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import OAI.Probability.InvariantIsing.Pressure.PressureDifferenceConcentration
namespace OAI
/-! Transfer the actual three-disorder pressure moments to one frozen Gaussian CGF. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal ENNReal
namespace InvariantIsing
private lemma joint_map_eq_of_conditional_map_eq {Ω G E : Type*}
[MeasurableSpace Ω] [MeasurableSpace G] [MeasurableSpace E]
(P : Measure Ω) (Q : Measure G) [SFinite Q]
(f g : Ω × G → E) (hf : Measurable f) (hg : Measurable g)
(he : ∀ ω, Q.map (fun z => f (ω, z)) = Q.map (fun z => g (ω, z))) :
(P.prod Q).map f = (P.prod Q).map g := by
apply Measure.ext
intro s hs
rw [Measure.map_apply hf hs, Measure.map_apply hg hs,
Measure.prod_apply (hf hs), Measure.prod_apply (hg hs)]
apply lintegral_congr_ae
refine ae_of_all P fun ω => ?_
have h := congrArg (fun ν : Measure E => ν s) (he ω)
have hfω : Measurable (fun z => f (ω, z)) := hf.comp (measurable_const.prodMk measurable_id)
have hgω : Measurable (fun z => g (ω, z)) := hg.comp (measurable_const.prodMk measurable_id)
rw [Measure.map_apply hfω hs, Measure.map_apply hgω hs] at h
exact h
private lemma tensorNamespacedLog_conditional_law {N m k : ℕ} (eig c : Fin N → ℝ)
(U : SpecialOrthogonal N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (T : LabeledTree n) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) :
gaussianCoordinates.map (fun g => tensorFlatLog eig (specialRotation U) c I degree amplitude n
(tensorPathProfile I degree n treeDegree h) (T, g)) =
gaussianCoordinates.map (fun g => tensorNamespacedLog eig c I degree amplitude n treeDegree h ((U, T), g)) := by
let ν := labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) T
let H := fun x : Spin N × LabeledLeaf n => rotatedEnergy eig (specialRotation U) x.1 + fieldEnergy c x.1
let K := fun F : Spin N × LabeledLeaf n → ℝ => Real.log (∫ x, Real.exp (H x + F x) ∂ν)
have : ∀ F : (Spin N × LabeledLeaf n → ℝ), IsProbabilityMeasure ((fun _ => ν) F) :=
fun _ => by
change IsProbabilityMeasure (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) T)
infer_instance
have hE : Measurable (fun p : (Spin N × LabeledLeaf n → ℝ) × (Spin N × LabeledLeaf n) => H p.2 + p.1 p.2) := by
apply measurable_from_prod_countable_left
intro x
have hc : Measurable (fun _ : Spin N × LabeledLeaf n → ℝ => H x) := measurable_const
exact hc.add (measurable_pi_apply x)
have hK : Measurable K := (measurable_random_referencePartition
(ν := fun _ : Spin N × LabeledLeaf n → ℝ => ν) measurable_const hE).log
let v : Fin (n + 1) → SpinTensorIndex I degree → ℝ≥0 :=
fun i => tensorPathProfile I degree n treeDegree h i
have hleft : Measurable (fun g x => cylinderField
(tensorLeafCoefficients (specialRotation U) I degree amplitude n v x) g) :=
Measurable.of_eval (fun _ => measurable_cylinderField _)
have hright : Measurable (fun g x => cylinderField
(tensorNamespacedCoefficients (specialRotation U) I degree amplitude n v x) g) :=
Measurable.of_eval (fun _ => measurable_cylinderField _)
change gaussianCoordinates.map (K ∘ (fun g x => cylinderField
(tensorLeafCoefficients (specialRotation U) I degree amplitude n v x) g)) =
gaussianCoordinates.map (K ∘ (fun g x => cylinderField
(tensorNamespacedCoefficients (specialRotation U) I degree amplitude n v x) g))
rw [← Measure.map_map hK hleft, ← Measure.map_map hK hright, tensorNamespacedFields_law]
private lemma measurable_tensorLeafField_joint {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (v : Fin (n + 1) → SpinTensorIndex I degree → ℝ≥0)
(x : Spin N × LabeledLeaf n) :
Measurable (fun p : SpecialOrthogonal N × (ℕ → ℝ) =>
cylinderField (tensorLeafCoefficients (specialRotation p.1) I degree amplitude n v x) p.2) := by
simp only [tensorLeafCoefficients, cylinderField_feature]
apply Finset.measurable_sum
intro i _
exact (((measurable_spinTensorFeature I degree amplitude x.1 i.2).comp measurable_fst).const_mul _).mul
((measurable_pi_apply (treeFeatureTag n x.2 i)).comp measurable_snd)
private lemma measurable_tensorJointLog {N : ℕ} (eig c : Fin N → ℝ) (n : ℕ)
(Y : (((SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) × (Spin N × LabeledLeaf n)) → ℝ)
(hY : Measurable Y) :
Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
Real.log (∫ x : Spin N × LabeledLeaf n,
Real.exp (rotatedEnergy eig (specialRotation p.1.1) x.1 + fieldEnergy c x.1 + Y (p, x))
∂labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.2)) := by
have hbase : Measurable (fun p : (((SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) ×
(Spin N × LabeledLeaf n)) => rotatedEnergy eig (specialRotation p.1.1.1) p.2.1 + fieldEnergy c p.2.1) := by
apply measurable_from_prod_countable_left
intro x
have hr : Measurable (fun U : SpecialOrthogonal N => rotatedEnergy eig (specialRotation U) x.1) := by
unfold rotatedEnergy
exact (Finset.measurable_sum _ fun i _ =>
((measurable_specialRotation_eval (spinVector x.1) i).pow_const 2).const_mul (eig i)).const_mul (1 / 2 : ℝ)
exact (hr.comp measurable_fst.fst).add_const (fieldEnergy c x.1)
let ν := fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.2
have : ∀ p, IsProbabilityMeasure (ν p) := fun p => by
change IsProbabilityMeasure (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.2)
infer_instance
exact (measurable_random_referencePartition (ν := ν)
((measurable_labeledSpinReference_general n (uniformSpinPrior N : Measure (Spin N))).comp measurable_fst.snd)
(hbase.add hY)).log
lemma measurable_tensorNamespacedLog {N m k : ℕ} (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) :
Measurable (tensorNamespacedLog eig c I degree amplitude n treeDegree h) := by
let v : Fin (n + 1) → SpinTensorIndex I degree → ℝ≥0 :=
fun i => tensorPathProfile I degree n treeDegree h i
have hY : Measurable (fun p : (((SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) ×
(Spin N × LabeledLeaf n)) =>
cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1.1) I degree amplitude n v p.2) p.1.2) := by
apply measurable_from_prod_countable_left
intro x
have hp : Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
(p.1.1, p.2)) := measurable_fst.fst.prodMk measurable_snd
convert (measurable_tensorNamespacedFields_joint I degree amplitude n v x).comp hp using 1
rfl
exact measurable_tensorJointLog eig c n _ hY
private lemma measurable_tensorJointFlatLog {N m k : ℕ} (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) :
Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
tensorFlatLog eig (specialRotation p.1.1) c I degree amplitude n
(tensorPathProfile I degree n treeDegree h) (p.1.2, p.2)) := by
let v : Fin (n + 1) → SpinTensorIndex I degree → ℝ≥0 :=
fun i => tensorPathProfile I degree n treeDegree h i
have hY : Measurable (fun p : (((SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) ×
(Spin N × LabeledLeaf n)) =>
cylinderField (tensorLeafCoefficients (specialRotation p.1.1.1) I degree amplitude n v p.2) p.1.2) := by
apply measurable_from_prod_countable_left
intro x
have hp : Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
(p.1.1, p.2)) := measurable_fst.fst.prodMk measurable_snd
convert (measurable_tensorLeafField_joint I degree amplitude n v x).comp hp using 1
rfl
exact measurable_tensorJointLog eig c n _ hY
def tensorNamespacedPressure {N m k : ℕ} (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ)
(p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) : ℝ :=
(N : ℝ)⁻¹ * tensorNamespacedLog eig c I degree amplitude n treeDegree h p
/-- The tensor pressure has the marked-cascade pressure law,
including the Haar rotation. This is a law identity, stronger than matching means. -/
theorem tensorNamespacedPressure_law {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (hb : CascadeExponents n b) :
HasLaw (tensorNamespacedPressure eig c I degree amplitude n treeDegree h)
((μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0))).map
(tensorDisorderPressure eig c I degree amplitude n))
((μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))).prod gaussianCoordinates) := by
let P := μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))
let F := fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
tensorFlatLog eig (specialRotation p.1.1) c I degree amplitude n
(tensorPathProfile I degree n treeDegree h) (p.1.2, p.2)
let G := tensorNamespacedLog eig c I degree amplitude n treeDegree h
have hF : Measurable F := measurable_tensorJointFlatLog eig c I degree amplitude n treeDegree h
have hG : Measurable G := measurable_tensorNamespacedLog eig c I degree amplitude n treeDegree h
have he : (P.prod gaussianCoordinates).map F = (P.prod gaussianCoordinates).map G :=
joint_map_eq_of_conditional_map_eq P gaussianCoordinates F G hF hG
(fun ω => tensorNamespacedLog_conditional_law eig c ω.1 I degree amplitude n ω.2 treeDegree h)
let S := fun x : ℝ => (N : ℝ)⁻¹ * x
have hS : Measurable S := measurable_id.const_mul _
have hes : (P.prod gaussianCoordinates).map (S ∘ F) = (P.prod gaussianCoordinates).map (S ∘ G) := by
rw [← Measure.map_map hS hF, ← Measure.map_map hS hG, he]
have hactual := (hasLaw_map (measurable_tensorDisorderPressure eig c I degree amplitude n).aemeasurable).comp
(tensorFlatToDisorder_measurePreserving μ I degree n b (tensorPathProfile I degree n treeDegree h)).hasLaw
have hflat := hactual.congr (tensorFlatPressure_eq_disorder_ae μ eig c I degree amplitude n b
(tensorPathProfile I degree n treeDegree h) hb)
refine ⟨(hG.const_mul (N : ℝ)⁻¹).aemeasurable, ?_⟩
exact hes.symm.trans hflat.map_eq
/-- Transport both the second moment and the exact variance and mean. -/
theorem tensorNamespacedPressure_moments {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (hb : CascadeExponents n b)
(hF : MemLp (tensorDisorderPressure eig c I degree amplitude n) 2
(μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0)))) :
let Q := (μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))).prod gaussianCoordinates
let R := μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0))
let F := tensorNamespacedPressure eig c I degree amplitude n treeDegree h
MemLp F 2 Q ∧ variance F Q = variance (tensorDisorderPressure eig c I degree amplitude n) R ∧
(∫ p, F p ∂Q) = ∫ p, tensorDisorderPressure eig c I degree amplitude n p ∂ R := by
intro Q R F
have hlaw := tensorNamespacedPressure_law μ eig c I degree amplitude n b treeDegree h hb
have hid : MemLp id 2 (R.map (tensorDisorderPressure eig c I degree amplitude n)) :=
(memLp_map_measure_iff aestronglyMeasurable_id hF.aemeasurable).mpr hF
refine ⟨hlaw.memLp hid, hlaw.variance_eq.trans (variance_id_map hF.aemeasurable), ?_⟩
exact hlaw.integral_eq.trans (integral_map hF.aemeasurable aestronglyMeasurable_id)
/-- Freezing one Gaussian namespace preserves the pressure law at every
value of that namespace's amplitude, including zero. -/
theorem tensorFrozenPressure_moments {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (hb : CascadeExponents n b)
(j : Fin k) (t : ℝ)
(hF : MemLp (tensorDisorderPressure eig c I degree (Function.update amplitude j t) n) 2
(μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0)))) :
let Q := (tensorFrozenLaw μ n b j).prod gaussianCoordinates
let R := μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0))
let F := fun p => (N : ℝ)⁻¹ * tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j t p
MemLp F 2 Q ∧
variance F Q = variance (tensorDisorderPressure eig c I degree (Function.update amplitude j t) n) R ∧
(∫ p, F p ∂Q) = ∫ p, tensorDisorderPressure eig c I degree (Function.update amplitude j t) n p ∂ R := by
intro Q R F
have hp := tensorFrozenInsertion_measurePreserving μ n b j
have hm := tensorNamespacedPressure_moments μ eig c I degree (Function.update amplitude j t)
n b treeDegree h hb hF
refine ⟨hm.1.comp_measurePreserving hp, ?_, ?_⟩
· exact (hp.variance_fun_comp hm.1.aemeasurable).trans hm.2.1
· exact (hp.hasLaw.integral_comp hm.1.aestronglyMeasurable).trans hm.2.2
lemma tensorFrozenCGF_eq_scaled_difference {N m k : ℕ} (hN : 0 < N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ)
(hh : Monotone h) (h0 : 0 ≤ h 0) (j : Fin k) (t : ℝ) :
tensorFrozenCGF eig c I degree amplitude n treeDegree h j t =ᵐ[
(tensorFrozenLaw μ n b j).prod gaussianCoordinates]
(fun p => (N : ℝ) *
((N : ℝ)⁻¹ * tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j t p -
(N : ℝ)⁻¹ * tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j 0 p)) := by
filter_upwards [tensorFrozenCGF_eq_log_difference μ eig c I degree amplitude n b treeDegree h hh h0 j t]
with p hp
rw [hp]
have hn : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hN)
field_simp
/-- The expected frozen CGF is the volume times the difference of the
two actual all-disorder pressures. -/
theorem tensorFrozenCGF_mean {N m k : ℕ} (hN : 0 < N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ)
(hh : Monotone h) (h0 : 0 ≤ h 0) (hb : CascadeExponents n b) (j : Fin k) (t : ℝ)
(hF : ∀ a : ℝ, a = t ∨ a = 0 →
MemLp (tensorDisorderPressure eig c I degree (Function.update amplitude j a) n) 2
(μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0)))) :
let Q := (tensorFrozenLaw μ n b j).prod gaussianCoordinates
let R := μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0))
(∫ p, tensorFrozenCGF eig c I degree amplitude n treeDegree h j t p ∂Q) =
(N : ℝ) * ((∫ p, tensorDisorderPressure eig c I degree (Function.update amplitude j t) n p ∂ R) -
∫ p, tensorDisorderPressure eig c I degree (Function.update amplitude j 0) n p ∂ R) := by
intro Q R
have hmt := tensorFrozenPressure_moments μ eig c I degree amplitude n b treeDegree h hb j t (hF t (Or.inl rfl))
have hmz := tensorFrozenPressure_moments μ eig c I degree amplitude n b treeDegree h hb j 0 (hF 0 (Or.inr rfl))
rw [integral_congr_ae (tensorFrozenCGF_eq_scaled_difference hN μ eig c I degree amplitude n b treeDegree h hh h0 j t),
integral_const_mul, integral_sub (hmt.1.integrable (by norm_num)) (hmz.1.integrable (by norm_num)),
hmt.2.2, hmz.2.2]
/-- The actual full-disorder pressure variance controls the centered
selected-field CGF. The two pressures may be dependent. -/
theorem tensorFrozenCGF_centered_L1_bound {N m k : ℕ} (hN : 0 < N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ)
(hh : Monotone h) (h0 : 0 ≤ h 0) (hb : CascadeExponents n b) (j : Fin k) (t B : ℝ)
(hF : ∀ a : ℝ, a = t ∨ a = 0 →
MemLp (tensorDisorderPressure eig c I degree (Function.update amplitude j a) n) 2
(μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0))))
(hv : ∀ a : ℝ, a = t ∨ a = 0 →
variance (tensorDisorderPressure eig c I degree (Function.update amplitude j a) n)
(μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0))) ≤ B) :
let Q := (tensorFrozenLaw μ n b j).prod gaussianCoordinates
let Z := tensorFrozenCGF eig c I degree amplitude n treeDegree h j t
MemLp Z 2 Q ∧ (∫ p, |Z p - ∫ p', Z p' ∂Q| ∂Q) ≤ 2 * N * Real.sqrt B := by
intro Q Z
let F := fun a p => (N : ℝ)⁻¹ * tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j a p
have ht : t = t ∨ t = 0 := Or.inl rfl
have hz : (0 : ℝ) = t ∨ (0 : ℝ) = 0 := Or.inr rfl
have hmt := tensorFrozenPressure_moments μ eig c I degree amplitude n b treeDegree h hb j t (hF t ht)
have hmz := tensorFrozenPressure_moments μ eig c I degree amplitude n b treeDegree h hb j 0 (hF 0 hz)
have he : Z =ᵐ[Q] fun p => (N : ℝ) * (F t p - F 0 p) := by
exact tensorFrozenCGF_eq_scaled_difference hN μ eig c I degree amplitude n b treeDegree h hh h0 j t
simpa only [abs_of_nonneg (show (0 : ℝ) ≤ N from Nat.cast_nonneg N)] using
centered_scaled_difference_L1_bound Q hmt.1 hmz.1
(hmt.2.1.trans_le (hv t ht)) (hmz.2.1.trans_le (hv 0 hz)) he
end InvariantIsing
end
end OAI
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