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In PDE-constrained optimization, one aims to find design parameters that minimize some objective, subject to the satisfaction of a partial differential equation. A major challenges is computing gradients of the objective to the design parameters, as applying the chain rule requires computing the Jacobian of the design parameters to the PDE's state. The adjoint method avoids this Jacobian by computing partial derivatives of a Lagrangian. Evaluating these derivatives requires the solution of a second PDE with the adjoint differential operator to the constraint, resulting in a backwards-in-time simulation. Particle-based Monte Carlo solvers are often used to compute the solution to high-dimensional PDEs. However, such solvers have the drawback of introducing noise to the computed results, thus requiring stochastic optimization methods. To guarantee convergence in this setting, both the constraint and adjoint Monte Carlo simulations should simulate the same particle trajectories. For large simulations, storing full paths from the constraint equation for re-use in the adjoint equation becomes infeasible due to memory limitations. In this paper, we provide a reversible extension to the family of permuted congruential pseudorandom number generators (PCG). We then use such a generator to recompute these time-reversed paths for the heat equation, avoiding these memory issues.

The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. On the basis of the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information.

The proliferation of data generation has spurred advancements in functional data analysis. With the ability to analyze multiple variables simultaneously, the demand for working with multivariate functional data has increased. This study proposes a novel formulation of the epigraph and hypograph indexes, as well as their generalized expressions, specifically tailored for the multivariate functional context. These definitions take into account the interrelations between components. Furthermore, the proposed indexes are employed to cluster multivariate functional data. In the clustering process, the indexes are applied to both the data and their first and second derivatives. This generates a reduced-dimension dataset from the original multivariate functional data, enabling the application of well-established multivariate clustering techniques that have been extensively studied in the literature. This methodology has been tested through simulated and real datasets, performing comparative analyses against state-of-the-art to assess its performance.

The information diffusion prediction on social networks aims to predict future recipients of a message, with practical applications in marketing and social media. While different prediction models all claim to perform well, general frameworks for performance evaluation remain limited. Here, we aim to identify a performance characteristic curve for a model, which captures its performance on tasks of different complexity. We propose a metric based on information entropy to quantify the randomness in diffusion data, then identify a scaling pattern between the randomness and the prediction accuracy of the model. Data points in the patterns by different sequence lengths, system sizes, and randomness all collapse into a single curve, capturing a model's inherent capability of making correct predictions against increased uncertainty. Given that this curve has such important properties that it can be used to evaluate the model, we define it as the performance characteristic curve of the model. The validity of the curve is tested by three prediction models in the same family, reaching conclusions in line with existing studies. Also, the curve is successfully applied to evaluate two distinct models from the literature. Our work reveals a pattern underlying the data randomness and prediction accuracy. The performance characteristic curve provides a new way to systematically evaluate models' performance, and sheds light on future studies on other frameworks for model evaluation.

Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms. The matrix multiplication, a key component of many more advanced numerical algorithms, has so far proven tricky: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance. In this article, a new and fairly general algorithm for multiplying $\mathcal{H}^2$-matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application. The complexity and accuracy are analysed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing pseudo-Boolean functions. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three.

In this work, we consider the problem of goodness-of-fit (GoF) testing for parametric models -- for example, testing whether observed data follows a logistic regression model. This testing problem involves a composite null hypothesis, due to the unknown values of the model parameters. In some special cases, co-sufficient sampling (CSS) can remove the influence of these unknown parameters via conditioning on a sufficient statistic -- often, the maximum likelihood estimator (MLE) of the unknown parameters. However, many common parametric settings (including logistic regression) do not permit this approach, since conditioning on a sufficient statistic leads to a powerless test. The recent approximate co-sufficient sampling (aCSS) framework of Barber and Janson (2022) offers an alternative, replacing sufficiency with an approximately sufficient statistic (namely, a noisy version of the MLE). This approach recovers power in a range of settings where CSS cannot be applied, but can only be applied in settings where the unconstrained MLE is well-defined and well-behaved, which implicitly assumes a low-dimensional regime. In this work, we extend aCSS to the setting of constrained and penalized maximum likelihood estimation, so that more complex estimation problems can now be handled within the aCSS framework, including examples such as mixtures-of-Gaussians (where the unconstrained MLE is not well-defined due to degeneracy) and high-dimensional Gaussian linear models (where the MLE can perform well under regularization, such as an $\ell_1$ penalty or a shape constraint).

Given an observational study with $n$ independent but heterogeneous units, our goal is to learn the counterfactual distribution for each unit using only one $p$-dimensional sample per unit containing covariates, interventions, and outcomes. Specifically, we allow for unobserved confounding that introduces statistical biases between interventions and outcomes as well as exacerbates the heterogeneity across units. Modeling the conditional distribution of the outcomes as an exponential family, we reduce learning the unit-level counterfactual distributions to learning $n$ exponential family distributions with heterogeneous parameters and only one sample per distribution. We introduce a convex objective that pools all $n$ samples to jointly learn all $n$ parameter vectors, and provide a unit-wise mean squared error bound that scales linearly with the metric entropy of the parameter space. For example, when the parameters are $s$-sparse linear combination of $k$ known vectors, the error is $O(s\log k/p)$. En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality. As an application of the framework, our results enable consistent imputation of sparsely missing covariates.

Combining ideas coming from Stone duality and Reynolds parametricity, we formulate in a clean and principled way a notion of profinite lambda-term which, we show, generalizes at every type the traditional notion of profinite word coming from automata theory. We start by defining the Stone space of profinite lambda-terms as a projective limit of finite sets of usual lambda-terms, considered modulo a notion of equivalence based on the finite standard model. One main contribution of the paper is to establish that, somewhat surprisingly, the resulting notion of profinite lambda-term coming from Stone duality lives in perfect harmony with the principles of Reynolds parametricity. In addition, we show that the notion of profinite lambda-term is compositional by constructing a cartesian closed category of profinite lambda-terms, and we establish that the embedding from lambda-terms modulo beta-eta-conversion to profinite lambda-terms is faithful using Statman's finite completeness theorem. Finally, we prove that the traditional Church encoding of finite words into lambda-terms can be extended to profinite words, and leads to a homeomorphism between the space of profinite words and the space of profinite lambda-terms of the corresponding Church type.

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.