We present a notion of bilinear stability, which is to numerical stability what bilinear complexity is to time complexity. In bilinear complexity, an algorithm for evaluating a bilinear operator \(\beta : {\mathbb {U}} \times {\mathbb {V}} \rightarrow {\mathbb {W}}\) is a decomposition \(\beta = \varphi _1 \otimes \psi _1 \otimes w_1 + \dots + \varphi _r \otimes \psi _r \otimes w_r \); the number of terms r captures the speed of the algorithm; and its smallest possible value, i.e., the tensor rank of \(\beta \), quantifies the speed of a fastest algorithm. Bilinear stability introduces norms to the mix: The growth factor of the algorithm \(\Vert \varphi _1 \Vert _* \Vert \psi _1 \Vert _* \Vert w_1 \Vert + \cdots + \Vert \varphi _r \Vert _* \Vert \psi _r \Vert _* \Vert w_r \Vert \) captures the accuracy of the algorithm; and its smallest possible value, i.e., the tensor nuclear norm of \(\beta \), quantifies the accuracy of a stablest algorithm. To substantiate this notion, we establish a bound for the forward error in terms of the growth factor and present numerical evidence comparing various fast algorithms for matrix and complex multiplications, showing that larger growth factors correlate with less accurate results. Compared to similar studies of numerical stability, bilinear stability is more general, applying to any bilinear operators and not just matrix or complex multiplications; is more simplistic, bounding forward error in terms of a single (growth) factor; and is truly tensorial like bilinear complexity, invariant under any orthogonal change of coordinates. As an aside, we study a new algorithm for computing complex multiplication in terms of real, much like Gauss’s, but is optimally fast and stable in that it attains both tensor rank and nuclear norm.

我们提出了双线性稳定性的概念，双线性稳定性之于数值稳定性就像双线性复杂度之于时间复杂度。在双线性复杂度中，评估双线性算子\(\beta : {\mathbb {U}} \times {\mathbb {V}} \rightarrow {\mathbb {W}}\) 的算法是分解 \( \ beta = \varphi _1 \otimes \psi _1 \otimes w_1 + \dots + \varphi _r \otimes \psi _r \otimes w_r \) ; 项数r反映了算法的速度；其最小可能值，即\(\beta \)的张量秩，量化了最快算法的速度。双线性稳定性引入了范数：算法的增长因子\(\Vert \varphi _1 \Vert _* \Vert \psi _1 \Vert _* \Vert w_1 \Vert + \cdots + \Vert \varphi _r \Vert _* \Vert \psi _r \Vert _* \Vert w_r \Vert \)捕获算法的准确性；其最小可能值，即\(\beta \)的张量核范数，量化了最稳定算法的准确性。为了证实这一概念，我们根据增长因子建立了前向误差的界限，并提供了比较矩阵和复杂乘法的各种快速算法的数值证据，表明较大的增长因子与不太准确的结果相关。与数值稳定性的类似研究相比，双线性稳定性更通用，适用于任何双线性算子，而不仅仅是矩阵或复数乘法；更简单，用单个（增长）因素来限制前向误差；并且是真正的张量，就像双线性复杂性一样，在任何坐标正交变化下都保持不变。顺便说一句，我们研究了一种新的算法，用于计算实数的复数乘法，很像高斯的算法，但速度最快且稳定，因为它同时获得了张量秩和核范数。