Unsupervised Learning - Representation learning - Kernel PCA
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Introduction
In the context of a given dataset \(\mathbf{X} \in \mathbb{R}^{d \times n}\), where \(\mathbf{C} \in \mathbb{R}^{d \times d}\) represents the covariance matrix, Principal Component Analysis (PCA) encounters two critical challenges:
Time Complexity: Computing the eigenvalues and eigenvectors of \(\mathbf{C}\) requires an algorithmic complexity of \(O(d^3)\). Consequently, as the dimensionality \(d\) increases, the computational time becomes prohibitively large.
Non-Linear Dataset: In situations where the dataset resides in a non-linear subspace, PCA’s attempt to derive linear combinations of Principal Components may yield suboptimal results.
To address these issues, we propose methods for reducing the time complexity of finding eigenvalues and eigenvectors and for handling non-linear relationships in PCA.
Reducing the Time Complexity to Find Eigenvalues and Eigenvectors
Let us consider a dataset \(\mathbf{X}\) with a large number of features (\(d\)), denoted as \([d >> n]\), where:
- \(d\): number of features
- \(n\): number of datapoints
The dataset \(\mathbf{X}\) is represented as follows: \[ \mathbf{X} = \left[ \begin{array}{ccccc} | & | & | & & | \\ \mathbf{x}_1 & \mathbf{x}_2 & \mathbf{x}_3 & \ldots & \mathbf{x}_n \\ | & | & | & & | \end{array} \right] \quad \mathbf{X} \in \mathbb{R}^{d \times n} \]
The covariance matrix \(\mathbf{C}\) of \(\mathbf{X}\) is given by: \[ \mathbf{C} = \frac{1}{n}(\mathbf{X} \mathbf{X}^T) \quad \text{where } \mathbf{C} \in \mathbb{R}^{d \times d} \quad \ldots[1] \]
Let \(\mathbf{w}_k\) be the eigenvector corresponding to the \(k\)-th largest eigenvalue \(\lambda_k\) of \(\mathbf{C}\). We know: \[ \mathbf{C}\mathbf{w}_k = \lambda_k \mathbf{w}_k \]
Substituting \(\mathbf{C}\) from equation \([1]\) into the above equation and solving for \(\mathbf{w}_k\), we obtain: \[ \mathbf{w}_k = \sum_{i=1}^n \left(\frac{\mathbf{x}_i^T \mathbf{w}_k}{n \lambda_k}\right) \cdot \mathbf{x}_i \]
Thus, \(\mathbf{w}_k\) is a linear combination of the datapoints. Consequently, we can express: \[ \mathbf{w}_k = \mathbf{X} \alpha_k \quad \text{for some } \alpha_k \quad \ldots [2] \]
Let \(\mathbf{X}^T \mathbf{X} = \mathbf{K}\), where \(\mathbf{K} \in \mathbb{R}^{n \times n}\). We can utilize this to solve for \(\alpha_k\). After some algebra (refer to relevant literature), we obtain: \[ \mathbf{K} \alpha_k = (n \lambda_k) \alpha_k \]
According to the Spectral Theorem, the non-zero eigenvalues of \(\mathbf{X}\mathbf{X}^T\) and \(\mathbf{X}^T \mathbf{X}\) are identical. Therefore, we can assert that if \(\lambda \ne 0\) is an eigenvalue of \(\mathbf{X}\mathbf{X}^T\) with eigenvector \(\mathbf{w}\), then \(\mathbf{X}\mathbf{X}^T\mathbf{w}=\lambda \mathbf{w}\). By multiplying both sides by \(\mathbf{X}^T\), we find \(\mathbf{X}^T \mathbf{X}(\mathbf{X}^T\mathbf{w})=\lambda \mathbf{X}^T\mathbf{w}\), which confirms that \(\lambda\) is also an eigenvalue for \(\mathbf{X}^T\mathbf{X}\), with the corresponding eigenvector being \(\mathbf{X}^T\mathbf{w}\).
Let \(\beta_k\) be the eigenvector corresponding to the \(k\)-th largest eigenvalue \(n \lambda_k\) of \(\mathbf{K}\). Solving the eigen equation for \(\mathbf{K}\) yields: \[ \alpha_k = \frac{\beta_k}{\sqrt{n \lambda_k}} \quad \ldots [3] \]
By employing equations \([2]\) and \([3]\), we can compute the eigenvalues and eigenvectors of \(\mathbf{C}\) using \(\mathbf{K}\), effectively reducing the time complexity from \(O(d^3)\) to \(O(n^3)\).
Finding PCA for Non-Linear Relationships
Transforming Features
To address non-linear relationships, we propose mapping the dataset to higher dimensions as follows: \[ \mathbf{x} \to \phi(\mathbf{x}) \quad \mathbb{R}^d \to \mathbb{R}^D \quad \text{where } [D >> d] \]
To compute \(D\), let \(\mathbf{x}=\left [ \begin{array} {cc} f_1 & f_2 \end{array} \right ]\) represent features of a dataset containing datapoints lying on a second-degree curve in a two-dimensional space.
To convert it from quadratic to linear, we map the features to: \[ \phi(\mathbf{x})=\left [ \begin{array} {cccccc} 1 & f_1^2 & f_2^2 & f_1f_2 & f_1 & f_2 \end{array} \right ] \]
Mapping \(d\) features to the polynomial power \(p\) results in \(^{d+p} C_d\) new features.
However, it is essential to note that finding \(\phi(\mathbf{x})\) may be computationally demanding.
To overcome this issue, we present the solution in the subsequent section.
Kernel Functions
A function \(k: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}\) is considered a “valid” Kernel Function if it maps data points to the real numbers.
Proof of a “Valid” Kernel: There are two methods to establish the validity of a kernel:
Method 1: Explicitly exhibit the mapping to \(\phi\), which may be challenging in certain cases.
Method 2: Utilize Mercer’s Theorem, which states that \(k: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}\) is a valid kernel if and only if:
- \(k\) is symmetric, i.e., \(k(\mathbf{x},\mathbf{x}') = k(\mathbf{x}',\mathbf{x})\)
- For any dataset \(\{\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n\}\), the matrix \(\mathbf{K} \in \mathbb{R}^{n \times n}\), where \(\mathbf{K}_{ij} = k(\mathbf{x}_i,\mathbf{x}_j)\), is Positive Semi-Definite.
Two popular kernel functions are:
Polynomial Kernel: \(k(\mathbf{x},\mathbf{x}') = (\mathbf{x}^T\mathbf{x} + 1)^p\)
Radial Basis Function Kernel or Gaussian Kernel: \(k(\mathbf{x},\mathbf{x}') = \exp\left(\displaystyle-\frac{\|\mathbf{x}-\mathbf{x}'\|^2}{2\sigma^2}\right)\)
Kernel PCA
Consider a dataset \(\mathbf{X}\) with a large number of features (\(d\)), represented as \([d >> n]\), where:
- \(d\): number of features
- \(n\): number of datapoints
The dataset \(\mathbf{X}\) is given by: \[ \mathbf{X} = \left[ \begin{array}{ccccc} | & | & | & & | \\ \mathbf{x}_1 & \mathbf{x}_2 & \mathbf{x}_3 & \ldots & \mathbf{x}_n \\ | & | & | & & | \end{array} \right] \]
To perform Kernel PCA, follow these steps:
Step 1: Calculate the kernel matrix \(\mathbf{K} \in \mathbb{R}^{n \times n}\) using a kernel function, where \(\mathbf{K}_{ij} = k(\mathbf{x}_i,\mathbf{x}_j)\).
Step 2: Center the kernel using the formula: \[ \mathbf{K}^C = \mathbf{K} - \mathbf{I}\mathbf{K} - \mathbf{K}\mathbf{I} + \mathbf{I}\mathbf{K}\mathbf{I} \] where \(\mathbf{K}^C\) is the centered kernel, and \(\mathbf{I} \in \mathbb{R}^{n \times n}\) is a matrix with all elements equal to \(\frac{1}{n}\).
Step 3: Compute the eigenvectors \(\{\beta_1, \beta_2, \ldots, \beta_l\}\) and eigenvalues \(\{n\lambda_1, n\lambda_2, \ldots, n\lambda_l\}\) of \(\mathbf{K}^C\) and normalize them to obtain: \[ \forall u \quad \alpha_u = \frac{\beta_u}{\sqrt{n \lambda_u}} \]
Step 4: Compute the transformed data points: \[ \phi(\mathbf{x}_i)^T\mathbf{w} \in \mathbb{R}^{d} \to \left [ \begin{array}{cccc} \displaystyle \sum_{j=1}^{n} \alpha_{1j} \mathbf{K}^C_{ij} & \displaystyle \sum_{j=1}^{n} \alpha_{2j} \mathbf{K}^C_{ij} & \ldots & \displaystyle \sum_{j=1}^{n} \alpha_{nj} \mathbf{K}^C_{ij} \end{array} \right ] \]