展开图拉普拉斯去噪器仅能实现多项式图滤波器的组合
Unrolling a Graph-Laplacian Denoiser Realizes Only Compositions of Polynomial Graph Filters
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中文总结 AI 辅助
该研究证明，展开图拉普拉斯去噪器所得映射为多项式图滤波器组合，实际应用的阶数会约束假设空间，且其谱响应存在非零下限，所需精度阶数高于实际应用阶数。
中文摘要 AI 辅助
近期，针对基于图的图像复原的展开网络构造方法，通过截断泰勒展开从图拉普拉斯去噪器构建系统矩阵，再用固定步数的共轭梯度法对其求逆，且两个阶段的系数均为可学习参数。本文证明，对于这些系数的任意取值，乃至训练过程中的任意时刻，所得映射均为去噪算子的多项式，其次数不超过两个截断阶数的乘积；可学习的步骤仅能选择克里洛夫子空间中的元素，而无法对其进行扩展。在实际应用的阶数下，可达集合更是网络自身次数预算的多项式类中的测度零子集，因此该组合会对假设空间形成约束，而非扩展。在标准初始化下，实现的谱响应可通过闭式形式得到，其在谱的内部区间内超出预期响应，并趋近于非零下限。随后，算子的条件数下界（源于图构造而非图像内容）使得达到规定精度所需的阶数远高于实际应用的阶数。该受限类恰好是长期以来已有直接凸参数化方法的谱图滤波器。
英文摘要
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.
