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Given a data set with many features observed in a large number of conditions,it is desirable to fuse and aggregate conditions which are similar to ease theinterpretation and extract the main characteristics of the data. This paperpresents a multidimensional fusion penalty framework to address this questionwhen the number of conditions is large. If the fusion penalty is encoded by an$\ell_q$-norm, we prove for uniform weights that the path of solutions is atree which is suitable for interpretability. For the $\ell_1$ and$\ell_\infty$-norms, the path is piecewise linear and we derive a homotopyalgorithm to recover exactly the whole tree structure. For weighted$\ell_1$-fusion penalties, we demonstrate that distance-decreasing weights leadto balanced tree structures. For a subclass of these weights that we call"exponentially adaptive", we derive an $\mathcal{O}(n\log(n))$ homotopyalgorithm and we prove an asymptotic oracle property. This guarantees that werecover the underlying structure of the data efficiently both from astatistical and a computational point of view. We provide a fast implementationof the homotopy algorithm for the single feature case, as well as an efficientembedded cross-validation procedure that takes advantage of the tree structureof the path of solutions. Our proposal outperforms its competing procedures onsimulations both in terms of timings and prediction accuracy. As an example weconsider phenotypic data: given one or several traits, we reconstruct abalanced tree structure and assess its agreement with the known taxonomy.

Fast tree inference with weighted fusion penalties