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Knowing the actual precipitation in space and time is critical in hydrological modelling applications, yet the spatial coverage with rain gauge stations is limited due to economic constraints. Gridded satellite precipitation datasets offer an alternative option for estimating the actual precipitation by covering uniformly large areas, albeit related estimates are not accurate. To improve precipitation estimates, machine learning is applied to merge rain gauge-based measurements and gridded satellite precipitation products. In this context, observed precipitation plays the role of the dependent variable, while satellite data play the role of predictor variables. Random forests is the dominant machine learning algorithm in relevant applications. In those spatial predictions settings, point predictions (mostly the mean or the median of the conditional distribution) of the dependent variable are issued. The aim of the manuscript is to solve the problem of probabilistic prediction of precipitation with an emphasis on extreme quantiles in spatial interpolation settings. Here we propose, issuing probabilistic spatial predictions of precipitation using Light Gradient Boosting Machine (LightGBM). LightGBM is a boosting algorithm, highlighted by prize-winning entries in prediction and forecasting competitions. To assess LightGBM, we contribute a large-scale application that includes merging daily precipitation measurements in contiguous US with PERSIANN and GPM-IMERG satellite precipitation data. We focus on extreme quantiles of the probability distribution of the dependent variable, where LightGBM outperforms quantile regression forests (QRF, a variant of random forests) in terms of quantile score at extreme quantiles. Our study offers understanding of probabilistic predictions in spatial settings using machine learning.

Algorithms for solving the linear classification problem have a long history, dating back at least to 1936 with linear discriminant analysis. For linearly separable data, many algorithms can obtain the exact solution to the corresponding 0-1 loss classification problem efficiently, but for data which is not linearly separable, it has been shown that this problem, in full generality, is NP-hard. Alternative approaches all involve approximations of some kind, including the use of surrogates for the 0-1 loss (for example, the hinge or logistic loss) or approximate combinatorial search, none of which can be guaranteed to solve the problem exactly. Finding efficient algorithms to obtain an exact i.e. globally optimal solution for the 0-1 loss linear classification problem with fixed dimension, remains an open problem. In research we report here, we detail the rigorous construction of a new algorithm, incremental cell enumeration (ICE), that can solve the 0-1 loss classification problem exactly in polynomial time. We prove correctness using concepts from the theory of hyperplane arrangements and oriented matroids. We demonstrate the effectiveness of this algorithm on synthetic and real-world datasets, showing optimal accuracy both in and out-of-sample, in practical computational time. We also empirically demonstrate how the use of approximate upper bound leads to polynomial time run-time improvements to the algorithm whilst retaining exactness. To our knowledge, this is the first, rigorously-proven polynomial time, practical algorithm for this long-standing problem.

In this paper, we investigate the impact of compression on stochastic gradient algorithms for machine learning, a technique widely used in distributed and federated learning. We underline differences in terms of convergence rates between several unbiased compression operators, that all satisfy the same condition on their variance, thus going beyond the classical worst-case analysis. To do so, we focus on the case of least-squares regression (LSR) and analyze a general stochastic approximation algorithm for minimizing quadratic functions relying on a random field. We consider weak assumptions on the random field, tailored to the analysis (specifically, expected H\"older regularity), and on the noise covariance, enabling the analysis of various randomizing mechanisms, including compression. We then extend our results to the case of federated learning. More formally, we highlight the impact on the convergence of the covariance $\mathfrak{C}_{\mathrm{ania}}$ of the additive noise induced by the algorithm. We demonstrate despite the non-regularity of the stochastic field, that the limit variance term scales with $\mathrm{Tr}(\mathfrak{C}_{\mathrm{ania}} H^{-1})/K$ (where $H$ is the Hessian of the optimization problem and $K$ the number of iterations) generalizing the rate for the vanilla LSR case where it is $\sigma^2 \mathrm{Tr}(H H^{-1}) / K = \sigma^2 d / K$ (Bach and Moulines, 2013). Then, we analyze the dependency of $\mathfrak{C}_{\mathrm{ania}}$ on the compression strategy and ultimately its impact on convergence, first in the centralized case, then in two heterogeneous FL frameworks.

We present Surjective Sequential Neural Likelihood (SSNL) estimation, a novel method for simulation-based inference in models where the evaluation of the likelihood function is not tractable and only a simulator that can generate synthetic data is available. SSNL fits a dimensionality-reducing surjective normalizing flow model and uses it as a surrogate likelihood function which allows for conventional Bayesian inference using either Markov chain Monte Carlo methods or variational inference. By embedding the data in a low-dimensional space, SSNL solves several issues previous likelihood-based methods had when applied to high-dimensional data sets that, for instance, contain non-informative data dimensions or lie along a lower-dimensional manifold. We evaluate SSNL on a wide variety of experiments and show that it generally outperforms contemporary methods used in simulation-based inference, for instance, on a challenging real-world example from astrophysics which models the magnetic field strength of the sun using a solar dynamo model.

In this paper, we make the first attempt to apply the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. BINNs combine artificial neural networks with the well-established boundary integral equations (BIEs) to effectively solve partial differential equations (PDEs). The BIEs are utilized to map all the unknowns onto the boundary, after which these unknowns are approximated using artificial neural networks and resolved via a training process. In contrast to traditional neural network-based methods, the current BINNs offer several distinct advantages. First, by embedding BIEs into the learning procedure, BINNs only need to discretize the boundary of the solution domain, which can lead to a faster and more stable learning process (only the boundary conditions need to be fitted during the training). Second, the differential operator with respect to the PDEs is substituted by an integral operator, which effectively eliminates the need for additional differentiation of the neural networks (high-order derivatives of neural networks may lead to instability in learning). Third, the loss function of the BINNs only contains the residuals of the BIEs, as all the boundary conditions have been inherently incorporated within the formulation. Therefore, there is no necessity for employing any weighing functions, which are commonly used in traditional methods to balance the gradients among different objective functions. Moreover, BINNs possess the ability to tackle PDEs in unbounded domains since the integral representation remains valid for both bounded and unbounded domains. Extensive numerical experiments show that BINNs are much easier to train and usually give more accurate learning solutions as compared to traditional neural network-based methods.

Nonlinear extensions to the active subspaces method have brought remarkable results for dimension reduction in the parameter space and response surface design. We further develop a kernel-based nonlinear method. In particular we introduce it in a broader mathematical framework that contemplates also the reduction in parameter space of multivariate objective functions. The implementation is thoroughly discussed and tested on more challenging benchmarks than the ones already present in the literature, for which dimension reduction with active subspaces produces already good results. Finally, we show a whole pipeline for the design of response surfaces with the new methodology in the context of a parametric CFD application solved with the Discontinuous Galerkin method.

In this paper, we develop a unified regression approach to model unconditional quantiles, M-quantiles and expectiles of multivariate dependent variables exploiting the multidimensional Huber's function. To assess the impact of changes in the covariates across the entire unconditional distribution of the responses, we extend the work of Firpo et al. (2009) by running a mean regression of the recentered influence function on the explanatory variables. We discuss the estimation procedure and establish the asymptotic properties of the derived estimators. A data-driven procedure is also presented to select the tuning constant of the Huber's function. The validity of the proposed methodology is explored with simulation studies and through an application using the Survey of Household Income and Wealth 2016 conducted by the Bank of Italy.

The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.

We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.