Fast Tree Inference With Weighted Fusion Penalties

Given a dataset with many features observed in a large number of conditions, it is desirable to fuse and aggregate conditions that are similar to ease the interpretation and extract the main characteristics of the data. This article presents a multidimensional fusion penalty framework to address this question when the number of conditions are large. If the fusion penalty is encoded by an ℓ_q-norm, we prove for uniform weights that the path of solutions is a tree that is suitable for interpretability. For the ℓ₁ and ℓ_∞-norms, the path is piecewise linear and we derive a homotopy algorithm to recover exactly the whole tree structure. For weighted ℓ₁-fusion penalties, we demonstrate that distance-decreasing weights lead to balanced tree structures. For a subclass of these weights that we call “exponentially adaptive,” we derive an O(n log (n)) homotopy algorithm and we prove an asymptotic oracle property. This guarantees that we recover the underlying structure of the data efficiently both from a statistical and a computational point of view. We provide a fast implementation of the homotopy algorithm for the single feature case, as well as an efficient embedded cross-validation procedure that takes advantage of the tree structure of the path of solutions. Our proposal outperforms its competing procedures on simulations both in terms of timings and prediction accuracy. As an example we consider phenotypic data: given one or several traits, we reconstruct a balanced tree structure and assess its agreement with the known taxonomy. Supplementary materials for this article are available online.