Moritz Poguntke, Eda Oktay, Yuanzhe Xi, Erin Carson, Martin Stoll
原文摘要
Training Gaussian processes (GPs) at large scale is computationally challenging because evaluating the log-marginal likelihood and its gradients involves large, dense kernel matrices. We develop a scalable low-rank GP training approach that combines AnchorNet landmark selection with a fully independent training conditional (FITC) correction. The resulting objective and its analytic gradients can be evaluated using low-rank factors, diagonal quantities, and small dense matrices, without forming the full kernel matrix or its derivatives. The complete training procedure is implemented in single precision and evaluated on nine large-scale regression datasets. Across these problems, the method reaches the reference predictive accuracy while substantially reducing training time, although the required approximation rank varies across datasets. We also study the numerical behavior of the full-kernel predictive solve and show that single precision can become limiting for ill-conditioned problems. Finally, we investigate further reducing selected matrix operations to fp16 or TF32. These lower-precision variants provide only modest additional speedups and can become unreliable when the reduced system is poorly conditioned.
In this paper, we propose a fast method for estimating the condition number of sparse matrices using graph neural networks (GNNs). For efficient deployment of GNNs, we introduce a graph feature construction with $\mathrm{O}(\mathrm{nnz} + n)$ complexity, where $\mathrm{nnz}$ is the number of non-zero elements in the matrix and $n$ denotes the matrix dimension. We propose two schemes for estimating the matrix condition number using GNNs; one follows by decomposing the condition number and predicts the relatively more computationally intensive part $\|\mathbf{A}^{-1}\|$, without explicitly forming the inverse, while the other is to predict the whole condition number $κ$. Our approach can be extended to an arbitrary norm. Extensive experiments are conducted for the estimation of the 1-norm and 2-norm condition numbers, which show that our method achieves a significant speedup over the traditional numerical estimation methods. Our software for GNN condition number estimator is made publicly available at https://github.com/inEXASCALE/sparse-kappa.
We present an adaptive bitwidth allocation method for quantizing low-rank factors in hierarchical matrices in order to decrease storage while meeting a given constraint on their reconstruction error. For the bitwidth allocation we formulate a discrete storage minimization problem and show that an optimal assignment can give larger singular values at least as many bits as smaller ones. We further develop a greedy algorithm that meets the specified max-norm quantization error budget. We investigate a hierarchical block partition in which all of the non-diagonal blocks are represented by quantized low-rank factors and only dense blocks remain dense; in this way a global reconstruction bound related to $η_k$ and $R_k$ is obtained, with the former and latter being bounds on the truncation and reconstruction arithmetic errors, respectively. Extensive simulations, carried out with kernel and BEM matrices under block-relative budgets, indicate that the proposed method achieves a reduction in storage of up to $44.3\times$ when compared to storing the complete matrix densely in FP64, and a reduction of up to $5.70\times$ when compared to the FP64 low-rank factors within the same partition, together with measured reconstruction errors that remained below the maximum norm bounds in all the hierarchical tests.