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A closed-form free-probability law for loss-induced conditioning in deep photonic mesh networks

Year: 2026
Type: preprint
Abstract: CORRECTION (28 September 2026): The central result of this preprint was derived earlier and is not original to this work. The log-variance law Var[log x] = v + v²/12 (equivalently the coefficient γ = 1/3), and the description of the logarithm of the free multiplicative central limit as a semicircle plus a uniform law, were obtained by K.-P. Ho, "Central limit for the product of free random variables," arXiv:1101.5220 (2011). The same law was applied to accumulated mode-dependent loss in multimode fiber, using the same model of cascaded random unitary mixing and diagonal loss, in K.-P. Ho, "Exact model for mode-dependent gains and losses in multimode fiber," J. Lightwave Technol. 30(23), 3603–3609 (2012), doi:10.1109/JLT.2012.2223657, and in K.-P. Ho and J. M. Kahn, "Mode-dependent loss and gain: statistics and effect on mode-division multiplexing," Opt. Express 19(17), 16612–16635 (2011), doi:10.1364/OE.19.016612. Version 1 of the PDF does not cite these works. Its statement that no closed-form theory had described this limit, and its attribution of the semicircle-plus-uniform decomposition to later papers, are incorrect. This preprint should therefore be read as an application of Ho's law to Mach–Zehnder-interferometer mesh networks. Its mesh-specific results (agreement of structured Clements and Reck meshes with the law, the path-length topology criterion, and the trainability study) were obtained with a lumped per-mode loss model and have not been validated with loss distributed inside the mesh components. The finite-depth formula χ(L) = χ(1)L[1 + χ(1)L/3] is an asymptotic approximation that does not hold exactly at small depth, and the trainability evidence is limited. The abstract of version 1 is reproduced below, unchanged. Abstract (version 1): Deep optical neural networks built from cascaded Mach–Zehnder-interferometer (MZI) meshes promise fast, low-energy linear algebra, but optical loss limits how many layers can be cascaded and no closed-form theory has described that limit. We show that for a product of independent Haar-random meshes the squared-singular-value distribution is the L-fold free multiplicative convolution of the single-layer loss spectrum, and from it we derive a closed-form law for the dynamical-isometry order parameter χ(L) = Var[log s_i(W)]: χ(L) = σ_loss² L (1 + σ_loss² L/3) + O(σ⁶). The correction coefficient γ = 1/3 is shape-independent to leading order and exact in the free multiplicative Brownian scaling limit. It is fixed by the second-order coefficient (1/12) of the log-variance under free multiplicative convolution—a probabilistic fact, the decomposition of log into a semicircle plus a uniform law, established by Auer and Voit; our contribution is not this coefficient but its identification as the law governing loss-limited depth in photonic meshes. Large-scale simulations show that real, structured, noisy Clements meshes follow the same law as the idealized Haar product to about 1–2% up to N = 2048 modes, the free correction (a ≈13% departure from a naive additive model at the depths tested) being required to fit them; because asymptotic freeness of deterministic meshes is an open problem, the real-mesh law is supported empirically rather than proven. The same conditioning controls optimization: in a deep linear network a lossless control isolates loss as the cause of a depth-dependent training collapse, which we show to be ill-conditioning rather than vanishing gradients. It also ranks mesh architectures by loss resilience through a single layout quantity, the variance of the per-mode path length—rectangular (Clements) meshes are predicted to tolerate about 85× more depth than triangular (Reck) meshes at N = 128—and carries its own regime of validity, crossing over to a classical additive law when the meshes mix weakly. At the moderate depths of present devices the measured coefficient is slightly smaller, ≈0.30, well described by the empirical finite-depth fit γ_eff = 1/3 + α/L with α ≈ −0.34 (a finite-size fit, not a derived identity). Files: the preprint (PDF, version 1); the simulation and derivation-verification code with result tables, notebooks and a README mapping each script to the results in the preprint (code-and-data ZIP; code under the MIT License); and the LaTeX source of the preprint (source ZIP). The use of AI tools in this work is disclosed in the preprint.
DOI: 10.5281/zenodo.22986393
Language: English
Open Access status: green
OpenAlex ID: W7214522207
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