[Paper Review] Fast L1-NMF for Multiple Parametric Model Estimation

Mariano Tepper, Guillermo Sapiro|arXiv (Cornell University)|Oct 18, 2016

NMR spectroscopy and applications1 references6 citations

TL;DR

This paper proposes a parameterless, accelerated L1-NMF-based biclustering framework for multiple parametric model estimation that leverages consistent patterns from RANSAC-generated model hypotheses. By reformulating RANSAC output as a consensus matrix and applying fast L1-NMF, the method robustly identifies intersecting inlier sets and model parameters without enforcing non-intersecting models, achieving up to 60× speedup on large datasets compared to standard RANSAC.

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 extit{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.

Motivation & Objective

• To address the challenge of estimating multiple parametric models in data with noise and outliers, especially when models may intersect.
• To overcome the limitations of standard RANSAC, which is designed for single-model fitting and struggles with pseudo-outliers in multi-model scenarios.
• To develop a parameterless biclustering method based on L1-NMF that identifies consistent inlier patterns across RANSAC-generated model hypotheses.
• To accelerate L1-NMF computation for scalability to large datasets, enabling broader applicability beyond the current framework.
• To eliminate the need for hard thresholding by exploring real-valued object-model distance matrices as an alternative to binary consensus matrices.

Proposed method

• The method transforms RANSAC-generated model hypotheses into a consensus matrix where each entry indicates whether a data point belongs to a model’s consensus set.
• It applies a parameterless L1-NMF algorithm to this consensus matrix to extract biclusters representing inlier sets and their associated models.
• The L1-NMF formulation promotes sparsity and robustness, naturally handling overlapping inlier sets without enforcing model disjointness.
• An accelerated algorithm for L1-NMF is introduced, using iterative thresholding and active set strategies to improve convergence speed on medium and large problems.
• The method estimates final model parameters via least-squares regression on the recovered inlier sets from the biclustering step.
• The framework is evaluated on 2D and 3D geometric fitting tasks, including line, circle, and plane estimation, using real-world datasets like Pozzoveggiani and Piazza Bra.

Experimental results

Research questions

• RQ1 Can L1-NMF be effectively used to cluster RANSAC-generated model hypotheses into coherent inlier-model pairs without prior knowledge of the number of models?
• RQ2 How can the inherent consistency in RANSAC output be exploited to improve robustness and reduce sensitivity to pseudo-outliers?
• RQ3 To what extent can an accelerated L1-NMF algorithm scale to large datasets while maintaining accuracy in multiple model estimation?
• RQ4 Does relaxing the non-intersecting model assumption improve performance on real-world data with overlapping structures?
• RQ5 Can the hard thresholding step in consensus matrix construction be replaced with a real-valued distance matrix to eliminate a critical hyperparameter?

Key findings

• The proposed method achieves up to 60× speedup over standard RANSAC when processing the full 108,750-point Pozzoveggiani 3D dataset, with the performance gap widening as dataset size increases.
• On the Piazza Bra dataset, ARSE (accelerated RANSAC with L1-NMF) with a 82,831×24,841 preference matrix outperforms RSE on a much smaller 16,567×2,040 matrix, demonstrating sublinear scaling with matrix size.
• The method successfully recovers 3D planes in complex urban scenes, including the bell tower in Pozzoveggiani, which is only correctly estimated on the full dataset due to sufficient point density.
• The accelerated L1-NMF algorithm scales sublinearly with preference matrix size, indicating strong computational efficiency and scalability to large problems.
• The framework naturally handles intersecting models, avoiding the artificial constraints of non-overlapping inlier sets common in existing methods.
• The method maintains high accuracy across multiple subsampled versions of datasets, showing robustness to data sparsity and noise.