Asymptotic spectra: Theory, applications, and extensions

In 1969, Strassen shocked the computational world with his subcubic algorithm for multiplying matrices. Attempting to understand the best possible algorithm for this problem, Strassen went on to develop his magnificent theory of <italic>asymptotic spectra</italic> in three papers between 1986–1991. Expressed in the great generality of partially ordered semirings, the centerpiece of this theory is a <italic>duality</italic> theorem between the asymptotic “rank” of elements and a topological space which is called asymptotic spectrum. This duality theorem is a vast generalization of linear programming duality (in which we have a semigroup rather than a semiring), and indeed also of certain versions of the Positivstellensatz, the duality theorem of polynomial inequalities over the reals. Focusing on understanding the structure of the asymptotic spectrum of matrix multiplication, the theory has provided surprising <italic>connectivity</italic> and <italic>convexity</italic> theorems for it. Strassen’s theory has led to many subsequent results, especially new algorithmic, structural, and barrier results on matrix multiplication, and, more generally, for the semiring of tensors (which includes the matrix multiplication tensors). Perhaps even more impressively, the generality of Strassen’s theory has been applied recently to the study of a variety of very different settings and parameters, in diverse fields including communication theory, graph theory, probability theory, quantum information theory and computational complexity. We feel that these developments call for an exposition of this growing field. This paper gives a comprehensive, self-contained, modern survey of Strassen’s theory of asymptotic spectra and its various old and new application areas. For accessibility we provide many examples and high-level discussions of definitions and techniques. The paper contains some new ingredients. We disentangle some proofs to make them more modular, and make each part as general as possible. We introduce some new notions, which sometimes lead to simpler, more intuitive proofs, as well as to some stronger or more general theorems. One such consequence is our connectivity theorem for the asymptotic spectrum of <italic>any</italic> tensor network, greatly generalizing Strassen’s connectivity theorem for the special case of matrix multiplication. Another consequence is progress on a conjecture of Strassen which generalizes Schönhage’s tau theorem.