| import OAI.Probability.InvariantIsing.Arrays.NSpinCascade |
| import OAI.Probability.InvariantIsing.Pressure.Concentration |
|
|
| namespace OAI |
|
|
| /-! Haar-direction comparison for the actual enriched Ising recursion. -/ |
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|
| noncomputable section |
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|
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal |
|
|
| namespace InvariantIsing |
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|
| 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)) |
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|
| 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 |
|
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| 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 |
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| end |
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| end OAI |
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