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Spectral clustering and its extensions usually consist of two steps: (1)constructing a graph and computing the relaxed solution; (2) discretizingrelaxed solutions. Although the former has been extensively investigated, thediscretization techniques are mainly heuristic methods, e.g., k-means, spectralrotation. Unfortunately, the goal of the existing methods is not to find adiscrete solution that minimizes the original objective. In other words, theprimary drawback is the neglect of the original objective when computing thediscrete solution. Inspired by the first-order optimization algorithms, wepropose to develop a first-order term to bridge the original problem anddiscretization algorithm, which is the first non-heuristic to the best of ourknowledge. Since the non-heuristic method is aware of the original graph cutproblem, the final discrete solution is more reliable and achieves thepreferable loss value. We also theoretically show that the continuous optimumis beneficial to discretization algorithms though simply finding its closestdiscrete solution is an existing heuristic algorithm which is also unreliable.Sufficient experiments significantly show the superiority of our method.

Discretize Relaxed Solution of Spectral Clustering via a Non-Heuristic Algorithm