提出方法本质是多项式图滤波器,限制了模型表达能力。
Unrolling a Graph-Laplacian Denoiser Realizes Only Compositions of Polynomial Graph Filters
- 通过泰勒展开与共轭梯度法构造图滤波器,本质为多项式组合
- 实际可达的模型空间仅为多项式类中的极小部分
- 适合研究图神经网络表达能力与优化机制的学者
近期基于图图像恢复的展开网络将图拉普拉斯去噪器通过截断泰勒展开生成系统矩阵,并用固定步数的共轭梯度法求逆,两阶段系数均学习。本文证明该映射始终是去噪算子的多项式,次数不超过两截断阶数的乘积,且对任意系数设置和训练点均成立:学习步骤仅在无法扩展的Krylov子空间中选择元素。在实践中常用阶数下,可到达的集合是网络自身阶数预算下多项式类的测度为零子集,因此组合反而约束了假设空间而非拓展。在标准初始化下,实现的谱响应可闭式表达,在谱内部超出预期响应并趋近于非零下界。内部构造导致的算子条件数下界表明,实现指定精度所需的阶数远高于实际使用阶数。该受限类正是长期存在直接凸参数化的谱图滤波器。
原文摘要 · Abstract (English)
A recent construction of unrolled networks for graph-based image restoration forms a system matrix from a graph-Laplacian denoiser through a truncated Taylor expansion, then inverts it with a fixed number of conjugate-gradient steps, with the coefficients of both stages learned. This paper shows the resulting map is a polynomial in the denoising operator, of degree at most the product of the two truncation orders, for every setting of those coefficients and therefore at every point of training: the learned steps select an element of a Krylov subspace they cannot enlarge. At the orders used in practice the reachable set is moreover a measure-zero subset of the polynomial class of the network's own degree budget, so the composition constrains the hypothesis space rather than enlarging it. At the standard initialization the realized spectral response is obtained in closed form, exceeding the intended response throughout the interior of the spectrum and approaching a nonzero floor. A lower bound on the operator's condition number, internal to the graph construction rather than to image content, then places the order required for a prescribed accuracy well above the order used in practice. The confining class is precisely the spectral graph filters for which a direct, convex parameterization has long been available.
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