Table 1.

Comparison of the quantum mechanics (QM) and machine learning (ML) formalism.

QM		ML
Consider an archetypal case, which does not reduce the generality, a solid with one orbital $∣i⟩$ per atomic site i. In tight-binding formalism using the hopping integrals tij, the probability of transition from the orbital $∣i⟩$ to orbital $∣j⟩$, the Hamiltonian reads: $\mathbf{H}={\sum }_{i,j}{t}_{ij}∣i⟩⟨j∣$. The energy levels of the system are the eigenvalues of Schrödinger equation: $\mathbf{H}∣m⟩={ϵ}_m∣m⟩$. We are interested in the estimation of the local energy ${ϵ}_{i_{\star }}$ associated with the atom i⋆.		Consider that we have learned the sample covariance matrix Σb of M data points, $x_m\in {\mathbb{R}}^D$. The data are centred to mean zero. The mth element of the descriptor space can be written in an initial basis as $\mid {\mathbf{x}}_m⟩=\sum x_m^i\mid i⟩$. The eigenelement of ${\mathbf{\Sigma }}_b$ is $\left\{{\lambda }_m,\mid {\mathbf{v}}_m⟩\right\}$. We are interested in the statistical distance $d_{i_{\star }}$ of the data point $\mid x_{i_{\star }}⟩$.
$\mathbf{H}={\sum }_{i,j}{t}_{ij}∣i⟩⟨j∣$	(t.1)	${\mathbf{\Sigma }}_b={\sum }_{i,j}\left(\frac{1}{M-1}{\sum }_m^M{x}_m^i{x}_m^j\right)∣i⟩⟨j∣$	(t.2)
$\mathbf{H}={\sum }_m{ϵ}_m∣m⟩⟨m∣$	(t.3)	${\mathbf{\Sigma }}_b={\sum }_m{\lambda }_m∣{\mathbf{v}}_m⟩⟨{\mathbf{v}}_m∣$	(t.4)
E = ∑m∫dϵn(ϵ)ϵδ(ϵ − ϵm)	(t.5)	Tr(Σb) = ∑m∫dλλδ(λ − λm)	(t.6)
${\rho }_{i_{\star }}\left(ϵ\right)={\sum }_m{∣⟨{\mathbf{i}}_{\star }\mid m⟩∣}^2\delta \left(ϵ-{ϵ}_m\right)$	(t.7)	${\rho }_{i_{\star }}\left(\lambda \right)={\sum }_m\mid ⟨{\mathbf{x}}_{i_{\star }}\mid {\mathbf{v}}_m⟩{\mid }^2\delta \left(\lambda -{\lambda }_m\right)$	(t.8)
${ϵ}_{i_{\star }}=\int dϵ{\rho }_{i_{\star }}\left(ϵ\right)ϵn\left(ϵ\right)$	(t.9)	$d_{i_{\star }}^2=\int d\lambda {\rho }_{i_{\star }}\left(\lambda \right)\frac{1}{\lambda }$	(t.10)
$p\left({ϵ}_{i_{\star }}\right)\propto \exp \left(-\beta {ϵ}_{i_{\star }}\right)\mathrm{for}\beta \to 0$	(t.11)	$p\left({\mathbf{x}}_{i_{\star }}\right)\propto \exp \left(-d_{i_{\star }}^2/2\right)$	(t.12)

Commonly used quantum mechanics (QM) formalism of local energies (on left) is compared with ML formalism of sample covariance matrix and statistical distances (on right). To emphasize the similarities between the two approaches, we adopt the QM bra-ket notation for statistical distance. The data points of the descriptor space are the ket vectors $∣\mathbf{x}⟩=\mathbf{x}\in {\mathbb{R}}^{D\times 1}$, whereas the bra vectors are the transposed vectors $⟨\mathbf{x}∣={\mathbf{x}}^T\in {\mathbb{R}}^{1\times D}$. ${\rho }_{i_{\star }}$ and ρ_λ are the local density of states and variance, respectively, for the state $∣i_{\star }⟩$ / data point $\mid {\mathbf{x}}_{i_{\star }}⟩$. $p\left({ϵ}_{i_{\star }}\right)$ is the probability of the state $∣i_{\star }⟩$ in the limit of high temperature, where the Fermi-Dirac distribution becomes classical Boltzmann distribution. $p\left({\mathbf{x}}_{i_{\star }}\right)$ is the marginal likelihood of the data point $\mid {\mathbf{x}}_{i_{\star }}⟩$.