## Data representation Before doing further processing (like using machine learning techniques), we should wash irrelevant data out. Representing them may help doing so. In the high dimensional space, we may use 3 ways to identify how $X_{}i$, the point of $\{X\}$ trajectory, related to each other. - A vector of fingerprint $X_i$, like recording $\vec{q}$, a vector could be trajectory itself. - dissimilarity matrix $D_{ij}$, like a distance of fingerprint, distance from the configuration $i, j$. - Gram matrix, like a scalar product of fingerprint, i.e., inner product. Then How can we do with these quantities? If we have a dissimilar matrix $\mathbf{D}$ as $\mathbf{D} = \mathbf{c} \mathbf{1}^T + \mathbf{C}^T \mathbf{1} - 2\mathbf{K}$ $\mathbf{C} = \left( \sum_{\alpha} (X_{\alpha}^1)^2, \ldots, \sum_{\alpha} (X_{\alpha}^m)^2 \right)$we could define a centering matrix like $\mathbf{J} = \mathbf{I} - \frac{1}{M} \mathbf{1} \mathbf{1}^T$ And if we apply this $\mathbf{J}$, centering matrix to $\mathbf{D}$, we can have $-\frac{1}{2} \mathbf{JDJ} = \mathbf{K} + \text{constant}$. So we can project the trajectory on the basis of eigenvectors $V$ of the Gram matrix. >[!Info] >See similar chapters for more on data processing and sampling: [[Landmarks]], [[Reweighting and enhanced sampling techniques]], [[Dimensionality reduction]].