Linear model trees are regression trees that incorporate linear models in the leaf nodes. This preserves the intuitive interpretation of decision trees and at the same time enables them to better capture linear relationships, which is hard for standard decision trees. But most existing methods for fitting linear model trees are time consuming and therefore not scalable to large data sets. In addition, they are more prone to overfitting and extrapolation issues than standard regression trees. In this paper we introduce PILOT, a new algorithm for linear model trees that is fast, regularized, stable and interpretable. PILOT trains in a greedy fashion like classic regression trees, but incorporates an $L^2$ boosting approach and a model selection rule for fitting linear models in the nodes. The abbreviation PILOT stands for $PI$ecewise $L$inear $O$rganic $T$ree, where `organic' refers to the fact that no pruning is carried out. PILOT has the same low time and space complexity as CART without its pruning. An empirical study indicates that PILOT tends to outperform standard decision trees and other linear model trees on a variety of data sets. Moreover, we prove its consistency in an additive model setting under weak assumptions. When the data is generated by a linear model, the convergence rate is polynomial.

We aim at improving reasoning on inconsistent and uncertain data. We focus on knowledge-graph data, extended with time intervals to specify their validity, as regularly found in historical sciences. We propose principles on semantics for efficient Maximum A-Posteriori inference on the new Temporal Markov Logic Networks (TMLN) which extend the Markov Logic Networks (MLN) by uncertain temporal facts and rules. We examine total and partial temporal (in)consistency relations between sets of temporal formulae. Then we propose a new Temporal Parametric Semantics, which may combine several sub-functions, allowing to use different assessment strategies. Finally, we expose the constraints that semantics must respect to satisfy our principles.

Conductance-based graph clustering has been recognized as a fundamental operator in numerous graph analysis applications. Despite the significant success of conductance-based graph clustering, existing algorithms are either hard to obtain satisfactory clustering qualities, or have high time and space complexity to achieve provable clustering qualities. To overcome these limitations, we devise a powerful \textit{peeling}-based graph clustering framework \textit{PCon}. We show that many existing solutions can be reduced to our framework. Namely, they first define a score function for each vertex, then iteratively remove the vertex with the smallest score. Finally, they output the result with the smallest conductance during the peeling process. Based on our framework, we propose two novel algorithms \textit{PCon\_core} and \emph{PCon\_de} with linear time and space complexity, which can efficiently and effectively identify clusters from massive graphs with more than a few billion edges. Surprisingly, we prove that \emph{PCon\_de} can identify clusters with near-constant approximation ratio, resulting in an important theoretical improvement over the well-known quadratic Cheeger bound. Empirical results on real-life and synthetic datasets show that our algorithms can achieve 5$\sim$42 times speedup with a high clustering accuracy, while using 1.4$\sim$7.8 times less memory than the baseline algorithms.