Extended Data Table 2 Result of applying AlphaTensor to the tensor representing the cyclic convolution operation

From: Discovering faster matrix multiplication algorithms with reinforcement learning

  1. AlphaTensor finds the discrete Fourier matrix (DFT) and the inverse DFT matrix in finite fields. The figure shows the decompositions found by AlphaTensor of the n × n × n tensor representing the cyclic convolution of two vectors, for three different values of n in the finite field of order 17. The action space, characterized by the number of possible factor triplets {u(r), v(r), w(r)}, is thus 173n, which is of the order of 1029 for n = 8. Despite the huge action space, AlphaTensor finds the optimal rank-n decompositions for the three values of n. The factors in the figure are stacked vertically, i.e., U = [u(1), …, u(n)]. For ease of visualization, the factor entries have been expressed in terms of powers of an n-th primitive root of unity in the finite field. Within each column, each colour uniquely represents one element of the field (e.g., for the column n = 4, we have depicted in grey 40 = 44 = 4−4 = 1). By inspecting the patterns in the decompositions, one could extrapolate the results for other values of n and other fields. Indeed, the factors u(r) and v(r) correspond to the DFT coefficients, since \({u}_{k}^{(r)}={v}_{k}^{(r)}={z}^{kr}\), whereas the factors w(r) correspond to the inverse DFT, since \({w}_{k}^{(r)}={z}^{-kr}/n\) for 0≤k, r < n, where z is an n-th primitive root of unity (i.e., zn = 1 and zj ≠ 1 for any 1≤j < n).
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