We propose the Lanczos network (LanczosNet), which uses the Lanczos algorithm
to construct low rank approximations of the graph Laplacian for graph
convolution. Relying on the tridiagonal decomposition of the Lanczos algorithm,
we not only efficiently exploit multi-scale information via fast approximated
computation of matrix power but also design learnable spectral filters. Being
fully differentiable, LanczosNet facilitates both graph kernel learning as well
as learning node embeddings. We show the connection between our LanczosNet and
graph based manifold learning methods, especially the diffusion maps. We
benchmark our model against several recent deep graph networks on citation
networks and QM8 quantum chemistry dataset. Experimental results show that our
model achieves the state-of-the-art performance in most tasks. Code is released
at: https://github.com/lrjconan/LanczosNetwork.

本研究提出 Lanczos 网络（LanczosNet），其使用 Lanczos 算法构建图 Laplacian 的低秩逼近，以用于图卷积，通过利用 Lanczos 算法的三对角分解，我们不仅通过矩阵幂的快速逼近计算高效地利用多尺度信息，还设计出可学习的谱滤波器。LanczosNet 既有助于图内核学习，又有助于学习节点嵌入，并且完全可微分，本文将我们的 LanczosNet 与基于图的流形学习方法，特别是扩散映射进行对比。我们在引用网络和 QM8 量子化学数据集上对我们的模型进行基准测试，实验结果表明，我们的模型在大多数任务中均达到了最先进的性能。

LanczosNet: 多尺度深度图卷积网络

LanczosNet: Multi-Scale Deep Graph Convolutional Networks

One of the most compelling features of Gaussian process (GP) regression is
its ability to provide well-calibrated posterior distributions. Recent advances
in inducing point methods have sped up GP marginal likelihood and posterior
mean computations, leaving posterior covariance estimation and sampling as the
remaining computational bottlenecks. In this paper we address these
shortcomings by using the Lanczos algorithm to rapidly approximate the
predictive covariance matrix. Our approach, which we refer to as LOVE (LanczOs
Variance Estimates), substantially improves time and space complexity. In our
experiments, LOVE computes covariances up to 2,000 times faster and draws
samples 18,000 times faster than existing methods, all without sacrificing
accuracy.

本文提出一种基于 Lanczos 算法的方法 LOVE（LanczOs Variance Estimates）来解决高斯过程回归的计算瓶颈，大大提高了协方差的计算效率和采样速度。

高斯过程的恒定时间预测分布

Constant-Time Predictive Distributions for Gaussian Processes

We show that the standard Lanczos algorithm can be efficiently implemented
statistically and self consistently improved, using the stochastic reconfigurat
ion method, which has been recently introduced to stabilize the Monte Carlo
sign problem instability. With this scheme a few Lanczos steps over a given
variational wavefunction are possible even for large size as a particular case
of a more general and more accurate technique that allows to obtain lower
variational energies. This method has been tested extensively for a strongly
correlated model like the t-J model. With the standard Lanczos technique it is
possible to compute any kind of correlation functions, with no particular
computational effort. By using that the variance $<H^2>-<H>^2$ is zero for an
exact eigenstate, we show that the approach to the exact solution with few
Lanczos ite rations is indeed possible even for $\sim 100$ electrons for
reasonably good initial wavefunctions.The variational stochastic
reconfiguration technique presented here allows in general a many-parameter
energy optimization of any computable many-body wavefunction, including for
inst ance generic long range Jastrow factors and arbitrary site dependent
orbital determinants. This scheme improves further the accuracy of the
calculation, especially for long distance correlation functions.

通过随机重构方法，可以实现标准 Lanczos 算法的稳定和自我修正，进而在强相关模型中进行计算，优化多体波函数的能量。