| import OAI.Probability.InvariantIsing.Fields.FieldCompensatedTangent |
|
|
| namespace OAI |
|
|
| /-! The adjacent-variance cancellation for a single field height. Its |
| derivative is the negative conditional squared spin mean. This proves |
| the local identity underlying `fld:backward-derivative` directly from |
| the Gaussian recursion. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
|
|
| namespace InvariantIsing.FieldSmoothFamily |
|
|
| theorem parameter_derivative {I : Set ℝ} (F : FieldSmoothFamily I) |
| (t z : ℝ) (ht : t ∈ I) : |
| HasDerivAt (fun s => F.U (s, z)) (F.T (t, z)) t := by |
| have hmap : HasDerivAt (fun s : ℝ => (s, z)) ((1 : ℝ), (0 : ℝ)) t := |
| (hasDerivAt_id t).prodMk (hasDerivAt_const t z) |
| have hd := (F.derivative (t, z) ht).comp_hasDerivAt t hmap |
| simpa [pairLinear, Function.comp_def] using hd |
|
|
| theorem negative_variance_compensation {I : Set ℝ} (F : FieldSmoothFamily I) |
| (hI : IsOpen I) (b η : ℝ) (hv : ∀ t ∈ I, 0 < b - t) |
| {p : ℝ × ℝ} (hp : p.1 ∈ I) (hT : ∀ y, F.T (p.1, y) = 0) : |
| (F.transform hI b (-1) η (by simpa only [neg_one_mul, sub_eq_add_neg] using hv)).T p + |
| (1 / 2 : ℝ) * |
| ((F.transform hI b (-1) η (by simpa only [neg_one_mul, sub_eq_add_neg] using hv)).XX p + η * |
| ((F.transform hI b (-1) η (by simpa only [neg_one_mul, sub_eq_add_neg] using hv)).X p) ^ 2) = 0 := by |
| have hc := F.compensated_tangent b 0 1 η η hp |
| have hz : (fun y => F.T (p.1, y) + (0 : ℝ) / 2 * |
| (F.XX (p.1, y) + η * (F.X (p.1, y)) ^ 2)) = fun _ => 0 := by |
| funext y |
| simp only [hT y, zero_div, zero_mul, add_zero] |
| have hm : Measurable (fun y => F.U (p.1, y)) := F.mU.comp (by fun_prop) |
| rw [hz, field_gaussianTiltAverage_const _ _ hm (F.growth p.1)] at hc |
| change F.tangent b (-1) η p + (1 / 2 : ℝ) * |
| (F.curvature b (-1) η p + η * (F.mean b (-1) η p) ^ 2) = 0 |
| simpa only [sub_self, sub_zero, zero_sub, zero_div, zero_mul] using hc |
|
|
| theorem adjacent_tangent {I : Set ℝ} (F : FieldSmoothFamily I) |
| (hI : IsOpen I) (a b ζ η : ℝ) |
| (hplus : ∀ t ∈ I, 0 < a + t) (hminus : ∀ t ∈ I, 0 < b - t) |
| {p : ℝ × ℝ} (hp : p.1 ∈ I) (hT : ∀ y, F.T (p.1, y) = 0) : |
| let G := F.transform hI b (-1) η (by simpa only [neg_one_mul, sub_eq_add_neg] using hminus) |
| let H := G.transform hI a 1 ζ (by simpa only [one_mul] using hplus) |
| H.T p = -(η - ζ) / 2 * gaussianTiltAverage (a + p.1) ζ |
| (fun y => G.U (p.1, y)) (fun y => (G.X (p.1, y)) ^ 2) p.2 := by |
| let G := F.transform hI b (-1) η (by simpa only [neg_one_mul, sub_eq_add_neg] using hminus) |
| change G.tangent a 1 ζ p = -(η - ζ) / 2 * |
| gaussianTiltAverage (a + p.1) ζ (fun y => G.U (p.1, y)) |
| (fun y => (G.X (p.1, y)) ^ 2) p.2 |
| have hc := G.compensated_tangent a 1 0 ζ η hp |
| have hz : (fun y => G.T (p.1, y) + (1 : ℝ) / 2 * |
| (G.XX (p.1, y) + η * (G.X (p.1, y)) ^ 2)) = fun _ => 0 := by |
| funext y |
| exact F.negative_variance_compensation hI b η hminus hp hT |
| have hm : Measurable (fun y => G.U (p.1, y)) := G.mU.comp (by fun_prop) |
| rw [hz, field_gaussianTiltAverage_const _ _ hm (G.growth p.1)] at hc |
| simp only [sub_zero, one_mul, zero_div, zero_mul, add_zero] at hc |
| calc |
| G.tangent a 1 ζ p = _ := hc |
| _ = _ := by ring |
|
|
| end InvariantIsing.FieldSmoothFamily |
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|
| end |
|
|
| end OAI |
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