Efficient training for optical computing

Diffractive optical information processors have demonstrated significant promise in delivering high-speed, parallel, and energy-efficient inference for scaling machine learning tasks. Training, however, remains a major computational bottleneck in achieving scalable state-of-the-art classification models. The linear transformations in such systems are inherently constrained to compositions of circulant and diagonal matrix factors, representing free-space propagation and phase and/or amplitude modulation of light, respectively. While theoretically established that such factors can generate arbitrary linear transformations, only experimentally unfeasible upper bounds on the number of factors exist. Additionally, physical parameters such as inter-layer distances, number of layers, and phase-only modulation restrict the solution space. As trainable elements occupy only a subset of the overall transformation, current constrained minimization techniques incur unnecessary computational overhead, limiting scalability. In this work, we introduce a backpropagation algorithm that provides a novel closed form solution to calculate gradients and incorporates plane wave decomposition via the Fourier transform, computing gradients across all trainable elements in a given layer simultaneously, using only change-of-basis operations and element-wise multiplication. Given the lack of tractable analytical decompositions, this method extends beyond machine learning to generation of arbitrary linear transformations, wavefront shaping, and other signal processing tasks.