Xiaobo Liu

Andrii Dmytryshyn, Massimiliano Fasi, Nicholas J. Higham et al.

We consider the solution of the Sylvester equation $AX+XB=C$ in mixed precision. We derive a new iterative refinement scheme to solve perturbed quasi-triangular Sylvester equations; our rounding error analysis provides sufficient conditions for convergence and a bound on the attainable relative residual. We leverage this iterative scheme to solve the general Sylvester equation. The new algorithms compute the Schur decomposition of the coefficient matrices $A$ and $B$ in lower than working precision, use the low-precision Schur factors to obtain an approximate solution to the perturbed quasi-triangular equation, and iteratively refine it to obtain a working-precision solution. In order to solve the original equation to working precision, the unitary Schur factors of the coefficient matrices must be unitary to working precision, but this is not the case if the Schur decomposition is computed in low precision. We propose two effective approaches to address this: one is based on re-orthonormalization in working precision, and the other on explicit inversion of the almost-unitary factors. The two mixed-precision algorithms thus obtained are tested on various Sylvester and Lyapunov equations from the literature. Our numerical experiments show that, for both types of equations, the new algorithms are at least as accurate as existing ones. Our cost analysis, on the other hand, suggests that they would typically be faster than mono-precision alternatives if implemented on hardware that natively supports low precision.

Pascal den Boef, Patrick Kürschner, Xiaobo Liu et al.

Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a fixed-point process for which, generally, the mapping function changes in each iteration. To enable acceleration of the iterative solution for these problems, we propose two novel generalizations of RRE. First, we show how to effectively compute RRE for sequences of low-rank matrices. Second, we derive a formulation of RRE that is suitable for fixed-point processes for which the mapping function changes each iteration. We demonstrate the potential of the methods on several numerical examples involving the iterative solution of large-scale Lyapunov and Riccati matrix equations.

Peter Benner, Xiaobo Liu

We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations $AX + XA^T + W =0$, where $W=LL^T$ or $W=LSL^T$. Via rounding error analysis of the algorithms we derive sufficient conditions for the attainable normwise residuals in different precision settings and show how the algorithmic parameters should be chosen. These conditions are independent of the choice of inner solver, provided that the prescribed residual accuracy is attained in the inner solves. Using the sign-function Newton iteration as the solver, we demonstrate that reduced precisions, such as half precision with unit roundoff $u_s$, can be used efficiently for Lyapunov equations with condition numbers of order $1/u_s$ without compromising the attainable solution quality. This provides an algorithmic framework towards exploiting native low-precision hardware to accelerate Lyapunov solvers without sacrificing accuracy.

Kun Qiu, Harry Chang, Ying Wang et al.

Sophisticated traffic analytics, such as the encrypted traffic analytics and unknown malware detection, emphasizes the need for advanced methods to analyze the network traffic. Traditional methods of using fixed patterns, signature matching, and rules to detect known patterns in network traffic are being replaced with AI (Artificial Intelligence) driven algorithms. However, the absence of a high-performance AI networking-specific framework makes deploying real-time AI-based processing within networking workloads impossible. In this paper, we describe the design of Traffic Analytics Development Kits (TADK), an industry-standard framework specific for AI-based networking workloads processing. TADK can provide real-time AI-based networking workload processing in networking equipment from the data center out to the edge without the need for specialized hardware (e.g., GPUs, Neural Processing Unit, and so on). We have deployed TADK in commodity WAF and 5G UPF, and the evaluation result shows that TADK can achieve a throughput up to 35.3Gbps per core on traffic feature extraction, 6.5Gbps per core on traffic classification, and can decrease SQLi/XSS detection down to 4.5us per request with higher accuracy than fixed pattern solution.

Yaoming Cai, Zijia Zhang, Zhihua Cai et al.

This paper presents FLGC, a simple yet effective fully linear graph convolutional network for semi-supervised and unsupervised learning. Instead of using gradient descent, we train FLGC based on computing a global optimal closed-form solution with a decoupled procedure, resulting in a generalized linear framework and making it easier to implement, train, and apply. We show that (1) FLGC is powerful to deal with both graph-structured data and regular data, (2) training graph convolutional models with closed-form solutions improve computational efficiency without degrading performance, and (3) FLGC acts as a natural generalization of classic linear models in the non-Euclidean domain, e.g., ridge regression and subspace clustering. Furthermore, we implement a semi-supervised FLGC and an unsupervised FLGC by introducing an initial residual strategy, enabling FLGC to aggregate long-range neighborhoods and alleviate over-smoothing. We compare our semi-supervised and unsupervised FLGCs against many state-of-the-art methods on a variety of classification and clustering benchmarks, demonstrating that the proposed FLGC models consistently outperform previous methods in terms of accuracy, robustness, and learning efficiency. The core code of our FLGC is released at https://github.com/AngryCai/FLGC.