Density estimation on an unknown submanifold

We investigate density estimation from a $n$-sample in the Euclidean space $\mathbb R^D$, when the data is supported by an unknown submanifold $M$ of possibly unknown dimension $d < D$ under a reach condition. We study nonparametric kernel methods for pointwise and integrated loss, with data-driven bandwidths that incorporate some learning of the geometry via a local dimension estimator. When $f$ has H\"older smoothness $\beta$ and $M$ has regularity $\alpha$ in a sense to be defined, our estimator achieves the rate $n^{-\alpha \wedge \beta/(2\alpha \wedge \beta+d)}$ and does not depend on the ambient dimension $D$ and is asymptotically minimax for $\alpha \geq \beta$. Following Lepski's principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case $\alpha \leq \beta$: by estimating in some sense the underlying geometry of $M$, we establish in dimension $d=1$ that the minimax rate is $n^{-\beta/(2\beta+1)}$ proving in particular that it does not depend on the regularity of $M$. Finally, a numerical implementation is conducted on some case studies in order to confirm the practical feasibility of our estimators

Statistical Geometry in Quantum Mechanics

A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the Hilbert space H. By consideration of the square-root density function we can regard M as a submanifold of the unit sphere in H. Therefore, H embodies the `state space' of the probability distributions, and the geometry of M can be described in terms of the embedding of in H. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer-Rao and Bhattacharyya inequalities. The statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.Comment: 32 pages, LaTex file, Extended version to include quantum measurement theor

K\"ahlerian information geometry for signal processing

We prove the correspondence between the information geometry of a signal filter and a K\"ahler manifold. The information geometry of a minimum-phase linear system with a finite complex cepstrum norm is a K\"ahler manifold. The square of the complex cepstrum norm of the signal filter corresponds to the K\"ahler potential. The Hermitian structure of the K\"ahler manifold is explicitly emergent if and only if the impulse response function of the highest degree in $z$ is constant in model parameters. The K\"ahlerian information geometry takes advantage of more efficient calculation steps for the metric tensor and the Ricci tensor. Moreover, $\alpha$-generalization on the geometric tensors is linear in $\alpha$. It is also robust to find Bayesian predictive priors, such as superharmonic priors, because Laplace-Beltrami operators on K\"ahler manifolds are in much simpler forms than those of the non-K\"ahler manifolds. Several time series models are studied in the K\"ahlerian information geometry.Comment: 24 pages, published versio

Quantum state tomography with non-instantaneous measurements, imperfections and decoherence

Tomography of a quantum state is usually based on positive operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multi-dimensional Laplace integrals appearing in Bayesian Mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum non-demolition measurements relying on Rydberg atoms; heterodyne fluorescence measurements of a superconducting qubit.Comment: 11 pages, 4 figures, submitte