{aligned}$$
(36)

From (36) and (35) it is clear that

$$\begin{aligned} E(\mathcal{S }^*,\mathcal{S }^{(\tau )}) \le E(\mathcal{S }^*,\mathcal{S }^{(\tau -1)}) . \end{aligned}$$
(37)

This proves Lemma 2.

The inequality in (37) implies that estimating the SVD using \(\varvec{X}^{\top }\varvec{A} \varvec{X}\) obtains PCA parameters which are closer to the optimal values than those obtained at the previous iteration. Therefore, estimating a new PCA model after each cluster re-assignment step never increases the objective function. Furthermore, as the recovered clustering becomes more accurate, by definition there are fewer influential observations within each cluster. This implies that \(\max (\varvec{\varPhi } ) \rightarrow 0\), and so \( E(\mathcal{S }^*,\mathcal{S }^{(\tau )}) \rightarrow 0\).

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McWilliams, B., Montana, G. Subspace clustering of high-dimensional data: a predictive approach. Data Min Knowl Disc 28, 736–772 (2014). https://doi.org/10.1007/s10618-013-0317-y

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