Fast L1-NMF for Multiple Parametric Model Estimation

Abstract

In this work we introduce a comprehensive algorithmic pipeline for multiple parametric model estimation. The proposed approach analyzes the information produced by a random sampling algorithm (e.g., RANSAC) from a machine learning/optimization perspective, using a parameterless biclustering algorithm based on L1 nonnegative matrix factorization (L1-NMF). The proposed framework exploits consistent patterns that naturally arise during the RANSAC execution, while explicitly avoiding spurious inconsistencies. Contrarily to the main trends in the literature, the proposed technique does not impose non-intersecting parametric models. A new accelerated algorithm to compute L1-NMFs allows to handle medium-sized problems faster while also extending the usability of the algorithm to much larger datasets. This accelerated algorithm has applications in any other context where an L1-NMF is needed, beyond the biclustering approach to parameter estimation here addressed. We accompany the algorithmic presentation with theoretical foundations and numerous and diverse examples.

Fitting Multiple Heterogeneous Models by Multi-Class Cascaded T-Linkage

“…Outlier rejection. Following the route of the a-contrario approaches, outlying models are pruned using the statistical validation technique described in [23], the main difference is that we use it as a post-processing outlier rejection criteria on the attained models rather than as a pre-processing refinement of sampled model hypotheses. The idea is to compute the distribution of the cardinality of the consensus set of a model varying its inlier threshold and to express its unlikeliness in terms of NFA.…”

A Goodness-of-fit Test on the Number of Biclusters in a Relational Data Matrix

“…Information criterion on the number of biclusters K. Aside from the statistical-test-based methods, some studies have proposed to determine K based on the minimum description length [44,50,57] and modified DIC for the biclustering problem [11,12]. Particularly, under the regular grid constraint of the bicluster structure, an information criterion called integrated completed likelihood (ICL) has been proposed for determining K [15,33,56], which approximates the maximum marginal likelihood of a given K. These methods aim to select the optimal number of biclusters K from a given set of candidates, in terms of some criterion (e.g., marginal likelihood).…”