Distance metric learning aims to learn a distance metric that best describe the distance between two data points. One of the most commonly used distance metric is Mahalanobis distance. It is of the form

\[d(\mathbf{x}, \mathbf{y})^{2}=(\mathbf{x}-\mathbf{y})^{T} \mathbf{M}(\mathbf{x}-\mathbf{y})\]

where \(\mathbf{x}\) and \(\mathbf{y}\) are feature vectors for two different data points. Now, supposing we have \(n\) training examples \(\left(\mathbf{x}_{i}, y_{i}ight),(i=1,2, \ldots, n)\), we aim to learn the matrix \(\mathbf{M}\) from the data. Research and describe one supervised method and one unsupervised method to learn the matrix \(\mathbf{M}\).