展开迭代Lanczos算法用于理想低通图滤波器逼近
Unrolling Lanczos for Ideal Low-pass Graph Filter Approximation
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- York University（约克大学）
- Tsinghua University（清华大学）
- Manulife（宏利金融）
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中文总结 AI 辅助
本文提出展开Lanczos网络，通过算法展开和数据驱动参数学习直接逼近理想低通滤波的低频特征子空间，放宽正交性约束并保证数值稳定性，实验证明其优于经典Lanczos和Chebyshev方法。
中文摘要 AI 辅助
低通（LP）滤波是图信号处理（GSP）中的一项基本操作。在有限阶节点域方法中，基于Lanczos的滤波比Chebyshev多项式方法能更精确地逼近理想LP滤波器。我们表明，通过算法展开和数据驱动的参数学习，可以进一步改进Lanczos滤波的逼近效果。关键洞察在于，由于理想LP滤波是向低频特征子空间$\cS_K$的投影操作，而非像经典Lanczos那样逼近图拉普拉斯算子$Ł$的各个特征对，展开的Lanczos网络可以直接逼近$\cS_K$。具体来说，我们首先建立一个定理，识别Lanczos三对角矩阵$\T_m$的性质，这些性质有助于准确逼近低频特征子空间$\cS_K$。在此理论的指导下，我们放宽了Lanczos向量的正交性约束，从而得到能更好地张成$\cS_K$的Ritz向量。为确保数值稳定性，我们约束$\T_m$与对称矩阵相似，从而保证实特征值。实验结果表明，与经典Lanczos和Chebyshev方法相比，我们的展开Lanczos网络在理想LP滤波器逼近方面取得了更优的性能。
英文摘要
Low-pass (LP) filtering is a fundamental operation in graph signal processing (GSP). Among finite-order nodal-domain methods, Lanczos-based filtering provides more accurate approximations of ideal LP filters than Chebyshev polynomial methods. We show that the approximation of Lanczos filtering can be further improved through algorithm unrolling and data-driven parameter learning. The key insight is that, because ideal LP filtering is a projection operation into the low-frequency eigen-subspace $\cS_K$, instead of approximating individual eigen-pairs of a graph Laplacian $Ł$ as done in classical Lanczos, an unrolled Lanczos network can directly approximate $\cS_K$. Specifically, we first establish a theorem identifying properties of the Lanczos tridiagonal matrix $\T_m$ that promote accurate approximation of the low-frequency eigen-subspace $\cS_K$. Guided by this theory, we relax the orthogonality constraint on Lanczos vectors, resulting in Ritz vectors that better span $\cS_K$. To ensure numerical stability, we constrain $\T_m$ to be similar to a symmetric matrix, thereby guaranteeing real-valued eigenvalues. Experimental results on random graphs and learned graphs in two natural language processing (NLP) tasks show that our unrolled Lanczos network achieves superior ideal LP filter approximation compared to the classical Lanczos method.
