Searching for Efficient Linear Layers over a Continuous Space of Structured Matrices

Dense linear layers are the dominant computational bottleneck in large neural networks, presenting a critical need for more efficient alternatives. Previous efforts to develop alternatives have focused on a small number of hand-crafted structured matrices, and have neglected to investigate whether these structures can surpass dense layers in terms of compute-optimal scaling laws when both the model size and training examples are optimally allocated. In this work, we present a unifying framework that enables searching among all linear operators expressible via an Einstein summation. This framework encompasses many previously proposed structures, such as low-rank, Kronecker, Tensor-Train, and Monarch, along with many novel structures. We develop a taxonomy of all such operators based on their computational and algebraic properties, which provides insights into their scaling laws.