File size: 2,228 Bytes
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"assessment": "verified",
"claim": 2,
"destructive_control": {
"effect": "functions converge uniformly while derivative amplitudes diverge",
"mutation": "drop shared semiconvexity: add sin(nx)/sqrt(n)",
"result": [
{
"function_error": 0.25,
"gradient_amplitude": 4.0,
"n": 16
},
{
"function_error": 0.125,
"gradient_amplitude": 8.0,
"n": 64
},
{
"function_error": 0.0625,
"gradient_amplitude": 16.0,
"n": 256
},
{
"function_error": 0.03125,
"gradient_amplitude": 32.0,
"n": 1024
}
]
},
"gradient_error_vs_spacing_exponent": 1.0000000000000002,
"gradient_grid_rows": [
{
"atoms": 5,
"differentiable_cells_checked": 16,
"exact_gradient_supremum_bound": 0.25,
"sampled_max_gradient_error": 0.2495,
"spacing": 0.5
},
{
"atoms": 9,
"differentiable_cells_checked": 32,
"exact_gradient_supremum_bound": 0.125,
"sampled_max_gradient_error": 0.12475,
"spacing": 0.25
},
{
"atoms": 17,
"differentiable_cells_checked": 64,
"exact_gradient_supremum_bound": 0.0625,
"sampled_max_gradient_error": 0.062375,
"spacing": 0.125
},
{
"atoms": 33,
"differentiable_cells_checked": 128,
"exact_gradient_supremum_bound": 0.03125,
"sampled_max_gradient_error": 0.0311875,
"spacing": 0.0625
},
{
"atoms": 65,
"differentiable_cells_checked": 256,
"exact_gradient_supremum_bound": 0.015625,
"sampled_max_gradient_error": 0.01559375,
"spacing": 0.03125
},
{
"atoms": 129,
"differentiable_cells_checked": 512,
"exact_gradient_supremum_bound": 0.0078125,
"sampled_max_gradient_error": 0.007796875,
"spacing": 0.015625
}
],
"literal_claim": "Theorem 2 (Section V.A) shows that, under semiconvexity conditions on \u03a6, gradients of finitely \u1ef8-convex functions densely approximate gradients of all \u1ef8-convex functions.",
"semiconvex_native_instance": "the same quadratic Phi and f; every finite transform shares semiconvexity constant K=0"
}
|