The poster sessions will be held every afternoon after the sessions and after the diners in the Lounge room, such that discussions are encouraged:

Goffredo Chirco (Albert Einstein Institute Potsdam, Germany) We study the extension of Souriau's formalism of Lie group thermodynamics to the case of reparametrization invariant systems, in the perspective of a general covariant reformulation of statistical mechanics compatible with the conceptual scheme of General Relativity. We explore the extension of such a formalism in the quantum regime, in the context of the group field theory approach to quantum gravity.

Maël Dugast (Univ Lyon, UJM-Saint-Etienne, INSA Lyon), Guillaume Bouleux and Eric Marcon. The geometry of periodically correlated stochastic processes induced by the dilation. We investigate hidden structure of non-stationary signals, and describes them in a geometrical frame- work. Using the specific structure of positive-definite matrices, that arise as correlation matrices, we arrive to some dilation type results. The dilation theory states that the coefficients of a positive-definite matrice can be expressed in term of rotation matrices. These dilations matrices are composed by partial correlation coefficients, and not by the correlations coefficients. Therefore, we can map the set of correlation matrices to the Lie group which consist of rotation matrices, namely the special orthogonal or unitary group. In this settings, a stochastic process can be represented in SO(n), and methods such as classification can be reformulated in SO(n). The main strength in obtaining dilation matrices resides in its stationarity independence assumption. When dealing with stationary processes, the dilation result is the so-called Naimark dilation and a single dilation matrix is obtained. When the process is non-stationary, the dilation leads to the Kolmogorov decomposition, and a set of dilation matrices is provided. An interesting case is when the process is periodically correlated. In this case, we can show that only a finite number of dilation matrices is required to represent the whole process. Features of the underlying process can then be explored through these dilation matrices. Typically, in non-stationary operations, the dilation matrices are time ordered and we can interpolate a trajectory between them, using splines. For periodically correlated processes, this trajectory is a closed loop in the Lie group SO(n), and can be characterized by its length, and other geometrical features, like the torsion of the curve. The idea is to show that a higher level of variability in the signal is associated to a more erratic curve in the associated Lie group.

Simon Fong (School of Computer Science, The University of Birmingham, UK) Joshua Knowles, and Peter Tiňo. Induced Dualistic Structure and Probability Densities on Riemannian Manifolds. Stochastic optimization algorithms over Euclidean spaces has been studied in the framework of Information Geometry [1, 4]. In this work we aim to establish a foundation for computable stochastic optimization algorithms on Riemannian manifolds M by analysing induced pobability densities over M . In particular we focus on the relationship between the manifold M of the search space and the manifold of probability densities, by constructing a locally dually flat family of probability distributions on Riemannian manifolds. This relates the notions of volume elements on manifolds, dually flat Hessian Riemannian statistical manifolds [7], and probability distributions on a measurable space.

We extend the construction of Pennec [5] and Said et al. [6] to density bundle over M and to dually flat families of probability densities. We begin with dually flat family of probability densities supported on the closed ball of injectivity radius in tangent spaces of M . Amari [2] showed that such a family of probability densities supported on a subset of a topological vector space is a Riemannian manifold equipped with Fisher-Rao metric. A family of probability densities over M can thus be defined locally via the pullback line bundle induced by the inverse of Riemannian exponential map, which then allows us to construct a globally supported family of probability densities over M as mixture densities using partition of unity. The pulled-back family of densities locally satisfies Amari’s classical model of statistical manifolds [2], while it also satisfies globally the conditions of k- integrable parametrized measure models by Ay et al. [3] when M has finite Riemannian volume. Finally we also discuss the pullback dualistic structure between Hessian Riemannian manifolds. This allows us to analyse and compute induced probability densities on Riemannian manifolds explicitly, and study convergence results in stochastic optimization algorithms via the induced dualistic structure.

The contents of this abstract have been included in an extended paper submitted to

GSI2017 (currently under review): Induced Dualistic Structure and Locally Dually Flat Densities on Riemannian Manifolds.

References:

[1] Akimoto, Y., Nagata, Y., Ono, I., Kobayashi, S.: Theoretical foundation for CMA-ES from information geometry perspective. Algorithmica 64(4), 698–716 (2012)

[2] Amari, S., Nagaoka, H.: Methods of Information Geometry, Translations of Mathematical monographs, vol. 191. Oxford University Press (2000)

[3] Ay, N., Jost, J., Vân Lê, H., Schwachhöfer, L.: Information geometry and sufficient statistics. Probability Theory and Related Fields 162(1–2), 327– 364 (2015)

[4] Malagò, L., Matteucci, M., Pistone, G.: Towards the geometry of estimation of distribution algorithms based on the exponential family. In: Proceedings of the 11th Workshop Proceedings on Foundations of Genetic Algorithms. pp. 230–242. ACM (2011)

[5] Pennec, X.: Probabilities and statistics on Riemannian manifolds: A geometric approach. Ph.D. thesis, INRIA (2004)

[6] Said, S., Bombrun, L., Berthoumieu, Y., Manton, J.H.: Riemannian Gaussian distributions on the space of symmetric positive definite matrices. IEEE Transactions on Information Theory 63(4), 2153–2170 (2017)

[7] Shima, H., Yagi, K.: Geometry of Hessian manifolds. Differential Geometry and its Applications 7(3), 277–290 (1997)

Antonia M Frassino (Frankfurt Institute for Advanced Studies, Germany) and Dimitri Marinelli (Romanian Institute of Science and Technology): Deep learning and the universal quantum simulator, as meant by Feynmann, are machines able to build representations and can be used as efficient generative models. The representational power of the deep neural networks has been experimentally verified while its theoretical reasons behind their power are still unclear. The generative power of the universal quantum simulators has been proved rigorously, while their actual construction is a topic of active research in Physics. Quantum states and the relative entropy (KL-divergence) allow unveiling the geometrical structure of the two computational models and how information shapes their representation space. Here we introduce the two systems, addressing some problems and the open questions. In particular, we focus on an example presented in the literature that shows the representational power of neural network for quantum systems. The purpose of the analysis is to study how information geometry models their links with Nature.

Edgar Guzman-Gonzalez (Institute of Nuclear Sciences, UNAM, Mexico) Given a spin j, we define the spin shape space as the quotient of the space of the spin space under the action of the rotation group SO(3). This permits us to decompose the spin space as a fiber bundle, where the base is shape space an the fiber is generically isomorphic to SO(3). Using the Fubini-Study metric, we can define a connection and a metric in the fibers and in shape space in a natural way. Here we will present some applications of this decomposition to quantum mechanics, particularly, to the Berry's curvature, along with some mathematical results of the induced geometry in spin shape space.

Hideyuki Ishi (Nagoya University, Japan) We present a new class of convex cones consisting of real symmetric matrices. The class contains both affine homogeneous cones and cones corresponding to decomposable graphs. The affine homogeneous cone is a natural generalization of a symmetric cone. Harmonic analysis and differential geometry on the cone has been developed over the homogeneous cones, and the results can be applied to information geometry, convex programming, and statistics. On the other hand, the convex cones of positive definite matrices with prescribed zeros are intensively studied in the so-called graphical model theory, where the location of zeros ared assigned by a simple graph. If the graph is decomposable (chordal), analysis over the corresponding cone becomes quite feasible. In particular, the cone admits some analytic formulas which are quite similar to the ones for homogeneous cones. In our previous works, a new method for the study of homogeneous cone is established, where the cone is realized as the set of positive definite symmetric matrices with specific block decomposition. Extending this method, we now succeed to find a new class of convex cones admitting matrix realizations, and as is stated at the beginning, the class includes the cones corresponding to decomposable graphs. Then we can discuss the formulas mentioned above in a unified way. In particular, we have explicit formulas for Siegel-type integrals as well as the Koszul-Vinberg characteristic function of the cone in this class. Moreover, we introduce a family of Hessian metrics over the cone with explicit Legendre transforms. These results will be applied to statistics and information geometry.

Marco Laudato (University of Naples Federico II) Quantum and Tomographic Metrics from Relative Entropies. Starting form the relative Tsallis q-entropy as a potential function for the metric, we derive a one parameter family of quantum metrics for N-level systems and analyze in detail the cases N = 2,3. By using a two-parameters generalization of the Tsallis relative entropy, we derive a general expression for quantum metrics depending on the parameters of the entropy. Then we construct explicitly the Fisher-Rao tomographic metric for qubit and qutrit states in different reference frames on the Hilbert space of quantum states. We thus address the problem of reconstructing quantum metrics from tomographic ones, in relation to the uniqueness properties of the latter. Finally, we show that there exists a bijective relation between the choice of the tomographic scheme and the particular operator monotone function identifying a unique quantum metric tensor.

Leo Liberti (CNRS LIX, Ecole Polytechnique, France) Random projections in optimization. The well-known Johnson-Lindenstrauss lemma (JLL) shows that Euclidean distances can be preserved with low distortion via random projections. We show that such projections also preserve many other quantities, such as angles, separating hyperplanes, and distances to cones and convex sets. This makes it possible to approximately solve very large linear programs with a much smaller number of constraints (or variables). Some of our work in progress aims at solving harder problems, such as trust region subproblems and integer linear programs. Ultimately, we hope to show that the practical application of JLL-type random projections will exceed the realm of algorithms which are purely based on Euclidean distances, such as k-means or k-nearest neighbors.

Anton Mallasto (University of Copenhagen, Danemark) : We introduce a novel framework for statistical analysis of populations of non-degenerate Gaussian processes (GPs), which are natural representations of uncertain curves. This allows inherent variation or uncertainty in function-valued data to be properly incorporated in the population analysis. Using the 2-Wasserstein metric we geometrize the space of GPs with L^2 mean and covariance functions over compact index spaces. We present results on existence and uniqueness of the barycenter of a population of GPs, as well as convergence of the metric and the barycenter of their finite-dimensional counterparts. This justifies practical computations. Finally, we demonstrate our framework through experimental validation on GP datasets representing brain connectivity and climate development.

Fabio Maria Mele (Naples University Federico II, Italy) Fisher Metric, Geometric Entanglement and Spin Networks. Motivated by the idea that, in the background-independent framework of a quantum theory of gravity, entanglement is expected to play a key role in the reconstruction of spacetime geometry, we introduce the geometric formulation of Quantum Mechanics in the quantum gravity context, and we use it to give a tensorial characterization of entanglement on spin network states of quantum geometry. Starting from the simplest case of a single-link graph (Wilson line), we define a dictionary to construct a Riemannian metric tensor and a symplectic structure on the space of spin network states. The manifold of (pure) quantum states is then stratified in terms of orbits of equally entangled states and the block-coefficient matrices of the corresponding pulled-back tensors fully encode the information about separability and entanglement. In particular, the off-diagonal blocks define an entanglement monotone interpreted as a distance with respect to the separable state. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We finally extend then such analysis to the study of non-local correlations between two non-adjacent regions of a generic spin network. Our analysis shows that the same spin network graph can be understood as an information graph whose connectivity encodes, both at the local and non-local level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.

Hiroshi Matsuzoe (Nagoya Institute of Technology, Japan) : A survey on infinite dimensional affine differential geometry and information geometry. The main object of affine differential geometry is to study hypersurfaces or immersions that are affinely congruent in an affine space. It is known that dual affine connections and statistical manifold structures naturally arise in this framework. For example, a statistical manifold structure of an exponential family is realized by an affine hypersurface immersion, and the Kullback-Leibler divergence coincides with the affine support function. The Legendre transformation can be discussed by the codimension two affine immersion of a special kind. Therefore, the geometry of dually flat spaces can be generalized by affine differential geometry. In this presentation, we would like to give a short survey about the relations between the infinite dimensional framework of affine differential geometry and information geometry. In particular, we would like to discuss the infinite dimensional affine differential geometry of the maximal exponential families and the alpha-families.

Stéphane Puechmorel (Ecole nationale de l'aviation civile, France) Natural metrics on the tangent bundle to a statistical manifold. The Fischer information metric provides statistical manifolds with a riemanian structure and has been intensively studied in information geometry. The tangent bundle TM of a statistical manifold M is itself a manifold, and one may ask about metrics that can be defined on it. Generally speaking, a connection D on TM can be used to split the tangent space to TM into an horizontal HTM and a vertical VTM distribution, with TTM=HTM+VTM. Any vector field X on TM admits an horizontal (resp. vertical) lift Xh (resp. Xv) on HTM (resp. VTM). Assuming D to be the levi-civita connection with respect to the metric g on M, natural metrics G on TM are those satisfying G(Xh,Yh) = g(X,Y), G(Xh,Yv) = 0 for abritratry vector fields X,Y on M. Special cases are the Sasaki metric, for which G(Xv,Yv) = g(X,Y) and the Cheeger-Gromoll metric. In the context of statistical manifolds, the Levi-Civita connection is the 0-connection in the so-called family of alpha-connections of Amari. The first part of this work is devoted to the study of its associated Sasaki and Cheeger-Gromoll metrics, and their relations with derivatives of the score functions. In a second part, the question of general a-connections is discussed. In particular, e and m connections, which are non-metric but of major importance in information geometry, cannot be directly associated with a natural metric on TM due to the requirement G(Xh,Yv) = 0 that makes sense only with the Levi-Civita connection (orthogonality relation). Adapted metrics on TM will thus be considered, which fit within the frame of alpha-connections. Finally, a statistical interpretation will be given for the objects introduced.

Ha Quang Minh (Istituto Italiano di Tecnologia, Italy) Infinite-dimensional Log-determinant Divergences between positive definite trace class operators. We present a novel parametrized family of Alpha Log-Determinant (Log-Det) divergences between positive definite unitized trace class operators on an infinite-dimensional Hilbert space. This is a generalization of the Alpha Log-Det diver-

gences between symmetric, positive definite (SPD) matrices [1] to the infinite-dimensional setting. The generalization is carried out via the introduction of the extended Fredholm determinant for unitized trace class operators and the accompanying generalization of the log-concavity for determinants of SPD matrices. For the Log-Det divergences between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), we obtain closed form formulas via the corresponding Gram matrices. By employing the Log-Det divergences between positive definite unitized trace class operators, we then generalize the Bhattacharyya and Hellinger distances and the Kullback-Leibler and Rényi divergences between multivariate normal distributions to Gaussian measures on an infinite-dimensional Hilbert space. The current infinite-dimensional Log-Det formulation is appearing in [3]. It has recently been generalized to [4], which is the infinite-dimensional generalization of the Alpha-Beta Log-Det divergences between SPD matrices [2].

References

[1] Z. Chebbi and M. Moakher. Means of Hermitian positive-definite matrices based on the Log-Determinant α-divergence function. Linear Algebra and its Applications, 436(7):1872–1889, 2012.

[2] A. Cichocki, S. Cruces, and S. Amari. Log-Determinant divergences revisited: Alpha-Beta and

Gamma Log-Det divergences. Entropy, 17(5):2988–3034, 2015.

[3] H.Q. Minh. Infinite-dimensional Log-Determinant divergences between positive definite trace class operators. Linear Algebra and Its Applications, In Press, 2016.

[4] H.Q. Minh. Infinite-dimensional Log-Determinant divergences II: Alpha-Beta divergences. arXiv preprint arXiv:1610.08087v2, 2016.

Eduardo Serrano-Ensástiga (Institute for Nuclear Sciences, UNAM, Mexico) Study of the FS metric with the Majorana's stellar representation. The action of the group SO(3) in the Hilbert space of spin j H allows a principal fiber bundle structure, and split the tangent space in each point in two vectorial subspaces, the vectical and horizontal space, where the horizontal vectors are defined as the ortogonal vectors of the vertical ones with the FS metric. By the other hand, the Majorana’s stellar representation gives us a geometric view of H and its tangent bundle TH in terms of moving points on the sphere. In this representation, the vertical vectors have a clear interpretation, but the horizontal vectors don't. We expose some properties of the FS metric, the characteristics of the vertical vectors and a study of the horizontality condition in terms of the stellar representation.