Clustering Affine Subspaces: Algorithms and Hardness

Lee, 2012

Category: Algorithms

Overall Rating

2.0/5 (14/35 pts)

This paper presents a theoretically distinct perspective on clustering incomplete data by framing it as a geometric problem on affine subspaces, supported by novel geometric tools like a Helly-type theorem for axis-parallel flats.
However, the practical algorithms derived from this framework... face a critical limitation: exponential time complexity in the number of clusters (k) and the dimension of the flats (Delta).
This inherent computational barrier, acknowledged by the paper's own hardness results, significantly curtails its applicability to problem sizes commonly encountered in modern data analysis.
more flexible, scalable, and widely adopted methods for handling incomplete data and clustering have emerged since 2012, likely offering superior performance and practicality for contemporary research goals.

This paper approaches the problem of clustering from a geometric perspective, specifically by modeling data objects with missing features as affine subspaces ('flats').
could the geometric insights from the axis-parallel case... be leveraged to analyze the structure of high-dimensional latent spaces learned by deep networks when trained on incomplete data?
specialized geometric algorithms derived from this paper's principles could potentially be used for clustering, anomaly detection, or downstream tasks more effectively than standard techniques.
the difficulty highlighted for general flats and the need for better initial center finding (mentioned in conclusions) could inspire new iterative geometric optimization techniques or even differentiable geometric layers within deep learning architectures

The proposed PTAS for general k and Delta (axis-parallel case) has a running time dependency of 2^(O(Deltaklog k)), which is exponential in both k (number of clusters) and Delta (dimension of flats). This is a severe bottleneck, making the algorithm impractical for even moderately sized k or Delta.
The assumption that "incomplete data objects in Rd can be modeled as affine subspaces"... is a very specific geometric interpretation of missing values; real-world missingness patterns can be far more complex and potentially not align with simple linear subspaces...
Current methods for dealing with incomplete data in clustering tasks have largely moved past this specific geometric formulation.
Attempting to directly apply this framework to cutting-edge areas like large-scale AI/ML applications... would likely lead to academic dead-ends or significant inefficiencies.

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