Documentation

THEORY.md

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+# Background
+
+## Density
+
+The density is represented as a sum of atomic contributions (density fitting):
+
+$$
+\rho(\vec r) = \sum_i c_i \chi_i,
+$$
+
+where $[\chi_i]$ is a density-fitting basis such as `cc-pvqz-jkfit` and $[c_i]$ is a coefficients vector.
+The index $i$ is a shorthand for $(\ell,n,m)$, where $\ell$ is the angular momentum, 
+$m$ is the magnetic quantum number (for a real spherical harmonic), 
+and $n$ is the radial channel index (since in a basis there can be several functions
+with the same $(\ell, m)$ but different exponents).
+
+## Prediction
+
+The coefficients are obtained from kernel $K$ and weights $\vec x$:
+
+$$
+c_i^{\mathrm{ML}}(\vec x) = \sum_j K_{ij} x_j.
+$$
+
+## Kernel 
+
+We use the $λ$-SOAP kernel. It is computed as a dot product of power spectra (representations).
+The kernel can be interpreted as a function comparing orbitals of the same $\ell$ and $n$ but possibly different $m$
+sitting on atoms (environments) with the same nuclear charge $q$.
+Thus the full matrix $K$ has a block structure and consists of small $(2\ell+1) \times (2\ell+1)$ matrices.
+
+## Reference environments
+
+There are too many orbitals in the training set to compute and invert the regression matrix.
+Thus the sum index $j$ in the predictions goes not over all the training set orbitals,
+but over the orbitals sitting on a selected subset of atoms (reference environments).
+Normally, there are $\sim 1000$ environments selected by farthest point sampling (FPS).
+
+
+## Regression
+
+The loss function is a sum of 
+squared coefficient errors norms 
+computed with a metrix matrix $O$
+for each molecule:
+
+$$
+\sum_{\mathrm{train}} (\vec c-\vec c^{\mathrm{ML}})^T O (\vec c-\vec c^{\mathrm{ML}}).
+$$
+
+The matrix $O$ guides the optimization target (e.g. residue self-overlap or self-repulsion).
+The regression equation becomes
+
+$$
+B \vec x = \vec A,
+$$
+
+where for historical reasons we call the coefficient matrix "B matrix" and the constant vector "A vector".
+The problem dimensionality equals to the number of orbitals in the reference environments,
+and $A$ and $B$ are computed as a sum of sparse matrix multiplications:
+
+$$
+B = \sum_{\mathrm{mol} \in \mathrm{train}} K_{\mathrm{refs}/\mathrm{mol}} O_{\mathrm{mol}} K_{\mathrm{mol}/\mathrm{refs}}
+$$
+
+$$
+\vec A = \sum_{\mathrm{mol} \in \mathrm{train}} K_{\mathrm{refs}/\mathrm{mol}} O_{\mathrm{mol}}  \vec c_{\mathrm{mol}} 
+$$
+
+The computation of the $B$ matrix is the most computationally expensive step.
+
+The weight are obtained by solving the linear system. For regularization, 
+a unit matrix as well as a sparse kernel matrix $K_{refs/refs}$ are added to the $B$ matrix,
+each multiplied by its own parameter (a non-negative number $\ll 1$).