| import OAI.Probability.InvariantIsing.Arrays.NSpinTensorFrozen |
| import OAI.Probability.InvariantIsing.Pressure.RandomCoefficientIBP |
|
|
| namespace OAI |
|
|
| /-! Actual frozen exponential integrability and selected-field log partition identities. -/ |
|
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| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators NNReal |
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| namespace InvariantIsing |
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| lemma tensorNamespacedCoefficients_variance_eq {N m k : ℕ} (U : Rotation 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) : |
| (tensorNamespacedCoefficients U I degree amplitude n v x).sum (fun _ c => c ^ 2) = |
| (tensorLeafCoefficients U I degree amplitude n v x).sum (fun _ c => c ^ 2) := by |
| have hi : Function.Injective (fun i : Fin (n + 1) × SpinTensorIndex I degree => |
| tensorNamespaceTag I degree (i.1.1, (labeledAddress n x.2).take i.1, i.2)) := by |
| intro i j hij |
| apply treeFeatureTag_injective n x.2 |
| exact congrArg Encodable.encode ((tensorNamespaceTag_injective I degree) hij) |
| rw [tensorNamespacedCoefficients, featureCoefficients_variance _ hi, |
| tensorLeafCoefficients, featureCoefficients_variance _ (treeFeatureTag_injective n x.2)] |
|
|
| lemma tensorNamespacedPath_variance_le {N m k : ℕ} (U : Rotation N) |
| (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) (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (x : Spin N × LabeledLeaf n) : |
| (tensorNamespacedCoefficients U I degree amplitude n |
| (fun i => tensorPathProfile I degree n treeDegree h i) x).sum (fun _ c => c ^ 2) ≤ |
| h n * N + ∑ j, amplitude j ^ 2 := by |
| rw [tensorNamespacedCoefficients_variance_eq] |
| exact tensorPathProfile_variance_cap U I degree amplitude n treeDegree h hh h0 x |
|
|
| private lemma measurable_tensorBaseEnergy {N : ℕ} (eig c : Fin N → ℝ) (n : ℕ) : |
| Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (Spin N × LabeledLeaf n) => |
| rotatedEnergy eig (specialRotation p.1.1) p.2.1 + fieldEnergy c p.2.1) := by |
| apply measurable_from_prod_countable_left |
| intro x |
| have hrot : 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 (hrot.comp measurable_fst).add_const (fieldEnergy c x.1) |
|
|
| private lemma tensorBase_exp_integrable {N : ℕ} (eig c : Fin N → ℝ) (n : ℕ) |
| (ω : SpecialOrthogonal N × LabeledTree n) : |
| Integrable (fun x : Spin N × LabeledLeaf n => |
| Real.exp (rotatedEnergy eig (specialRotation ω.1) x.1 + fieldEnergy c x.1)) |
| (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.2) := |
| (Integrable.of_finite : Integrable |
| (fun σ : Spin N => Real.exp (rotatedEnergy eig (specialRotation ω.1) σ + fieldEnergy c σ)) |
| (uniformSpinPrior N : Measure (Spin N))).comp_fst (labeledLeafLaw n ω.2) |
|
|
| private lemma measurable_namespacedPathField {N m k : ℕ} |
| (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)) × |
| (Spin N × LabeledLeaf n) => |
| cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1.1) I degree amplitude n |
| (fun i => tensorPathProfile I degree n treeDegree h i) p.2) p.1.2) := by |
| apply measurable_from_prod_countable_left |
| intro x |
| let v : Fin (n + 1) → SpinTensorIndex I degree → ℝ≥0 := |
| fun i => tensorPathProfile I degree n treeDegree h i |
| have hm : Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) => |
| (p.1.1, p.2)) := measurable_fst.fst.prodMk measurable_snd |
| have hf := (measurable_tensorNamespacedFields_joint I degree amplitude n v x).comp hm |
| convert hf using 1 |
| rfl |
|
|
| private lemma measurable_frozenSelectedField {N m k : ℕ} |
| (I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) |
| (n : ℕ) (treeDegree : Fin k → ℕ) (j : Fin k) : |
| Measurable (fun p : (TensorFrozenData N n j × (ℕ → ℝ)) × (Spin N × LabeledLeaf n) => |
| cylinderField (jointSpectralMonomialCoefficients |
| (specialRotation p.1.1.1.1) I (degree j) n (treeDegree j) p.2) p.1.2) := by |
| apply measurable_from_prod_countable_left |
| intro x |
| have hp : Measurable (fun p : TensorFrozenData N n j × (ℕ → ℝ) => (p.1.1.1, p.2)) := |
| measurable_fst.fst.fst.prodMk measurable_snd |
| have hf := (measurable_jointSpectralMonomialField I (degree j) n (treeDegree j) x).comp hp |
| convert hf using 1 |
| rfl |
|
|
| /-- Exponentials of the tensor Hamiltonian are integrable for |
| almost every joint rotation/tree/Gaussian disorder. No concentration or |
| limiting-pressure hypothesis is used. -/ |
| theorem tensorNamespaced_exp_integrable_ae {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) (hh : Monotone h) (h0 : 0 ≤ h 0) : |
| ∀ᵐ p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) |
| ∂(μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))).prod gaussianCoordinates, |
| Integrable (fun x : Spin N × LabeledLeaf n => Real.exp |
| (rotatedEnergy eig (specialRotation p.1.1) x.1 + fieldEnergy c x.1 + |
| cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1) I degree amplitude n |
| (fun i => tensorPathProfile I degree n treeDegree h i) x) p.2)) |
| (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.2) := by |
| let Ω := SpecialOrthogonal N × LabeledTree n |
| let ν := fun ω : Ω => labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.2 |
| let H := fun (ω : Ω) (x : Spin N × LabeledLeaf n) => |
| rotatedEnergy eig (specialRotation ω.1) x.1 + fieldEnergy c x.1 |
| let A := fun (ω : Ω) => tensorNamespacedCoefficients (specialRotation ω.1) I degree amplitude n |
| (fun i => tensorPathProfile I degree n treeDegree h i) |
| have : ∀ ω, IsProbabilityMeasure (ν ω) := fun ω => by |
| change IsProbabilityMeasure (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.2) |
| infer_instance |
| have hν : Measurable ν := |
| (measurable_labeledSpinReference_general n (uniformSpinPrior N : Measure (Spin N))).comp measurable_snd |
| have hbase (ω : Ω) : Integrable (fun x => Real.exp (H ω x)) (ν ω) := tensorBase_exp_integrable eig c n ω |
| have : ∀ ω, IsProbabilityMeasure ((ν ω).tilted (H ω)) := fun ω => isProbabilityMeasure_tilted (hbase ω) |
| have htilt : Measurable (fun ω => (ν ω).tilted (H ω)) := |
| measurable_random_tilted_measure (H := Function.uncurry H) hν (measurable_tensorBaseEnergy eig c n) |
| have hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × (Spin N × LabeledLeaf n) => |
| cylinderField (A p.1.1 p.2) p.1.2) := measurable_namespacedPathField I degree amplitude n treeDegree h |
| have hA (ω : Ω) (x : Spin N × LabeledLeaf n) : |
| (A ω x).sum (fun _ c => c ^ 2) ≤ h n * N + ∑ j, amplitude j ^ 2 := |
| tensorNamespacedPath_variance_le (specialRotation ω.1) I degree amplitude n treeDegree h hh h0 x |
| have he := randomCoefficient_all_exp_ae |
| (P := μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))) htilt A hAm hA |
| filter_upwards [he] with p hp |
| have hi := hp 1 |
| rw [integrable_tilted_iff (hbase p.1)] at hi |
| simpa only [H, A, ν, one_mul, smul_eq_mul, ← Real.exp_add] using hi |
|
|
| /-- The frozen base energy is the actual Hamiltonian with the selected |
| amplitude set to zero, and is exponentially integrable almost surely. -/ |
| theorem tensorFrozenBase_exp_integrable_ae {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) (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (j : Fin k) : |
| ∀ᵐ ω : TensorFrozenData N n j ∂tensorFrozenLaw μ n b j, |
| Integrable (fun x => Real.exp (tensorFrozenBaseEnergy eig c I degree amplitude n treeDegree h j ω x)) |
| (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.1.2) := by |
| have hp := tensorFrozenInsertion_measurePreserving μ n b j |
| have he := hp.quasiMeasurePreserving.ae |
| (tensorNamespaced_exp_integrable_ae μ eig c I degree (Function.update amplitude j 0) |
| n b treeDegree h hh h0) |
| have he' : ∀ᵐ p : TensorFrozenData N n j × (ℕ → ℝ) |
| ∂(tensorFrozenLaw μ n b j).prod gaussianCoordinates, |
| Integrable (fun x => Real.exp (tensorFrozenBaseEnergy eig c I degree amplitude n treeDegree h j p.1 x)) |
| (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.1.2) := by |
| filter_upwards [he] with p hp |
| convert hp using 1 |
| funext x |
| congr 1 |
| dsimp only [tensorFrozenBaseEnergy] |
| rw [tensorNamespacedField_frozen (specialRotation p.1.1.1) I degree amplitude n treeDegree h hh h0 j x p.2 p.1.2] |
| filter_upwards [Measure.ae_ae_of_ae_prod he'] with ω hω |
| obtain ⟨z, hz⟩ := hω.exists |
| exact hz |
|
|
|
|
| def 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) |
| (p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) : ℝ := |
| Real.log (∫ x : Spin N × LabeledLeaf n, Real.exp |
| (rotatedEnergy eig (specialRotation p.1.1) x.1 + fieldEnergy c x.1 + |
| cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1) I degree amplitude n |
| (fun i => tensorPathProfile I degree n treeDegree h i) x) p.2) |
| ∂labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.2) |
|
|
| def tensorFrozenPartitionLog {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) (j : Fin k) (t : ℝ) |
| (p : TensorFrozenData N n j × (ℕ → ℝ)) : ℝ := |
| tensorNamespacedLog eig c I degree (Function.update amplitude j t) n treeDegree h |
| (p.1.1, gaussianNamespaceJoin (j + 1) (p.2, p.1.2)) |
|
|
| def tensorFrozenCGF {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) (j : Fin k) (t : ℝ) |
| (p : TensorFrozenData N n j × (ℕ → ℝ)) : ℝ := |
| cgf (fun x => cylinderField (jointSpectralMonomialCoefficients |
| (specialRotation p.1.1.1) I (degree j) n (treeDegree j) x) p.2) |
| (tensorFrozenReference eig c I degree amplitude n treeDegree h j p.1) t |
|
|
| lemma measurable_tensorFrozenCGF {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) (j : Fin k) (t : ℝ) : |
| Measurable (tensorFrozenCGF eig c I degree amplitude n treeDegree h j t) := by |
| let A := fun (ω : TensorFrozenData N n j) => |
| jointSpectralMonomialCoefficients (specialRotation ω.1.1) I (degree j) n (treeDegree j) |
| let ν := tensorFrozenReference eig c I degree amplitude n treeDegree h j |
| have hm : Measurable (fun p : (TensorFrozenData N n j × (ℕ → ℝ)) × (Spin N × LabeledLeaf n) => |
| cylinderField (A p.1.1 p.2) p.1.2) := measurable_frozenSelectedField I degree n treeDegree j |
| have hf := measurable_randomCoefficient_cgf (ν := ν) |
| (measurable_tensorFrozenReference eig c I degree amplitude n treeDegree h j) A hm t |
| convert hf using 1 |
| rfl |
|
|
| /-- The selected-field CGF is an actual square-integrable observable on |
| the frozen external disorder and independent Gaussian factor. -/ |
| theorem tensorFrozenCGF_memLp_two {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) (j : Fin k) (t : ℝ) : |
| MemLp (tensorFrozenCGF eig c I degree amplitude n treeDegree h j t) 2 |
| ((tensorFrozenLaw μ n b j).prod gaussianCoordinates) := by |
| have hm := measurable_tensorFrozenCGF eig c I degree amplitude n treeDegree h j t |
| apply (memLp_two_iff_integrable_sq hm.aestronglyMeasurable).mpr |
| exact joint_integrable_square_of_uniform_conditionals |
| (P := tensorFrozenLaw μ n b j) hm |
| (fun ω => (cylinder_cgf_second |
| (tensorFrozenReference eig c I degree amplitude n treeDegree h j ω) |
| (jointSpectralMonomialCoefficients (specialRotation ω.1.1) I (degree j) n (treeDegree j)) |
| (jointSpectralMonomialCoefficients_variance_le_one (specialRotation ω.1.1) I (degree j) n (treeDegree j)) t).1) |
| (fun ω => (cylinder_cgf_second |
| (tensorFrozenReference eig c I degree amplitude n treeDegree h j ω) |
| (jointSpectralMonomialCoefficients (specialRotation ω.1.1) I (degree j) n (treeDegree j)) |
| (jointSpectralMonomialCoefficients_variance_le_one (specialRotation ω.1.1) I (degree j) n (treeDegree j)) t).2) |
|
|
| /-- The CGF is exactly the difference of the actual log partitions at |
| selected amplitude `t` and selected amplitude zero, almost surely. -/ |
| theorem tensorFrozenCGF_eq_log_difference {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) (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 => tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j t p - |
| tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j 0 p) := by |
| have hp := tensorFrozenInsertion_measurePreserving μ n b j |
| have hbase := (measurePreserving_fst (μ := tensorFrozenLaw μ n b j) (ν := gaussianCoordinates)).quasiMeasurePreserving.ae |
| (tensorFrozenBase_exp_integrable_ae μ eig c I degree amplitude n b treeDegree h hh h0 j) |
| have hfull := hp.quasiMeasurePreserving.ae |
| (tensorNamespaced_exp_integrable_ae μ eig c I degree (Function.update amplitude j t) |
| n b treeDegree h hh h0) |
| filter_upwards [hbase, hfull] with p hb hf |
| let ν := labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.1.2 |
| let H := tensorFrozenBaseEnergy eig c I degree amplitude n treeDegree h j p.1 |
| let Y := fun x : Spin N × LabeledLeaf n => cylinderField (jointSpectralMonomialCoefficients |
| (specialRotation p.1.1.1) I (degree j) n (treeDegree j) x) p.2 |
| have henergy (a : ℝ) (x : Spin N × LabeledLeaf n) : |
| rotatedEnergy eig (specialRotation p.1.1.1) x.1 + fieldEnergy c x.1 + |
| cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1.1) I degree |
| (Function.update amplitude j a) n (fun i => tensorPathProfile I degree n treeDegree h i) x) |
| (gaussianNamespaceJoin (j + 1) (p.2, p.1.2)) = H x + a * Y x := by |
| have he := tensorFrozenEnergy_insert eig c I degree (Function.update amplitude j a) |
| n treeDegree h hh h0 j p.1 p.2 x |
| simpa only [tensorFrozenBaseEnergy, Function.update_idem, Function.update_self, H, Y] using he |
| have hpart (a : ℝ) : tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j a p = |
| Real.log (∫ x, Real.exp (H x + a * Y x) ∂ν) := by |
| apply congrArg Real.log |
| apply integral_congr_ae |
| exact ae_of_all _ fun x => congrArg Real.exp (henergy a x) |
| have hi : Integrable (fun x => Real.exp (H x + t * Y x)) ν := by |
| convert hf using 1 |
| funext x |
| exact congrArg Real.exp (henergy t x).symm |
| have hz : 0 < ∫ x, Real.exp (H x) ∂ν := integral_exp_pos hb |
| have hz' : 0 < ∫ x, Real.exp (H x + t * Y x) ∂ν := integral_exp_pos hi |
| rw [hpart t, hpart 0] |
| simp only [zero_mul, add_zero] |
| change Real.log (∫ x, Real.exp (t * Y x) ∂gibbsProbability ν H) = _ |
| rw [gibbsProbability_eq_tilted ν H hb, integral_exp_tilted H (fun x => t * Y x)] |
| change Real.log ((∫ x, Real.exp (H x + t * Y x) ∂ν) / ∫ x, Real.exp (H x) ∂ν) = _ |
| exact Real.log_div hz'.ne' hz.ne' |
|
|
| end InvariantIsing |
|
|
| end |
|
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| end OAI |
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|
