Obtaining sparse, interpretable representations of observable data is crucial
in many machine learning and signal processing tasks. For data representing
flows along the edges of a graph, an intuitively interpretable way to obtain
such representations is to lift the graph structure to a simplicial complex:
The eigenvectors of the associated Hodge-Laplacian, respectively the incidence
matrices of the corresponding simplicial complex then induce a Hodge
decomposition, which can be used to represent the observed data in terms of
gradient, curl, and harmonic flows. In this paper, we generalize this approach
to cellular complexes and introduce the cell inference optimization problem,
i.e., the problem of augmenting the observed graph by a set of cells, such that
the eigenvectors of the associated Hodge Laplacian provide a sparse,
interpretable representation of the observed edge flows on the graph. We show
that this problem is NP-hard and introduce an efficient approximation algorithm
for its solution. Experiments on real-world and synthetic data demonstrate that
our algorithm outperforms current state-of-the-art methods while being
computationally efficient.

本文将利用相关的 Hodge 拉普拉斯矩阵的特征向量以及对应的单纯复形的关联矩阵进行 Hodge 分解，为观测数据提供梯度、旋度和谐波流形式的稀疏、可解释表示，从而解决了在图的边缘流中获得稀疏、可解释表示的问题，并通过引入一种高效的近似算法来解决本文介绍的细胞推断优化问题。实验结果表明，该算法在真实数据和合成数据上表现优于当前最先进的方法，并且计算效率高。

通过稀疏胞复合物在图上表示边流

Representing Edge Flows on Graphs via Sparse Cell Complexes

Interactive visualizations are crucial in ad hoc data exploration and
analysis. However, with the growing number of massive datasets, generating
visualizations in interactive timescales is increasingly challenging. One
approach for improving the speed of the visualization tool is via data
reduction in order to reduce the computational overhead, but at a potential
cost in visualization accuracy. Common data reduction techniques, such as
uniform and stratified sampling, do not exploit the fact that the sampled
tuples will be transformed into a visualization for human consumption.
We propose a visualization-aware sampling (VAS) that guarantees high quality
visualizations with a small subset of the entire dataset. We validate our
method when applied to scatter and map plots for three common visualization
goals: regression, density estimation, and clustering. The key to our sampling
method's success is in choosing tuples which minimize a visualization-inspired
loss function. Our user study confirms that optimizing this loss function
correlates strongly with user success in using the resulting visualizations. We
also show the NP-hardness of our optimization problem and propose an efficient
approximation algorithm. Our experiments show that, compared to previous
methods, (i) using the same sample size, VAS improves user's success by up to
35% in various visualization tasks, and (ii) VAS can achieve a required
visualization quality up to 400 times faster.

提出了可视化感知采样（VAS）方法，通过优化可视化启发式损失函数，从而在保证可视化质量的基础上，实现对大规模数据集的快速采样。实验表明，VAS 相比现有的方法具有更高的成功率和更快的速度，尤其在回归、密度估计和聚类等场景下。