Problem
Research questions and friction points this paper is trying to address.
universal approximation
diffractive optical processors
approximation error
physical limits
statistical learnability
Universal Function Approximation via Diffractive Optical Processors: Physical Limits, Error Bounds, and Learnability
🤖 AI Summary
This work investigates the theoretical limits and learnability of diffractive optical processors for universal function approximation. By integrating universal approximation theory, Fourier feature expansions, and diffractive optical models, it establishes the first rigorous mathematical foundation for such systems and systematically analyzes the quantitative relationships among expressive power, optical degrees of freedom, statistical learnability, and physical realizability. The methodology encompasses phase-encoded diffractive architectures, optimization of spatially varying coherent point spread functions, Fourier truncation analysis, photon-statistical modeling, and finite-class statistical learning theory. Key contributions include derived bounds on approximation error, resource scaling laws, constraints imposed by photon budgets, and fundamental learnability limits, collectively offering principled design guidelines for large-scale analog optical computing systems.
📝 Abstract
We present a unified theoretical framework connecting classical universal approximation theory, Fourier-feature approximation, and diffractive optical processors. We show that phase-encoded diffractive processors implement finite Fourier-feature expansions whose mathematical completeness follows from Fourier/Stone-Weierstrass arguments, while their physical realizability is governed by finite coefficient synthesis through optimized spatially varying coherent point-spread functions (PSFs). Our analyses derive approximation-error bounds that separate Fourier truncation, PSF-synthesis, input phase error, optical hardware, readout, and noise contributions; establish scaling relationships linking approximation complexity to optical degrees of freedom and input/output space-bandwidth products; derive photon-budget and throughput limits imposed by photon statistics; formulate finite-class statistical learnability bounds for phase-quantized diffractive function approximators; and analyze the impact of spatially incoherent illumination. We further analyze coherent optical cascadability and show that quadratic feature expansion through coherent mixing and optical readout provides a mechanism for enhanced representation while remaining fundamentally distinct from the depth-separation results established for digital neural networks. Our analyses provide a rigorous theoretical foundation for diffractive nonlinear function approximation and establish quantitative relationships among mathematical expressivity, optical hardware resources, statistical learning, and physical performance limits, thereby offering general design principles for large-scale analog optical computing systems.
Innovation
Methods, ideas, or system contributions that make the work stand out.
diffractive optical processors
universal approximation
Fourier-feature expansion
optical learnability
error bounds
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