In the main tables of the paper, we did not include the performances of α-TIM in the standard balanced setting. Here, we emphasize that α-TIM is a generalization of TIM [1] as when α 1 (i.e., the α-entropies tend to the Shannon entropies), α-TIM tends to TIM. Therefore, in the standard setting, where optimal hyper-parameter αis obtained over validation tasks that are balanced (as in the standard validation tasks of the original TIM and the other existing methods), the performance of α-TIM is the same as TIM. When αis tuned on balanced validation tasks, we obtain an optimal value of αvery close to 1, and our α-mutual information approaches the standard mutual information. When the validation tasks are uniformly random, as in our new setting and in the validation plots we provided in the main figure, one can see that the performance of α-TIM remains competitive when we tend to balanced testing tasks (i.e., when a is increasing), but is significantly better than TIM when we tend to uniformly-random testing tasks (a = 1).

Hierarchical clustering studies a recursive partition of a data set into clusters of successively smaller size, and is a fundamental problem in data analysis. In this work we study the cost function for hierarchical clustering introduced by Dasgupta [12], and present two polynomial-time approximation algorithms: Our first result is an O(1)-approximation algorithm for graphs of high conductance. Our simple construction bypasses complicated recursive routines of finding sparse cuts known in the literature (e.g., [6, 11]). Our second and main result is an O(1)approximation algorithm for a wide family of graphs that exhibit a well-defined structure of clusters. This result generalises the previous state-of-the-art [10], which holds only for graphs generated from stochastic models. The significance of our work is demonstrated by the empirical analysis on both synthetic and real-world data sets, on which our presented algorithm outperforms the previously proposed algorithm for graphs with a well-defined cluster structure [10].

Feature Selection and Functional Data Analysis are two dynamic areas of research, with important applications in the analysis of large and complex data sets. Straddling these two areas, we propose a new highly efficient algorithm to perform Group Elastic Net with application to function-on-scalar feature selection, where a functional response is modeled against a very large number of potential scalar predictors. First, we introduce a new algorithm to solve Group Elastic Net in ultrahigh dimensional settings, which exploits the sparsity structure of the Augmented Lagrangian to greatly reduce computational burden. Next, taking advantage of the properties of Functional Principal Components, we extend our algorithm to the function-on-scalar regression framework. We use simulations to demonstrate the CPU time gains afforded by our approach compared to its best existing competitors, and present an application to data from a Genome Wide Association Study on childhood obesity.

Providing a very low critical probability pc means that certification occurs when the simulation ends after a large number of iterations m. We introduce `c the threshold associated to pc s.t. Table 5 shows that this approximation is excellent even for large pc. This shows that mis a little larger than mc = log(pc)/log(1 1/N). This section assumes that X = xo + σ X with X N(0n; In) and that h(x) = x>g τ with g Rn and kgk= 1 (w.l.o.g.).

Modeling directed graphs with differentiable representations is a fundamental requirement for performing machine learning on graph-structured data.