Median K-flats for hybrid linear modeling with many outliers (0909.3123v1)

Abstract: We describe the Median K-Flats (MKF) algorithm, a simple online method for hybrid linear modeling, i.e., for approximating data by a mixture of flats. This algorithm simultaneously partitions the data into clusters while finding their corresponding best approximating l1 d-flats, so that the cumulative l1 error is minimized. The current implementation restricts d-flats to be d-dimensional linear subspaces. It requires a negligible amount of storage, and its complexity, when modeling data consisting of N points in D-dimensional Euclidean space with K d-dimensional linear subspaces, is of order O(n K d D+n d^2 D), where n is the number of iterations required for convergence (empirically on the order of 10^4). Since it is an online algorithm, data can be supplied to it incrementally and it can incrementally produce the corresponding output. The performance of the algorithm is carefully evaluated using synthetic and real data.

An Overview of the Median $K$-Flats Algorithm for Hybrid Linear Modeling with Outliers

The paper "Median $K$-Flats for Hybrid Linear Modeling with Many Outliers" introduces a novel algorithm called Median $K$-Flats (MKF), targeted at hybrid linear modeling wherein data is approximated by a mixture of linear subspaces or "flats." The MKF algorithm emphasizes robust handling of datasets afflicted with substantial outliers, advancing beyond traditional methods such as the $K$-Flats algorithm, especially in scenarios with high-dimensional data or pronounced noise.

Key Methodological Innovations

MKF distinguishes itself primarily by employing an $\ell_1$ norm for minimizing cumulative error, diverging from the conventional $\ell_2$ norm used in the $K$-Flats method. This choice reflects a strategic pivot aiming to bolster the robustness of data clustering against outliers. Data partitioning and optimal subspace determination are executed concurrently, guided by the minimization of the cumulative $\ell_1$ error across identified clusters. This methodological choice theoretically dampens the sensitivity to outliers, as opposed to the inherently larger influence they cast under the $\ell_2$ criterion.

The algorithm's complexity comes in at $$O(n_s \cdot K \cdot d \cdot D+n_s \cdot d^2 \cdot D)$$, with $n_s$ representing the convergence iterations—empirically capped at around 10,000, establishing MKF as computationally feasible for practical implementation scenarios. Critically, MKF operates as an online algorithm, processing data in an incremental fashion which suits real-time data streams or datasets too voluminous for static treatment.

Empirical Evaluation and Results

The paper evaluates the MKF algorithm using both synthetic datasets and the real-world Hopkins 155 database, benchmarking its performance against other algorithms like GPCA, LSA, and MoPPCA. Across various instances of linear subspaces, the MKF algorithm demonstrates superior classification accuracy, particularly in settings fraught with high percentages of outliers or data of higher intrinsic dimensionality. On the Hopkins 155 database, even under less noisy conditions with lower dimensions, MKF competes commendably, surpassing several traditional and contemporary algorithms.

Implications and Prospects

The practical implications of MKF are notable: it offers a scalable and robust solution for clustering tasks imbued with noise and outliers—common in dynamic environments like video sequence analysis or high-throughput bioinformatics. Methodologically, it suggests that the shift to $\ell_1$ norms in objective functions might be a principle worth exploring across more dimensions of computational mathematics and algorithm design, particularly wherever outlier influence is a concern.

Looking ahead, refinements to MKF could focus on adapting the algorithm for affine subspace modeling or exploiting its potential for semi-supervised learning applications. Additionally, the foundational robustness afforded by $\ell_1$ minimization warrants broader theoretical exploration to understand its potential pitfalls and boundaries in varied non-linear or mixed-dimension scenarios.

This paper roots its contributions firmly in the concrete mathematical underpinnings of hybrid modeling, advancing a toolset that aligns closely with modern data processing needs, making it a valuable reference point for future research pursuits in algorithm design and applied data science.