Эксперимент: Intrinsic Schrödinger Bridge Diffusion (#1055)

{ "worked": true, "confidence": 8, "verdict": "Built and numerically verified a sphere-manifold intrinsic diffusion MVP with tangent projection, tangent noise, and normalization retraction. Post-retraction constraint error stayed at 3.33e-16 across all step sizes, unconstrained one-step violation scaled as h^0.570 versus the predicted h^0.5, and the stated control cost scaled quadratically with amplitude with exponent 2.0. Intrinsic and projection-at-every-step updates are equivalent in this toy setting, so this validates constraint preservation but not a generative-quality win over frequent projection.", "metrics": { "baseline": "Ambient Euler diffusion: median one-step norm violations 0.1596, 0.1043, 0.0706, 0.0487 for h=0.04, 0.02, 0.01, 0.005; fitted exponent 0.570. Short endpoint mean final norm error was 0.4220 and alignment was 0.9739.", "idea": "Intrinsic/retracted diffusion: maximum intermediate norm error was 3.33e-16 for every tested step size; short endpoint mean final norm error was 5.05e-17 and alignment was 0.6095. Control-cost sweep exponent was 2.0, with costs 0.02738, 0.10953, 0.43812, 1.75249, and 7.00997." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py", "files": [ "experiment.py", "results.json" ], "limitations": "Only the sphere manifold and a simple linear energy were tested. No learned neural Schrödinger-bridge controller, endpoint KL estimation, multimodal target, Stiefel manifold, likelihood evaluation, or wall-clock/FLOP comparison was implemented. The sphere intrinsic update uses normalization retraction and is equivalent to projection after every Euler step, so the result validates the geometric mechanism rather than superiority over a well-designed frequent-projection baseline." }