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b317718 | 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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import OAI.Probability.InvariantIsing.Arrays.NSpinCascade
import OAI.Probability.InvariantIsing.Pressure.Concentration
namespace OAI
/-! Haar-direction comparison for the actual enriched Ising recursion. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace InvariantIsing
private theorem vectorCascade_terminal_comparison {N : ℕ} (hN : 0 < N) (n : ℕ)
(b : ℕ → ℝ) (v : ℕ → ℝ≥0) (hb : ∀ i < n, 0 < b i)
{F G : (Fin N → ℝ) → ℝ} (hF : Measurable F) (hG : Measurable G)
(hFg : IsingPerceptron.HasLinearGrowth F) (hGg : IsingPerceptron.HasLinearGrowth G)
{C : ℝ} (hFG : ∀ z, |F z - G z| ≤ C) (z : Fin N → ℝ) :
|IsingPerceptron.cascadeRecursion n b (fun i => vectorGaussianLaw N (v i))
(fun _ p => p.1 + p.2) F z -
IsingPerceptron.cascadeRecursion n b (fun i => vectorGaussianLaw N (v i))
(fun _ p => p.1 + p.2) G z| ≤ C := by
induction n generalizing b v z with
| zero => exact hFG z
| succ n ih =>
let bs := fun i => b (i + 1)
let vs := fun i => v (i + 1)
let F' := IsingPerceptron.cascadeRecursion n bs (fun i => vectorGaussianLaw N (vs i))
(fun _ p => p.1 + p.2) F
let G' := IsingPerceptron.cascadeRecursion n bs (fun i => vectorGaussianLaw N (vs i))
(fun _ p => p.1 + p.2) G
have hbs : ∀ i < n, 0 < bs i := fun i hi => hb (i + 1) (by omega)
have hmF : Measurable F' := IsingPerceptron.measurable_cascadeRecursion n bs
(fun i => vectorGaussianLaw N (vs i)) (fun _ => measurable_fst.add measurable_snd) hF
have hmG : Measurable G' := IsingPerceptron.measurable_cascadeRecursion n bs
(fun i => vectorGaussianLaw N (vs i)) (fun _ => measurable_fst.add measurable_snd) hG
have hgF : IsingPerceptron.HasLinearGrowth F' :=
IsingPerceptron.cascadeRecursion_linearGrowth n bs (fun i => vectorGaussianLaw N (vs i))
(fun i _ => vectorGaussianLaw_moments hN (vs i)) hF hFg hbs
have hgG : IsingPerceptron.HasLinearGrowth G' :=
IsingPerceptron.cascadeRecursion_linearGrowth n bs (fun i => vectorGaussianLaw N (vs i))
(fun i _ => vectorGaussianLaw_moments hN (vs i)) hG hGg hbs
have hiF : Integrable (fun g => Real.exp (b 0 * F' (z + g)))
(vectorGaussianLaw N (v 0) : Measure (Fin N → ℝ)) :=
IsingPerceptron.integrable_exp_of_linearGrowth _ (vectorGaussianLaw_moments hN (v 0))
(hmF.comp (measurable_const.add measurable_id)) (hgF.add_left z) (b 0)
have hiG : Integrable (fun g => Real.exp (b 0 * G' (z + g)))
(vectorGaussianLaw N (v 0) : Measure (Fin N → ℝ)) :=
IsingPerceptron.integrable_exp_of_linearGrowth _ (vectorGaussianLaw_moments hN (v 0))
(hmG.comp (measurable_const.add measurable_id)) (hgG.add_left z) (b 0)
exact IsingPerceptron.logMean_abs_sub_le _ (hb 0 (by omega)) hiF hiG
(fun g => ih bs vs hbs (z + g))
theorem abs_rotatedCascadeValue_sub_rotation_le {N : ℕ} (hN : 0 < N)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (hb : IsingPerceptron.CascadeExponents n b)
(eig : Fin N → ℝ) (U V : SpecialOrthogonal N) (c z : Fin N → ℝ)
(K : ℝ) (hK : 0 ≤ K) (heig : ∀ i, |eig i| ≤ K) :
|rotatedCascadeValue n b v eig (specialRotation U) c z -
rotatedCascadeValue n b v eig (specialRotation V) c z| ≤
K * N * frobeniusDistance U V := by
apply vectorCascade_terminal_comparison hN n b v (fun i hi => (hb.1 i hi).1)
(measurable_rotatedFieldTerminal eig (specialRotation U) c)
(measurable_rotatedFieldTerminal eig (specialRotation V) c)
(rotatedFieldTerminal_linearGrowth eig (specialRotation U) c)
(rotatedFieldTerminal_linearGrowth eig (specialRotation V) c)
intro y
apply abs_logPartition_sub_le
intro σ
simpa only [add_sub_add_right_eq_sub] using
abs_rotatedEnergy_sub_rotation_le eig U V K hK heig σ
theorem abs_rotatedEnrichedPressure_sub_rotation_le {N : ℕ} (hN : 0 < N)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (root : ℝ≥0)
(hb : IsingPerceptron.CascadeExponents n b) (eig : Fin N → ℝ)
(U V : SpecialOrthogonal N) (c : Fin N → ℝ)
(K : ℝ) (hK : 0 ≤ K) (heig : ∀ i, |eig i| ≤ K) :
|rotatedEnrichedPressure n b v root eig (specialRotation U) c -
rotatedEnrichedPressure n b v root eig (specialRotation V) c| ≤
K * frobeniusDistance U V := by
have hIU := integrable_rotatedCascadeValue hN n b v hb eig (specialRotation U) c root
have hIV := integrable_rotatedCascadeValue hN n b v hb eig (specialRotation V) c root
have hi := norm_integral_le_of_norm_le_const
(μ := (vectorGaussianLaw N root : Measure (Fin N → ℝ)))
(f := fun z => rotatedCascadeValue n b v eig (specialRotation U) c z -
rotatedCascadeValue n b v eig (specialRotation V) c z)
(C := K * N * frobeniusDistance U V) (ae_of_all _ fun z => by
simpa only [Real.norm_eq_abs] using
abs_rotatedCascadeValue_sub_rotation_le hN n b v hb eig U V c z K hK heig)
have hI : |∫ z, (rotatedCascadeValue n b v eig (specialRotation U) c z -
rotatedCascadeValue n b v eig (specialRotation V) c z)
∂(vectorGaussianLaw N root : Measure (Fin N → ℝ))| ≤ K * N * frobeniusDistance U V := by
simpa only [Real.norm_eq_abs, probReal_univ, mul_one] using hi
unfold rotatedEnrichedPressure
rw [← mul_sub, ← integral_sub hIU hIV, abs_mul,
abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg N))]
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
calc
_ ≤ (N : ℝ)⁻¹ * (K * N * frobeniusDistance U V) :=
mul_le_mul_of_nonneg_left hI (by positivity)
_ = _ := by field_simp
end InvariantIsing
end
end OAI
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