Approximate symmetries

To be more general let us consider a priori information related to some approximate symmetry [67]. In contrast to an exact symmetry where it is sufficient to restrict $v$ to be symmetric, approximate symmetries require the definition of a distance measuring the deviation from exact symmetry. In particular, consider a unitary symmetry operation $S$, i.e., $S^\dagger=S^{-1}$, $S^\dagger$ denoting the hermitian conjugate of $S$. Further, define an operator ${\bf S}$ acting on (local or nonlocal) potentials $V$, by ${\bf S} V$ = $S^\dagger V S$. In case of an exact symmetry $V$ commutes with $S$, i.e., $[V,S]$ = $0$ and thus ${\bf S} V$ = $S^\dagger V S$ = $V$. In case of an approximate symmetry we may choose a prior

$$p_0(V) \propto e^{-E_S} ,$$ (35)

with `symmetry energy' or `symmetry error'

$$E_S =\frac{1}{2}<\!V-{\bf S}V\,\vert\,{\bf K}_S\,\vert\,V-{\bf S}V\!> =\frac{1}{2}<\!V\,\vert\,{\bf K}_0\,\vert\,V\!> ,$$ (36)

some positive (semi-)definite ${\bf K}_S$, hence positive semi-definite ${\bf K}_0$ = $({\bf I}-{\bf S})^\dagger {\bf K}_S ({\bf I}-{\bf S})$, ${\bf I}$ denoting the identity. (Symmetric $V$ are within the Null space of ${\bf K}_S$.) If ${\bf S}$ belongs to a Lie group it can be expressed by a Lie group parameter $\theta$ and the generator ${\bf s}$ of the corresponding infinitesimal symmetry operation as ${\bf S}(\theta)$ = $\exp(\theta {\bf s})$. Hence, we can define an error with respect to the infinitesimal operation ${\bf s}$ with, say, ${{\bf K}_S}$ = ${\bf I}$,

$$E_s =\lim_{\theta\rightarrow0} \frac{1}{2}\mbox{$<\!\frac{V... ...\frac{1}{2}<\!V\,\vert\,{\bf s}^\dagger {\bf s}\,\vert\,V\!> .$$ (37)

Choosing, for instance, ${\bf s}$ as the derivative operator (for vanishing or periodic boundary terms) results in the typical Laplacian smoothness prior which measures the degree of symmetry of $v$ under infinitesimal translations.

Another possibility to implement approximate symmetries is given by a prior with symmetric reference potential $V_S$ = ${\bf S}V_S$

$$E_{V_S}=\frac{1}{2}\mbox{$<\!V-V_S\,\vert\,V-V_S\!>$} .$$ (38)

In contrast to Eq. (36) which is minimized by any symmetric $V$, this term is minimized only by $V$ = $V_S$. Note, that also in Eq. (36) an explicit non-zero reference potential $V_0$ can be included, meaning that not $V$ but the difference $V-V_0$ is expected to be approximately symmetric.

Finally, a certain deviation $\kappa$ from exact symmetry might even be expected. This can be implemented by including `generalized data terms' [32]

$$E_{S,\kappa} =\frac{1}{2}(E_S(V)-\kappa)^2 =\frac{1}{2}(\frac{1}{2}\mbox{$<\!V-{\bf S}V\,\vert\,V-{\bf S}V\!>$}-\kappa)^2 ,$$ (39)

similar to the usual mean squared error terms used in regression.