File size: 3,319 Bytes
c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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
end
end OAI
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