To capture the extremal behaviour of complex environmental phenomena in practice, flexible techniques for modelling tail behaviour are required. In this paper, we introduce a variety of such methods, which were used by the Lancopula Utopiversity team to tackle the data challenge of the 2023 Extreme Value Analysis Conference. This data challenge was split into four sections, labelled C1-C4. Challenges C1 and C2 comprise univariate problems, where the goal is to estimate extreme quantiles for a non-stationary time series exhibiting several complex features. We propose a flexible modelling technique, based on generalised additive models, with diagnostics indicating generally good performance for the observed data. Challenges C3 and C4 concern multivariate problems where the focus is on estimating joint extremal probabilities. For challenge C3, we propose an extension of available models in the multivariate literature and use this framework to estimate extreme probabilities in the presence of non-stationary dependence. Finally, for challenge C4, which concerns a 50 dimensional random vector, we employ a clustering technique to achieve dimension reduction and use a conditional modelling approach to estimate extremal probabilities across independent groups of variables.
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Spatially distributed functional data are prevalent in many statistical applications such as meteorology, energy forecasting, census data, disease mapping, and neurological studies. Given their complex and high-dimensional nature, functional data often require dimension reduction methods to extract meaningful information. Inverse regression is one such approach that has become very popular in the past two decades. We study the inverse regression in the framework of functional data observed at irregularly positioned spatial sites. The functional predictor is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect, while the relation between the scalar response and the functional predictor is modeled using the inverse regression framework. For estimation, we consider local linear smoothing with a general weighting scheme, which includes as special cases the schemes under which equal weights are assigned to each observation or to each subject. This framework enables us to present the asymptotic results for different types of sampling plans over time such as non-dense, dense, and ultra-dense. We discuss the domain-expanding infill (DEI) framework for spatial asymptotics, which is a mix of the traditional expanding domain and infill frameworks. The DEI framework overcomes the limitations of traditional spatial asymptotics in the existing literature. Under this unified framework, we develop asymptotic theory and identify conditions that are necessary for the estimated eigen-directions to achieve optimal rates of convergence. Our asymptotic results include pointwise and $L_2$ convergence rates. Simulation studies using synthetic data and an application to a real-world dataset confirm the effectiveness of our methods.
We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.
One of the challenges of natural language understanding is to deal with the subjectivity of sentences, which may express opinions and emotions that add layers of complexity and nuance. Sentiment analysis is a field that aims to extract and analyze these subjective elements from text, and it can be applied at different levels of granularity, such as document, paragraph, sentence, or aspect. Aspect-based sentiment analysis is a well-studied topic with many available data sets and models. However, there is no clear definition of what makes a sentence difficult for aspect-based sentiment analysis. In this paper, we explore this question by conducting an experiment with three data sets: "Laptops", "Restaurants", and "MTSC" (Multi-Target-dependent Sentiment Classification), and a merged version of these three datasets. We study the impact of domain diversity and syntactic diversity on difficulty. We use a combination of classifiers to identify the most difficult sentences and analyze their characteristics. We employ two ways of defining sentence difficulty. The first one is binary and labels a sentence as difficult if the classifiers fail to correctly predict the sentiment polarity. The second one is a six-level scale based on how many of the top five best-performing classifiers can correctly predict the sentiment polarity. We also define 9 linguistic features that, combined, aim at estimating the difficulty at sentence level.
We consider the problem of optimising the expected value of a loss functional over a nonlinear model class of functions, assuming that we have only access to realisations of the gradient of the loss. This is a classical task in statistics, machine learning and physics-informed machine learning. A straightforward solution is to replace the exact objective with a Monte Carlo estimate before employing standard first-order methods like gradient descent, which yields the classical stochastic gradient descent method. But replacing the true objective with an estimate ensues a ``generalisation error''. Rigorous bounds for this error typically require strong compactness and Lipschitz continuity assumptions while providing a very slow decay with sample size. We propose a different optimisation strategy relying on a natural gradient descent in which the true gradient is approximated in local linearisations of the model class via (quasi-)projections based on optimal sampling methods. Under classical assumptions on the loss and the nonlinear model class, we prove that this scheme converges almost surely monotonically to a stationary point of the true objective and we provide convergence rates.
In a world of increasing closed-source commercial machine learning models, model evaluations from developers must be taken at face value. These benchmark results, whether over task accuracy, bias evaluations, or safety checks, are traditionally impossible to verify by a model end-user without the costly or impossible process of re-performing the benchmark on black-box model outputs. This work presents a method of verifiable model evaluation using model inference through zkSNARKs. The resulting zero-knowledge computational proofs of model outputs over datasets can be packaged into verifiable evaluation attestations showing that models with fixed private weights achieve stated performance or fairness metrics over public inputs. These verifiable attestations can be performed on any standard neural network model with varying compute requirements. For the first time, we demonstrate this across a sample of real-world models and highlight key challenges and design solutions. This presents a new transparency paradigm in the verifiable evaluation of private models.
Support Vector Machines (SVMs) are an important tool for performing classification on scattered data, where one usually has to deal with many data points in high-dimensional spaces. We propose solving SVMs in primal form using feature maps based on trigonometric functions or wavelets. In small dimensional settings the Fast Fourier Transform (FFT) and related methods are a powerful tool in order to deal with the considered basis functions. For growing dimensions the classical FFT-based methods become inefficient due to the curse of dimensionality. Therefore, we restrict ourselves to multivariate basis functions, each one of them depends only on a small number of dimensions. This is motivated by the well-known sparsity of effects and recent results regarding the reconstruction of functions from scattered data in terms of truncated analysis of variance (ANOVA) decomposition, which makes the resulting model even interpretable in terms of importance of the features as well as their couplings. The usage of small superposition dimensions has the consequence that the computational effort no longer grows exponentially but only polynomially with respect to the dimension. In order to enforce sparsity regarding the basis coefficients, we use the frequently applied $\ell_2$-norm and, in addition, $\ell_1$-norm regularization. The found classifying function, which is the linear combination of basis functions, and its variance can then be analyzed in terms of the classical ANOVA decomposition of functions. Based on numerical examples we show that we are able to recover the signum of a function that perfectly fits our model assumptions. We obtain better results with $\ell_1$-norm regularization, both in terms of accuracy and clarity of interpretability.
Bayesian model averaging is a practical method for dealing with uncertainty due to model specification. Use of this technique requires the estimation of model probability weights. In this work, we revisit the derivation of estimators for these model weights. Use of the Kullback-Leibler divergence as a starting point leads naturally to a number of alternative information criteria suitable for Bayesian model weight estimation. We explore three such criteria, known to the statistics literature before, in detail: a Bayesian analogue of the Akaike information criterion which we call the BAIC, the Bayesian predictive information criterion (BPIC), and the posterior predictive information criterion (PPIC). We compare the use of these information criteria in numerical analysis problems common in lattice field theory calculations. We find that the PPIC has the most appealing theoretical properties and can give the best performance in terms of model-averaging uncertainty, particularly in the presence of noisy data, while the BAIC is a simple and reliable alternative.
When training predictive models on data with missing entries, the most widely used and versatile approach is a pipeline technique where we first impute missing entries and then compute predictions. In this paper, we view prediction with missing data as a two-stage adaptive optimization problem and propose a new class of models, adaptive linear regression models, where the regression coefficients adapt to the set of observed features. We show that some adaptive linear regression models are equivalent to learning an imputation rule and a downstream linear regression model simultaneously instead of sequentially. We leverage this joint-impute-then-regress interpretation to generalize our framework to non-linear models. In settings where data is strongly not missing at random, our methods achieve a 2-10% improvement in out-of-sample accuracy.
A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallelism and differences between the weak and the strong cases. Then, we consider the special case of guillotine rectangulations, and prove that they can be characterized - under all known mappings between permutations and rectangulations - by avoidance of two mesh patterns that correspond to "windmills" in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations.
The recent proliferation of knowledge graphs (KGs) coupled with incomplete or partial information, in the form of missing relations (links) between entities, has fueled a lot of research on knowledge base completion (also known as relation prediction). Several recent works suggest that convolutional neural network (CNN) based models generate richer and more expressive feature embeddings and hence also perform well on relation prediction. However, we observe that these KG embeddings treat triples independently and thus fail to cover the complex and hidden information that is inherently implicit in the local neighborhood surrounding a triple. To this effect, our paper proposes a novel attention based feature embedding that captures both entity and relation features in any given entity's neighborhood. Additionally, we also encapsulate relation clusters and multihop relations in our model. Our empirical study offers insights into the efficacy of our attention based model and we show marked performance gains in comparison to state of the art methods on all datasets.
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