It is desirable for statistical models to detect signals of interest independently of their position. If the data is generated by some smooth process, this additional structure should be taken into account. We introduce a new class of neural networks that are shift invariant and preserve smoothness of the data: functional neural networks (FNNs). For this, we use methods from functional data analysis (FDA) to extend multi-layer perceptrons and convolutional neural networks to functional data. We propose different model architectures, show that the models outperform a benchmark model from FDA in terms of accuracy and successfully use FNNs to classify electroencephalography (EEG) data.
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Preference-based optimization algorithms are iterative procedures that seek the optimal calibration of a decision vector based only on comparisons between couples of different tunings. At each iteration, a human decision-maker expresses a preference between two calibrations (samples), highlighting which one, if any, is better than the other. The optimization procedure must use the observed preferences to find the tuning of the decision vector that is most preferred by the decision-maker, while also minimizing the number of comparisons. In this work, we formulate the preference-based optimization problem from a utility theory perspective. Then, we propose GLISp-r, an extension of a recent preference-based optimization procedure called GLISp. The latter uses a Radial Basis Function surrogate to describe the tastes of the decision-maker. Iteratively, GLISp proposes new samples to compare with the best calibration available by trading off exploitation of the surrogate model and exploration of the decision space. In GLISp-r, we propose a different criterion to use when looking for new candidate samples that is inspired by MSRS, a popular procedure in the black-box optimization framework. Compared to GLISp, GLISp-r is less likely to get stuck on local optima of the preference-based optimization problem. We motivate this claim theoretically, with a proof of global convergence, and empirically, by comparing the performances of GLISp and GLISp-r on several benchmark optimization problems.
Branching process inspired models are widely used to estimate the effective reproduction number -- a useful summary statistic describing an infectious disease outbreak -- using counts of new cases. Case data is a real-time indicator of changes in the reproduction number, but is challenging to work with because cases fluctuate due to factors unrelated to the number of new infections. We develop a new model that incorporates the number of diagnostic tests as a surveillance model covariate. Using simulated data and data from the SARS-CoV-2 pandemic in California, we demonstrate that incorporating tests leads to improved performance over the state-of-the-art.
Class imbalance (CI) in classification problems arises when the number of observations belonging to one class is lower than the other. Ensemble learning combines multiple models to obtain a robust model and has been prominently used with data augmentation methods to address class imbalance problems. In the last decade, a number of strategies have been added to enhance ensemble learning and data augmentation methods, along with new methods such as generative adversarial networks (GANs). A combination of these has been applied in many studies, but the true rank of different combinations would require a computational review. In this paper, we present a computational review to evaluate data augmentation and ensemble learning methods used to address prominent benchmark CI problems. We present a general framework that evaluates 10 data augmentation and 10 ensemble learning methods for CI problems. Our objective is to identify the most effective combination for improving classification performance on imbalanced datasets. The results indicate that combinations of data augmentation methods with ensemble learning can significantly improve classification performance on imbalanced datasets. Our study is vital for the development of novel models for handling imbalanced datasets.
We propose a method to construct a joint statistical model for mixed-domain data to analyze their dependence. Multivariate Gaussian and log-linear models are particular examples of the proposed model. It is shown that the functional equation defining the model has a unique solution under fairly weak conditions. The model is characterized by two orthogonal parameters: the dependence parameter and the marginal parameter. To estimate the dependence parameter, a conditional inference together with a sampling procedure is proposed and is shown to provide a consistent estimator. Illustrative examples of data analyses involving penguins and earthquakes are presented.
We introduce a flexible method to simultaneously infer both the drift and volatility functions of a discretely observed scalar diffusion. We introduce spline bases to represent these functions and develop a Markov chain Monte Carlo algorithm to infer, a posteriori, the coefficients of these functions in the spline basis. A key innovation is that we use spline bases to model transformed versions of the drift and volatility functions rather than the functions themselves. The output of the algorithm is a posterior sample of plausible drift and volatility functions that are not constrained to any particular parametric family. The flexibility of this approach provides practitioners a powerful investigative tool, allowing them to posit a variety of parametric models to better capture the underlying dynamics of their processes of interest. We illustrate the versatility of our method by applying it to challenging datasets from finance, paleoclimatology, and astrophysics. In view of the parametric diffusion models widely employed in the literature for those examples, some of our results are surprising since they call into question some aspects of these models.
Bayesian optimization (BO), while proved highly effective for many black-box function optimization tasks, requires practitioners to carefully select priors that well model their functions of interest. Rather than specifying by hand, researchers have investigated transfer learning based methods to automatically learn the priors, e.g. multi-task BO (Swersky et al., 2013), few-shot BO (Wistuba and Grabocka, 2021) and HyperBO (Wang et al., 2022). However, those prior learning methods typically assume that the input domains are the same for all tasks, weakening their ability to use observations on functions with different domains or generalize the learned priors to BO on different search spaces. In this work, we present HyperBO+: a pre-training approach for hierarchical Gaussian processes that enables the same prior to work universally for Bayesian optimization on functions with different domains. We propose a two-step pre-training method and analyze its appealing asymptotic properties and benefits to BO both theoretically and empirically. On real-world hyperparameter tuning tasks that involve multiple search spaces, we demonstrate that HyperBO+ is able to generalize to unseen search spaces and achieves lower regrets than competitive baselines.
Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.
Langevin dynamics are widely used in sampling high-dimensional, non-Gaussian distributions whose densities are known up to a normalizing constant. In particular, there is strong interest in unadjusted Langevin algorithms (ULA), which directly discretize Langevin dynamics to estimate expectations over the target distribution. We study the use of transport maps that approximately normalize a target distribution as a way to precondition and accelerate the convergence of Langevin dynamics. We show that in continuous time, when a transport map is applied to Langevin dynamics, the result is a Riemannian manifold Langevin dynamics (RMLD) with metric defined by the transport map. We also show that applying a transport map to an irreversibly-perturbed ULA results in a geometry-informed irreversible perturbation (GiIrr) of the original dynamics. These connections suggest more systematic ways of learning metrics and perturbations, and also yield alternative discretizations of the RMLD described by the map, which we study. Under appropriate conditions, these discretized processes can be endowed with non-asymptotic bounds describing convergence to the target distribution in 2-Wasserstein distance. Illustrative numerical results complement our theoretical claims.
The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations but not for nonlinear problems. We further show how to derive \emph{a priori} bounds on the size of the region of analyticity. This method is applied to the trypsin molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is sufficiently general enough to be applied to other nonlinear problems in uncertainty quantification.
The accurate representation of precipitation in Earth system models (ESMs) is crucial for reliable projections of the ecological and socioeconomic impacts in response to anthropogenic global warming. The complex cross-scale interactions of processes that produce precipitation are challenging to model, however, inducing potentially strong biases in ESM fields, especially regarding extremes. State-of-the-art bias correction methods only address errors in the simulated frequency distributions locally at every individual grid cell. Improving unrealistic spatial patterns of the ESM output, which would require spatial context, has not been possible so far. Here, we show that a post-processing method based on physically constrained generative adversarial networks (cGANs) can correct biases of a state-of-the-art, CMIP6-class ESM both in local frequency distributions and in the spatial patterns at once. While our method improves local frequency distributions equally well as gold-standard bias-adjustment frameworks, it strongly outperforms any existing methods in the correction of spatial patterns, especially in terms of the characteristic spatial intermittency of precipitation extremes.
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