Numerical stability and tensor nuclear norm

Abstract

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 $β: \mathbb{U} \times \mathbb{V} \to \mathbb{W}$ is a decomposition $β= φ_1 \otimes ψ_1 \otimes w_1 + \dots + φ_r \otimes ψ_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 $β$, quantifies the speed of a fastest algorithm. Bilinear stability introduces norms to the mix: The growth factor of the algorithm $\lVert φ_1 \rVert_* \lVert ψ_1 \rVert_* \lVert w_1 \rVert + \dots + \lVert φ_r \rVert_* \lVert ψ_r \rVert_* \lVert w_r \rVert$ captures the accuracy of the algorithm; and its smallest possible value, i.e., the tensor nuclear norm of $β$, 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.