Exploiting Dynamic Sparsity in Einsum

Einsum expressions specify an output tensor in terms of several input tensors. They offer a simple yet expressive abstraction for many computational tasks in artificial intelligence and beyond. However, evaluating einsum expressions poses hard algorithmic problems that depend on the representation of the tensors. Two popular representations are multidimensional arrays and coordinate lists. The latter is a more compact representation for sparse tensors, that is, tensors where a significant proportion of the entries are zero. So far, however, most of the popular einsum implementations use the multidimensional array representation for tensors. Here, we show on a non-trivial example that, when evaluating einsum expressions, coordinate lists can be exponentially more efficient than multidimensional arrays. In practice, however, coordinate lists can also be significantly less efficient than multidimensional arrays, but it is hard to decide from the input tensors whether this will be the case. Sparsity evolves dynamically in intermediate tensors during the evaluation of an einsum expression. Therefore, we introduce a hybrid solution where the representation is switched on the fly from multidimensional arrays to co-ordinate lists depending on the sparsity of the remaining tensors. In our experiments on established benchmark einsum expressions, the hybrid solution is consistently competitive with or outperforms the better of the two static representations.