Approximate symmetries

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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 to be symmetric, approximate symmetries require the definition of a distance measuring the deviation from exact symmetry. In particular, consider a unitary symmetry operation , i.e., $S^\dagger=S^{-1}$ , $S^\dagger$ denoting the hermitian conjugate of . Further, define an operator ${\bf S}$ acting on (local or nonlocal) potentials , by ${\bf S} V$ = $S^\dagger V S$ . In case of an exact symmetry commutes with , i.e., = and thus ${\bf S} V$ = $S^\dagger V S$ = . In case of an approximate symmetry we may choose a prior

$\begin{displaymath} p_0(V) \propto e^{-E_S} , \end{displaymath}$

(35)

with `symmetry energy' or `symmetry error'

$\begin{displaymath} E_S =\frac{1}{2}<\!V-{\bf S}V\,\vert\,{\bf K}_S\,\vert\,V-{\bf S}V\!> =\frac{1}{2}<\!V\,\vert\,{\bf K}_0\,\vert\,V\!> , \end{displaymath}$

(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

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, ${{\bf K}_S}$ = ${\bf I}$ ,

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

(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

under infinitesimal translations.

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

$\begin{displaymath} E_{V_S}=\frac{1}{2}\mbox{$<\!V-V_S\,\vert\,V-V_S\!>$} . \end{displaymath}$

(38)

In contrast to Eq. (36) which is minimized by any symmetric

, this term is minimized only by

. Note, that also in Eq. (36) an explicit non-zero reference potential

can be included, meaning that not

but the difference

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]

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

(39)

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

Next: Mixtures of Gaussian process Up: Prior models for potentials Previous: Gaussian processes and smooth Contents

Joerg_Lemm 2000-06-06