The development of simple and fast hypergraph spectral methods has been hindered by the lack of numerical algorithms for simulating heat diffusions and computing fundamental objects, such as Personalized PageRank vectors, over hypergraphs. In this paper, we overcome this challenge by designing two novel algorithmic primitives. The first is a simple, easy-to-compute discrete-time heat diffusion that enjoys the same favorable properties as the discrete-time heat diffusion over graphs. This diffusion can be directly applied to speed up existing hypergraph partitioning algorithms. Our second contribution is the novel application of mirror descent to compute resolvents of non-differentiable squared norms, which we believe to be of independent interest beyond hypergraph problems. Based on this new primitive, we derive the first nearly-linear-time algorithm that simulates the discrete-time heat diffusion to approximately compute resolvents of the hypergraph Laplacian operator, which include Personalized PageRank vectors and solutions to the hypergraph analogue of Laplacian systems. Our algorithm runs in time that is linear in the size of the hypergraph and inversely proportional to the hypergraph spectral gap $\lambda_G$, matching the complexity of analogous diffusion-based algorithms for the graph version of the problem.

翻译：简单快速的超图谱方法的开发受到缺乏数值算法的阻碍，这些算法用于模拟热扩散和计算基础对象（如超图上的个性化PageRank向量）。在本文中，我们通过设计两种新颖的算法原语克服了这一挑战。第一种是一种简单、易于计算的离散时间热扩散方法，它享有与图上的离散时间热扩散相同的优良性质。这种扩散方法可直接应用于加速现有的超图分割算法。我们的第二个贡献是新颖地应用镜像下降法来计算不可微平方范数的预解式，我们认为这一方法在超图问题之外也具有独立的研究兴趣。基于这一新原语，我们推导出首个近似线性时间算法，该算法模拟离散时间热扩散来近似计算超图拉普拉斯算子的预解式，其中包括个性化PageRank向量和拉普拉斯系统超图类比问题的解。我们的算法运行时间与超图规模呈线性关系，且与超图谱间隙$\lambda_G$成反比，这与图版本问题中模拟扩散算法的复杂度相匹配。