Statistical mirror symmetry
Motivated by recent developments in information geometry, the authors define and investigate the notion of statistical mirror symmetry between a Hermitian manifold structure and an almost Kähler manifold structure on the tangent bundle \(TM\) of an affine manifold \(M\) endowed with a \(\mathcal{P}\)-geometry. Specifically, let \(M\) be a real, smooth, finite-dimensional manifold endowed with a Riemannian metric tensor \(g\), and a couple \((\nabla,\nabla^{*})\) of affine connection which are dually related with respect to \(g\) in the sense that \[ X\left(g(Y,Z)\right)=g\left(\nabla_{X}Y,Z\right) + g\left(Y,\nabla^{*}_{X}Z\right) \] holds for all vector fields \(X,Y,Z\) on \(M\). Then, it is not hard to show that \[ g\left(R^{\nabla}(Z,W)X,Y\right) + g\left(R^{\nabla^{*}}(Z,W)X,Y\right)=0 , \] where \(R^{\nabla}\) and \(R^{\nabla^{*}}\) denote, respectively, the curvature tensor of \(\nabla\) and \(\nabla^{*}\), and \[ \begin{split} g\left(T^{\nabla^{*}}(Z,X) - T^{\nabla}(Z,X),Y\right)&=\left(\nabla_{Z}g\right)(X,Y)- \left(\nabla_{X}g\right)(Z,Y)= \\ &=\left(\nabla_{X}^{*}g\right)(Z,Y) - \left(\nabla_{Z}^{*}g\right)(X,Y), \end{split} \] where \(T^{\nabla}\) and \(T^{\nabla^{*}}\) denote, respectively, the torsion tensors of \(\nabla\) and \(\nabla^{*}\). When both connection are torsion-less, the data \((M,g,\nabla,\nabla^{*})\) is referred to as a statistical manifold. The name is justified because this type of mathematical structure arises naturally in the context of statistical models where the manifold \(M\) is some manifold of parametrized probability distributions on some measure space (e.g., Gaussian measures on \(\mathbb{R}\)), the Riemannian metric tensor \(g\) is the Fisher-Rao metric tensor, and the couple \((\nabla,\nabla^{*})\) is defined either exploiting the family of \(\alpha\)-connections, or using a suitable extraction algorithm from a preferred relative entropy function, or divergence function, or contrast function. In this context, of particular interest is the case of a statistical manifold \((M,g,\nabla,\nabla^{*})\) for which both \(\nabla\) and \(\nabla^{*}\) are flat, referred to as a dually flat statistical manifold. It is clear that there are topological obstructions to dually flatness because \(M\) must admit a flat connection, i.e., it must be an affine manifold. From a purely mathematical point of view (and perhaps also from the point of view of quantum information geometry), there is no need to consider only torsion-less affine connections, and in fact, in the work under review, the authors consider the case in which \(\nabla\) is a flat connection, while \(\nabla^{*}\) is ``only'' curvature-free. Note that \(R^{\nabla^{*}}=0\) actually follows from the fact that \(\nabla\) is flat and from the equation on curvature above, while the equation on torsion above imposes a sort of coupling between the torsion of \(\nabla^{*}\), the Riemannian metric \(g\), and the affine connection \(\nabla\). The data \((M,g,\nabla,\nabla^{*})\) thus obtained is referred to as a manifold with \(\mathcal{P}\)-geometry, which may be considered as an example of a statistical manifold admitting torsion. It is clear that, very much like it happens for dually flat statistical manifolds, there are topological obstructions for the existence of a \(\mathcal{P}\)-geometry on \(M\). Indeed, in order to admit a flat affine connection, \(M\) must be an affine manifold. The whole Section 2 is a very nice and detailed account of statistical manifolds, dually-flat manifolds, Hessian manifolds, manifolds admitting a \(\mathcal{P}\)-geometry, Hermitian manifolds, and almost Kähler manifolds. It is well written and clearly prepares the reader for the main object of investigation of the paper, namely the statistical mirror symmetry on the tangent bundle \(TM\) of a manifold \(M\) admitting a \(\mathcal{P}\)-geometry. While it may be argued that the material presented in Section 2 is not new, it must be said that the presentation itself deserves attention because of its clarity and the fact that the work is almost self-consistent. In Section 3, the authors recall how the data \((M,g,\nabla)\) can be used to define geometrical structures on the tangent bundle \(TM\). The idea is to first exploit the connection \(\nabla\) to decompose the tangent bundle \(T(TM)\) of the tangent bundle \(TM\) into a vertical and a horizontal part. Given a vector field \(X\in\mathcal{X}(M)\) it is possible to define a vertical lift \(X^{V}\in\mathfrak{X}(TM)\) and a horizontal lift \(X^{H}\in\mathfrak{X}(TM)\) which, in local coordinates adapted to \(TM\), read \[ X=X^{j}\frac{\partial}{\partial x^{j}}, \quad X^{V} =X^{j}\frac{\partial}{\partial v^{j}}, \quad X^{H}= X^{j}\frac{\partial}{\partial x^{j}} - \Gamma^{j}_{kl}X^{k}v^{l}\frac{\partial}{\partial v^{j}}, \] where \(\Gamma^{j}_{kl}\) are the connection coefficients. Vertical and horizontal lift provide a generating set for the module of vector fields on \(TM\) and thus can be used to define a \((1,1)\)-tensor field \(J\) on \(T(TM)\) setting \[J(X^{H}):=X^{V}, \qquad J(X^{V}):= - X^{H} . \] Direct computation shows that \(J^{2}=-\mathrm{id}\) so that \(J\) is an \textbf{almost complex structure}. Next, a metric tensor \(G\) on \(TM\) is defined setting \[ G\left(X^{V},Y^{H}\right)=G\left(Y^{H},X^{V}\right) := 0 ,\quad G\left(X^{V},Y^{V}\right)=G\left(X^{H},Y^{H}\right) := g(X,Y). \] It is then clear that \(J\) preserves \(G\) in the sense that \[ G\left(Z,W\right)=G\left(J(Z),J(W)\right) \] for all \(Z,W\in\mathfrak{X}(TM)\), so that \((TM,G,J)\) becomes an almost Hermitian manifold whose associated 2-form \(\omega\) reads \[ \omega(Z,W):=G(Z,J(W)) \] for all \(Z,W\in\mathfrak{X}(TM)\). An important result is that \((TM,G,J)\) is a fair Hermitian manifold (i.e., \(J\) is a fair complex structure) if and only if \(\nabla\) is flat. On the other hand, it holds that \((TM,G,J)\) is an almost Kähler manifold (i.e., \(\mathrm{d}\omega=0\)) if and only if \[ T^{\nabla^{*}}=0 , \] in which case it coincides with the pullback through \(g\) of the canonical symplectic form on \(T^{*}M\). At this point, the authors exploit all the reviewed material to introduce the notion of statistical mirror symmetry. Given \((M,g,\nabla,\nabla^{*})\) defining a manifold endowed with a \(\mathcal{P}\)-geometry, there always are an Hermitian manifold \((TM,G^{\nabla},J^{\nabla})\) built from \((M,g,\nabla)\) and an almost Kähler manifold \((TM, G^{\nabla^{*}},J^{\nabla^{*}})\) built from \((M,g,\nabla^{*})\) which both become fair Kähler manifold when the torsion of \(\nabla^{*}\) vanishes. These two manifolds are defined to be in a relation of statistical mirror symmetry. The rest of Section 3 is devoted to a brief comparison of statistical mirror symmetry with mirror symmetry of Calabi-Yau manifolds and semi-flat Calabi-Yau manifolds. Section 4 is devoted to the explicit ``computation'' of statistical mirror symmetry in the specific case of manifolds of parametrized probability distributions used in classical information geometry, and to investigate the role of reference-representation biduality with respect to the notion of statistical mirror symmetry (in this case, the important result is proposition 20 connecting, in the case of an Hessian manifold, the almost Kähler structure on \(TM\) with the Bregman divergence of \(M\)). The authors also comment on the use of contrast and pre-contrast functions to define statistical manifolds and manifolds with \(\mathcal{P}\)-geometry, and they also introduce the notion of super-contrast function in order to be able to define statistical manifolds admitting torsion for both g-related connections. Section 5 contains a general discussion on the results of the work, with particular attention to global obstructions. In the end, I believe that ``Statistical mirror symmetry'' by J. Zhang and G. Khan is an interesting and well-written paper that can be of interest for researchers working both in pure differential geometry and in classical and quantum information geometry, and I personally believe that the notion of manifold endowed with a \(\mathcal{P}\)-geometry will be of particular relevance in the context of quantum information geometry where dually-related connection with torsion appear naturally.
scientific article; zbMATH DE number 7315171
scientific article