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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 | import OAI.Probability.InvariantIsing.Arrays.NSpinConcentration
namespace OAI
/-! The Haar contribution to the enriched pressure variance. -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped NNReal ENNReal
namespace InvariantIsing
/-- Integrating the Gaussian tail of a deviation gives its second moment.
The proof applies layer cake to the squared deviation, so only the elementary
exponential integral on the positive half-line is needed. -/
lemma integral_sq_sub_le_of_gaussian_tail {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (f : Ω → ℝ) (hf : Measurable f)
(m C a : ℝ) (hC : 0 ≤ C) (ha : 0 < a)
(htail : ∀ r : ℝ, 0 < r →
μ.real {ω | r ≤ |f ω - m|} ≤ C * Real.exp (-a * r ^ 2)) :
Integrable (fun ω => (f ω - m) ^ 2) μ ∧
(∫ ω, (f ω - m) ^ 2 ∂μ) ≤ C / a := by
let g := fun ω => (f ω - m) ^ 2
have hgm : Measurable g := (hf.sub_const m).pow_const 2
have hgn : ∀ ω, 0 ≤ g ω := fun ω => sq_nonneg _
have hk : Integrable (fun t : ℝ => C * Real.exp (-a * t)) (volume.restrict (Ioi 0)) :=
(integrableOn_exp_mul_Ioi (by linarith : -a < 0) 0).const_mul C
have hkn : ∀ t : ℝ, 0 ≤ C * Real.exp (-a * t) := fun _ => mul_nonneg hC (Real.exp_pos _).le
have hki : (∫ t : ℝ in Ioi 0, C * Real.exp (-a * t)) = C / a := by
rw [integral_const_mul, integral_exp_mul_Ioi (by linarith : -a < 0) 0]
simp [div_eq_mul_inv]
have hlin : (∫⁻ ω, ENNReal.ofReal (g ω) ∂μ) ≤ ENNReal.ofReal (C / a) := by
calc
_ = ∫⁻ t : ℝ in Ioi 0, μ {ω | t ≤ g ω} :=
lintegral_eq_lintegral_meas_le μ (ae_of_all _ hgn) hgm.aemeasurable
_ ≤ ∫⁻ t : ℝ in Ioi 0, ENNReal.ofReal (C * Real.exp (-a * t)) := by
apply lintegral_mono_ae
filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht
have he : {ω | t ≤ g ω} = {ω | Real.sqrt t ≤ |f ω - m|} := by
ext ω
simp only [mem_ofPred_eq, g, Real.sqrt_le_left (abs_nonneg _), sq_abs]
rw [he, ← ofReal_measureReal]
apply ENNReal.ofReal_le_ofReal
simpa only [Real.sq_sqrt ht.le] using htail (Real.sqrt t) (Real.sqrt_pos.mpr ht)
_ = ENNReal.ofReal (C / a) := by
rw [← ofReal_integral_eq_lintegral_ofReal hk (ae_of_all _ hkn), hki]
have hgi : Integrable g μ :=
(lintegral_ofReal_ne_top_iff_integrable hgm.aestronglyMeasurable
(ae_of_all _ hgn)).mp (ne_top_of_le_ne_top ENNReal.ofReal_ne_top hlin)
refine ⟨hgi, ?_⟩
have hI : ENNReal.ofReal (∫ ω, g ω ∂μ) ≤ ENNReal.ofReal (C / a) := by
rw [ofReal_integral_eq_lintegral_ofReal hgi (ae_of_all _ hgn)]
exact hlin
exact (ENNReal.ofReal_le_ofReal_iff (by positivity : 0 ≤ C / a)).mp hI
lemma variance_le_of_gaussian_median_tail {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (f : Ω → ℝ) (hf : Measurable f)
(m C a : ℝ) (hC : 0 ≤ C) (ha : 0 < a)
(htail : ∀ r : ℝ, 0 < r →
μ.real {ω | r ≤ |f ω - m|} ≤ C * Real.exp (-a * r ^ 2)) :
variance f μ ≤ C / a := by
rw [← variance_sub_const hf.aestronglyMeasurable m]
exact (variance_le_expectation_sq (hf.sub_const m).aestronglyMeasurable).trans
(integral_sq_sub_le_of_gaussian_tail μ f hf m C a hC ha htail).2
/-- Haar variance of the actual enriched pressure is at most a universal
constant times the squared spectral bound divided by dimension. -/
theorem haar_enrichedPressure_variance (hpub : HaarConcentrationInput) :
∃ C : ℝ, 0 < C ∧
∀ N : ℕ, 3 ≤ N →
∀ μ : Measure (SpecialOrthogonal N),
IsProbabilityMeasure μ → μ.IsMulLeftInvariant →
∀ n : ℕ, ∀ b : ℕ → ℝ, ∀ v : ℕ → ℝ≥0, ∀ root : ℝ≥0,
IsingPerceptron.CascadeExponents n b →
∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) →
variance (fun U : SpecialOrthogonal N =>
rotatedEnrichedPressure n b v root eig (specialRotation U) c) μ ≤
C * K ^ 2 / N := by
obtain ⟨C, a, hC, ha, hp⟩ := haar_enrichedPressure_median_tail hpub
refine ⟨C / a, by positivity, ?_⟩
intro N hN μ hμ hμinv n b v root hb eig c K hK heig
let : IsProbabilityMeasure μ := hμ
obtain ⟨m, _, _, ht⟩ := hp N hN μ hμ hμinv n b v root hb eig c K hK heig
have hn : (0 : ℝ) < N := by exact_mod_cast (show 0 < N by omega)
have hbound := variance_le_of_gaussian_median_tail μ
(fun U : SpecialOrthogonal N => rotatedEnrichedPressure n b v root eig (specialRotation U) c)
(measurable_rotatedEnrichedPressure n b v root eig c) m C (a * N / K ^ 2) hC.le
(by positivity) (fun r hr => by
simpa only [show -a * (N : ℝ) * r ^ 2 / K ^ 2 = -(a * N / K ^ 2) * r ^ 2 by ring]
using ht r hr)
refine hbound.trans_eq ?_
field_simp
end InvariantIsing
end
end OAI
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