Optimization on manifolds: A symplectic approach

There has been great interest in using tools from dynamical systems and numerical analysis of differential equations to understand and construct new optimization methods. In particular, recently a new paradigm has emerged that applies ideas from mechanics and geometric integration to obtain accelerated optimization methods on Euclidean spaces. This has important consequences given that accelerated methods are the workhorses behind many machine learning applications. In this paper we build upon these advances and propose a framework for dissipative and constrained Hamiltonian systems that is suitable for solving optimization problems on arbitrary smooth manifolds. Importantly, this allows us to leverage the well-established theory of symplectic integration to derive "rate-matching" dissipative integrators. This brings a new perspective to optimization on manifolds whereby convergence guarantees follow by construction from classical arguments in symplectic geometry and backward error analysis. Moreover, we construct two dissipative generalizations of leapfrog that are straightforward to implement: one for Lie groups and homogeneous spaces, that relies on the tractable geodesic flow or a retraction thereof, and the other for constrained submanifolds that is based on a dissipative generalization of the famous RATTLE integrator

Fully probabilistic deep models for forward and inverse problems in parametric PDEs

We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to-parameter (inverse) maps of parametric partial differential equations (PDEs). Our formulation leverages conventional PDE discretization techniques, deep neural networks, probabilistic modelling, and variational inference to assemble a fully probabilistic coherent framework. In the posited probabilistic model, both the forward and inverse maps are approximated as Gaussian distributions with a mean and covariance parameterized by deep neural networks. The PDE residual is assumed to be an observed random vector of value zero, hence we model it as a random vector with a zero mean and a user-prescribed covariance. The model is trained by maximizing the probability, that is the evidence or marginal likelihood, of observing a residual of zero by maximizing the evidence lower bound (ELBO). Consequently, the proposed methodology does not require any independent PDE solves and is physics-informed at training time, allowing the real-time solution of PDE forward and inverse problems after training. The proposed framework can be easily extended to seamlessly integrate observed data to solve inverse problems and to build generative models. We demonstrate the efficiency and robustness of our method on finite element discretized parametric PDE problems such as linear and nonlinear Poisson problems, elastic shells with complex 3D geometries, and time-dependent nonlinear and inhomogeneous PDEs using a physics-informed neural network (PINN) discretization. We achieve up to three orders of magnitude speed-up after training compared to traditional finite element method (FEM), while outputting coherent uncertainty estimates

Anti-Hepatitis C Virus Serology in Immune Thrombocytopenia: A Retrospective Analysis in 101 Patients.

Hepatitis C virus (HCV), an RNA virus, is known to be the major cause of post-transfusion non-A, non-B hepatitis. HCV can induce several expressions of autoimmunity, including both serological abnormalities and clinical disorders. The relationship between the HCV infection and anti-platelet autoimmunity has been occasionally described, but is still far from well-defined. We retrospectively analysed 101 serum specimens, collected between 1988 and 1994, from patients with immune thrombocytopenia (ITP) for the presence of anti-HCV antibodies. Eighty-seven patients were classified as having idiopathic, and 14 secondary ITP (4 systemic lupus erythematosus, 9 non-Hodgkin's lymphoma and 1 Evan's syndrome). Anti-HCV antibodies were determined by second generation tests (ELISA + RIBA). A specimen was considered positive for HCV antibodies in the presence of ELISA reactivity (sample optical density/cut-off > 1.00) accompanied by RIBA reactivity to at least one HCV specific antigen. 20 sera (20%) were positive, with a prevalence higher in secondary than in idiopathic ITP (43% vs. 16%, p < 0.05). No differences were found between anti-HCV positive and negative patients regarding gender, platelet count, platelet associated immunoglobulins, hepatitis B virus serology and liver enzyme profile. On the contrary, mean age was higher in the HCV positive vs HCV negative ones (58±18SD vs. 44±20yrs, p < 0.01), in keeping with the increasing prevalence of HCV infection with ageing. HCV positive patients, showed a poor response to treatment (platelet count lower than 50,000/μl after conventional medical therapy for immune thrombocytopenia) compared to anti-HCV negative ones, (50% versus 7.3%, p < 0.001). When we excluded patients who were exposed to risk factors for HCV infection after ITP diagnosis and before the serum collection, the prevalence of anti-HCV antibodies was not very different (17.6%) from that found in the series as a whole (19.8%). Our results seem to indicate that HCV infection may play a role in triggering several cases ITP, and moreover might constitute a negative prognostic factor for therapy response

Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion

First, we revisit basic theory of functional It\uf4/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness

Measuring Gaussian quantum information and correlations using the Renyi entropy of order 2

We demonstrate that the Renyi-2 entropy provides a natural measure of information for any multimode Gaussian state of quantum harmonic systems, operationally linked to the phase-space Shannon sampling entropy of the Wigner distribution of the state. We prove that, in the Gaussian scenario, such an entropy satisfies the strong subadditivity inequality, a key requirement for quantum information theory. This allows us to define and analyze measures of Gaussian entanglement and more general quantum correlations based on such an entropy, which are shown to satisfy relevant properties such as monogamy.Comment: 6+5 pages, published in PRL. Typo in Eq. (1) correcte

Measurement-induced disturbances and nonclassical correlations of Gaussian states

We study quantum correlations beyond entanglement in two-mode Gaussian states of continuous variable systems, by means of the measurement-induced disturbance (MID) and its ameliorated version (AMID). In analogy with the recent studies of the Gaussian quantum discord, we define a Gaussian AMID by constraining the optimization to all bi-local Gaussian positive operator valued measurements. We solve the optimization explicitly for relevant families of states, including squeezed thermal states. Remarkably, we find that there is a finite subset of two-mode Gaussian states, comprising pure states, where non-Gaussian measurements such as photon counting are globally optimal for the AMID and realize a strictly smaller state disturbance compared to the best Gaussian measurements. However, for the majority of two--mode Gaussian states the unoptimized MID provides a loose overestimation of the actual content of quantum correlations, as evidenced by its comparison with Gaussian discord. This feature displays strong similarity with the case of two qubits. Upper and lower bounds for the Gaussian AMID at fixed Gaussian discord are identified. We further present a comparison between Gaussian AMID and Gaussian entanglement of formation, and classify families of two-mode states in terms of their Gaussian AMID, Gaussian discord, and Gaussian entanglement of formation. Our findings provide a further confirmation of the genuinely quantum nature of general Gaussian states, yet they reveal that non-Gaussian measurements can play a crucial role for the optimized extraction and potential exploitation of classical and nonclassical correlations in Gaussian states.Comment: 16 pages, 5 figures; new results added; to appear in Phys. Rev.

Large-Scale Distributed Bayesian Matrix Factorization using Stochastic Gradient MCMC

Despite having various attractive qualities such as high prediction accuracy and the ability to quantify uncertainty and avoid over-fitting, Bayesian Matrix Factorization has not been widely adopted because of the prohibitive cost of inference. In this paper, we propose a scalable distributed Bayesian matrix factorization algorithm using stochastic gradient MCMC. Our algorithm, based on Distributed Stochastic Gradient Langevin Dynamics, can not only match the prediction accuracy of standard MCMC methods like Gibbs sampling, but at the same time is as fast and simple as stochastic gradient descent. In our experiments, we show that our algorithm can achieve the same level of prediction accuracy as Gibbs sampling an order of magnitude faster. We also show that our method reduces the prediction error as fast as distributed stochastic gradient descent, achieving a 4.1% improvement in RMSE for the Netflix dataset and an 1.8% for the Yahoo music dataset

Measuring bipartite quantum correlations of an unknown state

We report the experimental measurement of bipartite quantum correlations of an unknown two-qubit state. Using a liquid state Nuclear Magnetic Resonance setup and employing geometric discord, we evaluate the quantum correlations of a state without resorting to prior knowledge of its density matrix. The method is applicable to any 2⊠- d system and provides, in terms of number of measurements required, an advantage over full state tomography scaling with the dimension d of the unmeasured subsystem. The negativity of quantumness is measured as well for reference. We also observe the phenomenon of sudden transition of quantum correlations when local phase and amplitude damping channels are applied to the state. © 2013 American Physical Society