Common Principal Component Subspace and Clustering Symmetric Positive Semi-Definite Matrices

Hanchao Zhang

Hanchao.Zhang@nyulangone.org

Division of Biostatistics, Department of Population Health, Grossman School of Medicine, New York University

Thaddeus Tarpey

Thaddeus.Tarpey@nyulangone.org

Division of Biostatistics, Department of Population Health, Grossman School of Medicine, New York University

Table of contents

• Refresh Clustering Algorithm: K-Means
• Symmetric Positive Semi-Definite (SPD) Matrix-Valued Data
• SPD Matrices and K-Tensors Algorithm: Clustering SPD Matrices
• Principal and Self-Consistent Tensors (Common Principal Component Subspace)
• Simulation Study
• Real Data Analysis

Refresh Clustering Algorithm: K-Means

K-Means Algorithm

\[\begin{align*} & \underset{\mathscr C}{\text{minimize }} \underbrace{\frac{1}{n}\sum_{j=1}^m \sum_{k \in \mathscr C_j} \big\Vert x_k - \bar x_{\mathscr C_j}\big\Vert^2}_{\text{within cluster sum of squares}} \\ &\textbf{ OR } \\ &\underset{\mathscr C}{\text{maximize}} \underbrace{\sum_{i=m} \frac{n_{\mathscr C_j}}{n}\cdot\big\Vert\bar x_{\mathscr C_j}\big\Vert^2 }_{\text{between cluster sum of squares}} \end{align*}\] Minimize the Euclidean distance between the data points to its closet centroid

K-Means Algorithm

\[\begin{align*} & \underset{\mathscr C}{\text{minimize }} \underbrace{\frac{1}{n}\sum_{j=1}^m \sum_{k \in \mathscr C_j} \big\Vert x_k - \bar x_{\mathscr C_j}\big\Vert^2}_{\text{within cluster sum of squares}} \\ &\textbf{ OR } \\ &\underset{\mathscr C}{\text{maximize}} \underbrace{\sum_{i=m} \frac{n_{\mathscr C_j}}{n}\cdot\big\Vert\bar x_{\mathscr C_j}\big\Vert^2 }_{\text{between cluster sum of squares}} \end{align*}\] Minimize the Euclidean distance between the data points to its closet centroid

Clustering Centers

\[\begin{align*} \mathbf u_j & = \mathscr E[\mathbf X \vert \mathbf X \in \mathscr C_j] \\ & = \frac{1}{n_j}\sum_{i=1}^{n_j} \mathbf x_i \cdot \delta(\mathbf x_i \in \mathscr C_j), \end{align*}\] where the centers of cluster \(\mathscr C_j\) is the average of the data points that is classified to the group \(\mathscr C_j\).

Symmetric Positive Semi-Definite (SPD) Matrix-Valued Data

Functional Connectivity Matrix

Functional Connectivity. [Gillebert et al., 2013]

Functional connectivity is defined as the temporal coincidence of spatially distant neurophysiological events.

For each participant \(i\), let \(\mathbf{y}_{ij} \in \mathbb{R}^{\mathscr T}\) be the longitudinal measurement of blood oxygen level-dependent (BOLD) signal on the region of interest \(j\), \(j = 1,2,\ldots,p\).

The functional connectivity matrix for participant \(i\) is the covariance matrix of \(\mathbf y_{i}\): \(\mathbf{\Sigma}_i = \textbf{Cov}(\mathbf{y}_i) \succeq \mathbf{0}\)

We want to link the functional connectivity matrix to the clinical outcomes using clustering methods.

SPD Matrices and K-Tensors Algorithm: Clustering SPD Matrices

Positive Semi-Definite Matrices

Examples of Positive Semi-Definite Matrices

• \(\mathbf \psi_{2\times 2} = \begin{bmatrix} 1 & 0.5 \\ 0.5 & 1 \end{bmatrix}\)
• \(\mathbf \psi_{3\times 3} = \begin{bmatrix} 1 & 0.5 & 0.3 \\ 0.5 & 1 & 0.2 \\ 0.3 & 0.2 & 1 \end{bmatrix}\)
• \(\mathbf \psi_{p\times p} = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1p} \\ x_{21} & x_{22} & \cdots & x_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{p1} & x_{p2} & \cdots & x_{pp} \end{bmatrix}\)

Visualization

A ellipsoid centered at origin is defined by the set of points \(\mathbf x \in \mathbb R^p\) such that \(\mathbf x^T \mathbf \psi \mathbf x = 1\).

Positive Semi-Definite Matrices Visualization