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

翻译：谱聚类及其扩展方法通常包含两个步骤：(1) 构建图并计算松弛解；(2) 对松弛解进行离散化。尽管第一步已被广泛研究，但离散化技术主要采用启发式方法，例如k-means、谱旋转。遗憾的是，现有方法的目标并非寻找最小化原始目标函数的离散解。换言之，其主要缺陷在于计算离散解时忽视了原始目标函数。受一阶优化算法的启发，我们提出构建一阶项来桥接原始问题与离散化算法——据我们所知，这是首个非启发式方法。由于该非启发式方法感知原始图割问题，最终得到的离散解更可靠且能获得更优的损失值。我们还从理论上证明，连续最优解对离散化算法具有重要价值——尽管仅寻找其最近离散解的现有启发式算法并不可靠。充分的实验显著证明了我们方法的优越性。