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Explainability of decisions made by AI systems is driven by both recent regulation and user demand. These decisions are often explainable only \emph{post hoc}, after the fact. In counterfactual explanations, one may ask what constitutes the best counterfactual explanation. Clearly, multiple criteria must be taken into account, although "distance from the sample" is a key criterion. Recent methods that consider the plausibility of a counterfactual seem to sacrifice this original objective. Here, we present a system that provides high-likelihood explanations that are, at the same time, close and sparse. We show that the search for the most likely explanations satisfying many common desiderata for counterfactual explanations can be modeled using mixed-integer optimization (MIO). In the process, we propose an MIO formulation of a Sum-Product Network (SPN) and use the SPN to estimate the likelihood of a counterfactual, which can be of independent interest., Comment: In review

Tame functions are a class of nonsmooth, nonconvex functions, which feature in a wide range of applications: functions encountered in the training of deep neural networks with all common activations, value functions of mixed-integer programs, or wave functions of small molecules. We consider approximating tame functions with piecewise polynomial functions. We bound the quality of approximation of a tame function by a piecewise polynomial function with a given number of segments on any full-dimensional cube. We also present the first mixed-integer programming formulation of piecewise polynomial regression. Together, these can be used to estimate tame functions. We demonstrate promising computational results.

In classification and forecasting with tabular data, one often utilizes tree-based models. Those can be competitive with deep neural networks on tabular data and, under some conditions, explainable. The explainability depends on the depth of the tree and the accuracy in each leaf of the tree. We point out that decision trees containing leaves with unbalanced accuracy can provide misleading explanations. Low-accuracy leaves give less valid explanations, which could be interpreted as unfairness among subgroups utilizing these explanations. Here, we train a shallow tree with the objective of minimizing the maximum misclassification error across all leaf nodes. The shallow tree provides a global explanation, while the overall statistical performance of the shallow tree can become comparable to state-of-the-art methods (e.g., well-tuned XGBoost) by extending the leaves with further models.

The purpose of this paper is to propose a continuum micromechanics model for the simulation of uniaxial compressive and tensile tests on lime-based mortars, in order to predict their stiffness, compressive and tensile strengths, and tensile fracture energy. In tension, we adopt an incremental strain-controlled form of the Mori-Tanaka scheme with a damageable matrix phase, while a simple $J_2$ yield criterion is employed in compression. To reproduce the behavior of lime-based mortars correctly, the scheme must take into account shrinkage cracking among aggregates. This phenomenon is introduced into the model via penny-shaped cracks, whose density is estimated on the basis of a particle size distribution combined with the results of finite element analyses of a single crack formation between two spherical inclusions. Our predictions show a good agreement with experimental data and explain the advantages of compliant crushed brick fragments, often encountered in ancient mortars, over stiff sand particles. The validated model provides a reliable tool for optimizing the composition of modern lime-based mortars with applications in conservation and restoration of architectural heritage., Comment: 21 pages, 12 figures, 1 table