Fig. 3: Comparison between the complexity of previously known matrix multiplication algorithms and the ones discovered by AlphaTensor.
From: Discovering faster matrix multiplication algorithms with reinforcement learning
Left: column (n, m, p) refers to the problem of multiplying n × m with m × p matrices. The complexity is measured by the number of scalar multiplications (or equivalently, the number of terms in the decomposition of the tensor). ‘Best rank known’ refers to the best known upper bound on the tensor rank (before this paper), whereas ‘AlphaTensor rank’ reports the rank upper bounds obtained with our method, in modular arithmetic (\({{\mathbb{Z}}}_{2}\)) and standard arithmetic. In all cases, AlphaTensor discovers algorithms that match or improve over known state of the art (improvements are shown in red). See Extended Data Figs. 1 and 2 for examples of algorithms found with AlphaTensor. Right: results (for arithmetic in \({\mathbb{R}}\)) of applying AlphaTensor-discovered algorithms on larger tensors. Each red dot represents a tensor size, with a subset of them labelled. See Extended Data Table 1 for the results in table form. State-of-the-art results are obtained from the list in ref. 64.
