DMS-EPSRC: Certifying Accuracy of Randomized Algorithms in Numerical Linear Algebra

在过去的二十年中，最令人兴奋的发展之一是随机数值线性代数（RNLA）的快速发展。这导致了计算科学的范式转变，并使矩阵计算达到了前所未有的规模。除了速度之外，RNLA通常还提供具有出色准确性和鲁棒性的解决方案。RNLA的成功案例包括低秩近似、最小二乘问题和迹估计。此外，该领域在线性系统、本征值问题和张量近似方面也取得了新的进展。尽管有这些成功的故事，但理论和实践之间往往存在巨大差距。为了说明，考虑“子空间草图”的任务，这是RNLA中的一个关键思想。有理论支持基于FFT的结构化嵌入（又名“快速约翰逊-林登施特劳斯变换”）提供几何保持子空间嵌入，如果O（dlogd）样本用于d维子空间。然而，这通常是一个悲观的估计，并且O（d），比如2d个样本，在实践中出现的大多数问题中已经足够了。类似的情况也适用于稀疏的草图，对于稀疏的草图，漂亮的理论是可用的，但在实践中，它们通常在特定的实例中表现得更好。即使是确定性算法也是如此，比如列枢轴QR分解，特别是它是否给出了一个显示秩的QR分解。虽然最坏情况分析表明该算法可能是指数不稳定的，但数十年的实际计算表明，这样的例子非常罕见。显然，在任何这些问题中，这将是非常有用的，有一种方法来评估的正确性的解决方案计算的特定problem hand.The目标的建议是开发技术计算上界的错误由一个特定的实例的随机算法解决线性代数问题。这样的界限将允许用户将随机算法的计算速度与经典确定性方法的可靠性联合收割机结合起来。它还将帮助用户安全地平衡速度和精度的需求。我们在这里提出的方法在精神上接近“负责任的鲁莽”算法的概念，最近由最新的图灵奖获得者Jack Dongarra创造。这个想法是尝试一种快速但可能不稳定的算法，以获得潜在的解决方案。然后我们以可靠的方式评估解决方案的质量。具体而言，该提案的目的是为该评估开发快速和鲁棒的算法，从而证明解决方案的正确性，或者以其他方式输出解决方案可能不准确的警告。我们将设计算法，开发理论，并在为现代计算平台优化的开源软件中实现它们。有限元方法的历史发展将有助于我们的项目。先验分析是第一位的，并且证明了在指导有限元空间的设计中非常宝贵的误差界限。然而，这些界限过于悲观，无法有效地评估任何特定实例化中产生的错误，因为界限必须覆盖最具对抗性的输入。随后出现了后验分析，因为它专门适用于手头的问题，并且可以通过利用计算后可用的数量来提供准确的估计或误差范围。在这个项目中，我们的目标是在RNLA上下文中实现同样的目标。我们将追求的具体方向包括：子空间嵌入，低秩近似，列枢轴QR分解，线性系统，特征值问题和SVD，最小二乘问题，以及有效地构建Krylov子空间。

Among the most exciting developments over the last two decades is the rapid advances in randomized numerical linear algebra (RNLA). This has caused a paradigmatic shift in the computational sciences, and has enabled matrix computations of unprecedented scale. In addition to speed, RNLA often provides solutions with exceptional accuracy and robustness. Success stories in RNLA include low-rank approximation, least-squares problems, and trace estimation. In addition, the field has witnessed recent progress in linear systems, eigenvalue problems, and tensor approximation. Despite these success stories, there is often a wide gap between theory and practice. To illustrate, consider the task of "subspace sketching", which is a key idea in RNLA. There is theory to support a claim that structured embeddings (aka "fast Johnson-Lindenstrauss transforms") based on FFTs provide a geometry preserving subspace embedding if O(dlogd) samples are used for a d-dimensional subspace. However, this is usually a pessimistic estimate, and O(d), say 2d samples, is known to suffice in most problems that arise in practice. A similar story holds for sparse sketches, for which beautiful theory is available, but in practice they often perform much better for a particular instance. The same is true even of deterministic algorithms, such as the column-pivoted QR factorization, specifically whether it gives a rank-revealing QR factorization. While the worst-case analysis suggests the algorithm can be exponentially unstable, decades of practical computation has shown that such examples are vanishingly rare. Clearly, in any of these problems, it would be highly useful to have a methodology to assess the correctness of a solution computed for the particular problem at hand.The objective of this proposal is to develop techniques for computing upper bounds on the error incurred by a particular instantiation of a randomized algorithm for solving a linear algebraic problem. Such bounds would allow users to combine the computational speed of randomized algorithms with the reliability of classical deterministic methods. It would also help users in safely balancing the needs of speed and precision. The approach we propose to follow here is close in spirit to the notion of "responsibly reckless" algorithms, recently coined by Jack Dongarra, the latest Turing Award laureate. The idea is to try a fast, but potentially unstable, algorithm, to obtain a potential solution. Then we assess the quality of the solution in a reliable fashion. Specifically, the purpose of this proposal is to develop fast and robust algorithms for this assessment, thereby certifying the correctness of the solution, or otherwise output a warning that the solution may be inaccurate. We will design algorithms, develop theory, and implement them in open-source software optimised for modern computing platforms.The historical development of finite element methods would help put our project in context. A priori analysis came first, and error bounds that were invaluable in guiding the design of finite element spaces were proven. However, these bounds were too pessimistic to usefully assess the error incurred in any particular instantiation, since the bound had to cover the most adversarial input. A posteriori analysis then emerged in response, as it applies specifically to a problem at hand and can, by drawing on quantities available post-computation, provide accurate estimates or bounds on the error. In this project we aim to achieve the same in the RNLA context.Specific directions we will pursue include: Subspace embedding, low-rank approximation, column-pivoted QR factorization, linear systems, eigenvalue problems and SVD, least-squares problems, and building Krylov subspaces efficiently.