The extended Gaussian family is the closure of the Gaussian family obtained by completing the Gaussian family with the counterpart elements induced by degenerate covariance or degenerate precision matrices, or a mix of both degeneracies. The parameter space of the extended Gaussian family forms a symmetric positive semi-definite matrix bicone, i.e. two partial symmetric positive semi-definite matrix cones joined at their bases. In this paper, we study the Hilbert geometry of such an open bounded convex symmetric positive-definite bicone. We report the closed-form formula for the corresponding Hilbert metric distance and study exhaustively its invariance properties. We also touch upon potential applications of this geometry for dealing with extended Gaussian distributions.

Schr\"{o}dinger bridge can be viewed as a continuous-time stochastic control problem where the goal is to find an optimally controlled diffusion process with a pre-specified terminal distribution $\mu_T$. We propose to generalize this stochastic control problem by allowing the terminal distribution to differ from $\mu_T$ but penalizing the Kullback-Leibler divergence between the two distributions. We call this new control problem soft-constrained Schr\"{o}dinger bridge (SSB). The main contribution of this work is a theoretical derivation of the solution to SSB, which shows that the terminal distribution of the optimally controlled process is a geometric mixture of $\mu_T$ and some other distribution. This result is further extended to a time series setting. One application of SSB is the development of robust generative diffusion models. We propose a score matching-based algorithm for sampling from geometric mixtures and showcase its use via a numerical example for the MNIST data set.

Reciprocal processes are acausal generalizations of Markov processes introduced by Bernstein in 1932. In the literature, a significant amount of attention has been focused on developing dynamical models for reciprocal processes. Recently, probabilistic graphical models for reciprocal processes have been provided. This opens the way to the application of efficient inference algorithms in the machine learning literature to solve the smoothing problem for reciprocal processes. Such algorithms are known to converge if the underlying graph is a tree. This is not the case for a reciprocal process, whose associated graphical model is a single loop network. The contribution of this paper is twofold. First, we introduce belief propagation for Gaussian reciprocal processes. Second, we establish a link between convergence analysis of belief propagation for Gaussian reciprocal processes and stability theory for differentially positive systems.

This paper is about the asymptotic behavior of the Kalman filter [11]. The Kalman-Bucy filter merges predictions from a trusted model of the dynamics of the system with incoming measurements in order to get an accurate, real-time estimate of the unknown internal state of the system. The estimation relies on the computation of a positive semidefinite matrix P, the covariance of the estimation error. The difference equation verified by P is a discrete-time algebraic Riccati equation. Kalman showed that, for a linear time-invariant system, under detectability conditions, the Riccati equation converges to a fixed point, which is unique under certain stabilizability conditions ([10], see also [9]). The classical convergence analysis requires several steps, showing that the error covariance is upper bounded, that, with zero initial value, it is monotone increasing, so that it admits a limit, and then proving that the corresponding filter is stable and that the limit is the same for all initial covariances. In [4] Bougerol proposed a more geometric convergence analysis by showing that the discrete-time Riccati iteration is a contraction for the Riemannian metric associated to the cone of positive definite matrices. Other authors elaborated along these lines (see e.g.