# Subspace Clustering via Optimal Direction Search

## TL;DR

This paper introduces a spectral clustering method for subspace clustering that uses convex optimization to find optimal directions, improving robustness to noise and closely spaced subspaces, especially in face clustering tasks.

## Contribution

The paper proposes a novel convex program for optimal direction search within spectral clustering, enhancing performance in noisy and complex subspace scenarios.

## Key findings

- Outperforms existing subspace clustering methods in noisy environments
- Achieves state-of-the-art results in face clustering tasks
- Efficiently identifies neighborhood sets for data points

## Abstract

This letter presents a new spectral-clustering-based approach to the subspace clustering problem. Underpinning the proposed method is a convex program for optimal direction search, which for each data point d finds an optimal direction in the span of the data that has minimum projection on the other data points and non-vanishing projection on d. The obtained directions are subsequently leveraged to identify a neighborhood set for each data point. An alternating direction method of multipliers framework is provided to efficiently solve for the optimal directions. The proposed method is shown to notably outperform the existing subspace clustering methods, particularly for unwieldy scenarios involving high levels of noise and close subspaces, and yields the state-of-the-art results for the problem of face clustering using subspace segmentation.