Fast Tree Inference With Weighted Fusion Penalties. Issue 1 (2nd January 2017)

• Record Type: Journal Article
• Authors: Chiquet, Julien; Gutierrez, Pierre; Rigaill, Guillem
• Is Part Of: Journal of computational and graphical statistics. Volume 26:Issue 1(2017)
• Page Start: 205
• Page End: 216
• Publication Date: 2017-01-02
• Subjects: Exponentially adaptive weights -- Fusion penalties -- Homotopy algorithm -- Tree reconstruction; Mathematical statistics -- Data processing -- Periodicals; Mathematical statistics -- Graphic methods -- Periodicals; 519.50285
• DOI: 10.1080/10618600.2015.1096789
• Languages: English
• ISSNs: 1061-8600

Abstract

ABSTRACT: 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 anO ( 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.