tensor_reduction_docs.md

SOAP Compression

The length of the standard SOAP vector (powerspectrum)

$$p^{\alpha\beta}{nn'l} = \sum_m c^\alpha{nlm} c^\beta_{n'lm} = \bf{c}^\alpha_{nl} \cdot \bf{c}^\beta_{n'l}$$

is $\frac{1}{2}NS(NS+1)(L+1) + 1$ where $N$=n_max, $S$=n_species, $L$=l_max and the final element is covariance_sigma0 (set after normalisation). Various compression strategies are available to reduce this $\mathcal{O}(N^2S^2)$ scaling.

Tensor-reduction

The tensor-reduction introudced in Tensor-reduced atomic density representations can be applied by first selecting which "density channels" (radial, species or both) are mixed together before selecting how the mixed channels should be coupled together (tensor product, element-wise). Tensor decomposition using radial-species mixing with element-wise coupling is written as

$$p_{kl} = \sum_m c_{klm} c_{klm} = \bf{c}{kl} \cdot \bf{c}{kl} $$

where $$c_{klm} = \sum_{\alpha n} W^k_{\alpha n} c^\alpha_{nlm} $$

Keyword	Values	Description
R_mix	T or F	mixes the radial channels
Z_mix	T or F	mixes the species channels
sym_mix	T or F	T means use "tensor-decomposition" whereas F means use "tensor-sketching" as described in the original article
K	int > 0	How many mixed channels to create
coupling	T or F	T means use tensor product coupling across mixed channels, F means use element-wise

Note that full tensor product coupling is always used across any "un-mixed" channels.

Examples

• R_mix=T Z_mix=T K=5 sym_mix=T coupling=F Means randomly mix the radial and species channels together to form 5 new channels then couple each of these channels to itself only. Length of final vector will be $K(L+1)$. If coupling=T then length would be $\frac{1}{2}K(K+1)(L+1)$ as each channel would be coupled to every other channel (per l).
• R_mix=F Z_mix=T K=5 sym_mix=F coupling=F Means randomly mix the species channels together so that there are n_max*K channels. Final length will be $\frac{1}{2}N(N+1)K(L+1)$ from element-wise coupling between the K mixed species channels and full tensor-product coupling between the radial channels.

Radial and species sensitive correlation orders

The radial and species sensitive correlation orders can be set independently using

Keyword	Values	Description
nu_R	0, 1, 2	radially sensitive correlation order
nu_S	0, 1, 2	species senstitive correlation order

The schematic below illustrates the physical interpretation and includes a correspondance between the nu_R nu_S notation used here and that used in the original article Compressing local atomic neighbouhood descriptors .

Other

• Setting diagonal_radial=T only includes terms where $n=n'$ so that the length becomes $\frac{1}{2}NS(S+1)(L+1) + 1$
• Different species can be grouped together using the Z_map keyword where commas separate groups of elements e.g. Z_map={8, 23 41 42} treats all the metals as identical and Oxygen as distinct. As the power spectrum is correlation order 2 it is also possible to specify two distinct groupings separated by a colon e.g. Z_map={8, 23, 41, 42: 8, 23 41 42} where in the first density every element is distinct whilst in the second again the metals are all treated as identical but Oxygen is distinct.