arXiv subjects

Andrea Zanoni

Publications and source records attributed to Andrea Zanoni.

24 records · Page 2Linked to original sources

arXiv2022

Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions

We propose a novel method for drift estimation of multiscale diffusion processes when a sequence of discrete observations is given. For the Langevin dynamics in a two-scale potential, our approach relies on the eigenvalues and the eigenfunctions of the homogenized dynamics. Our first estimator is derived from a martingale estimating function of the generator of the homogenized diffusion process. However, the unbiasedness of the estimator depends on the rate with which the observations are sampled. We therefore introduce a second estimator which relies also on filtering the data and we prove that it is asymptotically unbiased independently of the sampling rate. A series of numerical experiments illustrate the reliability and efficiency of our different estimators.

math.NA↗

arXiv2022

Robust Estimation of Effective Diffusions from Multiscale Data

We present a novel methodology based on filtered data and moving averages for estimating effective dynamics from observations of multiscale systems. We show in a semi-parametric framework of the Langevin type that our approach is asymptotically unbiased with respect to the theory of homogenization. Moreover, we demonstrate on a range of challenging numerical experiments that our method is accurate in extracting coarse-grained dynamics from multiscale data. In particular, the estimators we propose are more robust and require less knowledge of the full model than the standard technique of subsampling, which is widely employed in practice in this setting.

math.NA↗

arXiv2021

Drift Estimation of Multiscale Diffusions Based on Filtered Data

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

math.NA↗

arXiv2020

Minimax Classification with 0-1 Loss and Performance Guarantees

Supervised classification techniques use training samples to find classification rules with small expected 0-1 loss. Conventional methods achieve efficient learning and out-of-sample generalization by minimizing surrogate losses over specific families of rules. This paper presents minimax risk classifiers (MRCs) that do not rely on a choice of surrogate loss and family of rules. MRCs achieve efficient learning and out-of-sample generalization by minimizing worst-case expected 0-1 loss w.r.t. uncertainty sets that are defined by linear constraints and include the true underlying distribution. In addition, MRCs' learning stage provides performance guarantees as lower and upper tight bounds for expected 0-1 loss. We also present MRCs' finite-sample generalization bounds in terms of training size and smallest minimax risk, and show their competitive classification performance w.r.t. state-of-the-art techniques using benchmark datasets.

stat.ML↗

arXiv2020

Ensemble Kalman filter for multiscale inverse problems

We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analysed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is provided, and convergence of the approximate conditional posterior distribution is proved with respect to the Wasserstein distance. A numerical experiment validates our methodology, with a particular emphasis on modelling error and computational cost.

math.NA↗

arXiv2019

Supervised classification via minimax probabilistic transformations

Conventional techniques for supervised classification constrain the classification rules considered and use surrogate losses for classification 0-1 loss. Favored families of classification rules are those that enjoy parametric representations suitable for surrogate loss minimization, and low complexity properties suitable for overfitting control. This paper presents classification techniques based on robust risk minimization (RRM) that we call linear probabilistic classifiers (LPCs). The proposed techniques consider unconstrained classification rules, optimize the classification 0-1 loss, and provide performance bounds during learning. LPCs enable efficient learning by using linear optimization, and avoid overffiting by using RRM over polyhedral uncertainty sets of distributions. We also provide finite-sample generalization bounds for LPCs and show their competitive performance with state-of-the-art techniques using benchmark datasets.

stat.ML↗