openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Arrays /NSpinTensorHaarFluctuation.lean
| import OAI.Probability.InvariantIsing.Arrays.NSpinTensorHaar | |
| import OAI.Probability.InvariantIsing.Arrays.NSpinTensorFluctuation | |
| namespace OAI | |
| /-! Full Haar, Gaussian-root, and cascade fluctuations for the actual finite tensor model. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped BigOperators NNReal | |
| namespace InvariantIsing | |
| /-- Independent common Gaussian root and marked cascade. -/ | |
| def tensorRootTreeLaw {N m k : ℕ} (I : Fin m → Finset (Fin N)) | |
| (degree : Fin k → Fin m → ℕ) (n : ℕ) (b : ℕ → ℝ) | |
| (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (root : SpinTensorIndex I degree → ℝ≥0) : | |
| Measure ((SpinTensorIndex I degree → ℝ) × NoiseTree (SpinTensorIndex I degree → ℝ) n) := | |
| (tensorGaussianLaw I degree root : Measure (SpinTensorIndex I degree → ℝ)).prod | |
| (tensorCascadeLaw I degree n b v) | |
| instance tensorRootTreeLaw_probability {N m k : ℕ} (I : Fin m → Finset (Fin N)) | |
| (degree : Fin k → Fin m → ℕ) (n : ℕ) (b : ℕ → ℝ) | |
| (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (root : SpinTensorIndex I degree → ℝ≥0) : | |
| IsProbabilityMeasure (tensorRootTreeLaw I degree n b v root) := by | |
| let : IsProbabilityMeasure (tensorCascadeLaw I degree n b v) := | |
| (noiseCascadeLaw (SpinTensorIndex I degree → ℝ) n b | |
| (fun i => tensorGaussianLaw I degree (v i))).prop | |
| unfold tensorRootTreeLaw | |
| infer_instance | |
| /-- The normalized spin/leaf log partition retaining all three disorders. -/ | |
| def tensorDisorderPressure {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) | |
| (p : SpecialOrthogonal N × ((SpinTensorIndex I degree → ℝ) × | |
| NoiseTree (SpinTensorIndex I degree → ℝ) n)) : ℝ := | |
| (N : ℝ)⁻¹ * tensorCascadeLog eig (specialRotation p.1) c I degree amplitude n p.2.1 p.2.2 | |
| private lemma measurable_tensorLeafState {N m k : ℕ} (I : Fin m → Finset (Fin N)) | |
| (degree : Fin k → Fin m → ℕ) (n : ℕ) : | |
| Measurable (fun p : (SpinTensorIndex I degree → ℝ) × NoiseLeaf (SpinTensorIndex I degree → ℝ) n => | |
| tensorLeafState I degree n p.1 p.2) := by | |
| induction n with | |
| | zero => exact measurable_fst | |
| | succ n ih => | |
| exact ih.comp ((measurable_fst.add measurable_snd.snd.fst).prodMk measurable_snd.snd.snd) | |
| lemma measurable_tensorDisorderPressure {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) : | |
| Measurable (tensorDisorderPressure eig c I degree amplitude n) := by | |
| let Ω := SpecialOrthogonal N × ((SpinTensorIndex I degree → ℝ) × | |
| NoiseTree (SpinTensorIndex I degree → ℝ) n) | |
| have hs : Measurable (fun p : Ω × NoiseLeaf (SpinTensorIndex I degree → ℝ) n => | |
| (p.1.1, tensorLeafState I degree n p.1.2.1 p.2)) := | |
| measurable_fst.fst.prodMk ((measurable_tensorLeafState I degree n).comp | |
| (measurable_fst.snd.fst.prodMk measurable_snd)) | |
| have ht := ((measurable_spinTensorTerminal_joint eig c I degree amplitude).comp hs).exp | |
| have hi := (ht.stronglyMeasurable.integral_kernel_prod_right' | |
| (κ := (noiseLeafKernel (SpinTensorIndex I degree → ℝ) n).comap | |
| (fun p : Ω => p.2.2) measurable_snd.snd)).measurable.log | |
| have hlog : Measurable (fun p : Ω => tensorCascadeLog eig (specialRotation p.1) c | |
| I degree amplitude n p.2.1 p.2.2) := by | |
| simpa only [tensorCascadeLog, noiseLeafTerminal_spinTensorTerminal, Function.comp_apply, | |
| Kernel.comap_apply] using hi | |
| exact hlog.const_mul (N : ℝ)⁻¹ | |
| lemma tensorDisorderPressure_conditional_mean (hpub : GaussianLipschitzVarianceInput) | |
| {N m k : ℕ} (hN : 0 < 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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) | |
| (hb : CascadeExponents n b) (site : ℝ≥0) (monomial : Fin k → ℝ≥0) : | |
| (∫ q, tensorDisorderPressure eig c I degree amplitude n (U, q) | |
| ∂tensorRootTreeLaw I degree n b v (tensorVarianceProfile I degree site monomial)) = | |
| tensorEnrichedPressure eig (specialRotation U) c I degree amplitude n b v | |
| (tensorVarianceProfile I degree site monomial) := by | |
| let : IsProbabilityMeasure (tensorCascadeLaw I degree n b v) := | |
| (noiseCascadeLaw (SpinTensorIndex I degree → ℝ) n b | |
| (fun i => tensorGaussianLaw I degree (v i))).prop | |
| have h := tensorCascadeLog_root_tree_variance_le hpub hN eig (specialRotation U) c | |
| I degree amplitude n b v hb site monomial | |
| unfold tensorDisorderPressure tensorEnrichedPressure tensorRootTreeLaw | |
| rw [integral_const_mul, integral_prod _ (h.1.integrable (by norm_num))] | |
| congr 1 | |
| apply integral_congr_ae | |
| exact ae_of_all _ fun z => | |
| (tensorCascadeLog_recursion hN eig (specialRotation U) c I degree amplitude n b v hb z).2.1 | |
| /-- The full pressure variance is the sum of the root/cascade conditional | |
| contribution and the Haar variance of its conditional mean. -/ | |
| theorem tensorDisorderPressure_variance (hhaar : HaarConcentrationInput) | |
| (hgauss : GaussianLipschitzVarianceInput) : | |
| ∃ C : ℝ, 0 < C ∧ | |
| ∀ N : ℕ, 3 ≤ N → | |
| ∀ μ : Measure (SpecialOrthogonal N), IsProbabilityMeasure μ → μ.IsMulLeftInvariant → | |
| ∀ m k : ℕ, ∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) → | |
| ∀ I : Fin m → Finset (Fin N), ∀ degree : Fin k → Fin m → ℕ, | |
| ∀ amplitude : Fin k → ℝ, ∀ n : ℕ, ∀ b : ℕ → ℝ, CascadeExponents n b → | |
| ∀ site : ℕ → ℝ≥0, ∀ monomial : ℕ → Fin k → ℝ≥0, | |
| let v := fun i => tensorVarianceProfile I degree (site i) (monomial i) | |
| let P := tensorRootTreeLaw I degree n b (fun i => v (i + 1)) (v 0) | |
| let F := tensorDisorderPressure eig c I degree amplitude n | |
| let L := K + 2 * (N : ℝ)⁻¹ * | |
| (∑ i : Fin (n + 1), ∑ j, (monomial i j : ℝ) * amplitude j ^ 2 * ∑ a, (degree j a : ℝ)) | |
| MemLp F 2 (μ.prod P) ∧ variance F (μ.prod P) ≤ | |
| (4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2 | |
| ∂(rawCascadeLaw n b : Measure (RawTree n))) + | |
| (site 0 : ℝ) * N + ∑ j, (monomial 0 j : ℝ) * amplitude j ^ 2) / (N : ℝ) ^ 2 + | |
| C * L ^ 2 / N := by | |
| obtain ⟨C, hC, hHaar⟩ := haar_tensorEnrichedPressure_variance hhaar | |
| refine ⟨C, hC, ?_⟩ | |
| intro N hN μ hμ hμinv m k eig c K hK heig I degree amplitude n b hb site monomial v P F L | |
| let : IsProbabilityMeasure μ := hμ | |
| have hNm : 0 < N := by omega | |
| let X := fun U : SpecialOrthogonal N => tensorEnrichedPressure eig (specialRotation U) c | |
| I degree amplitude n b (fun i => v (i + 1)) (v 0) | |
| have hX := hHaar N hN μ hμ hμinv m k eig c K hK heig I degree amplitude n b hb site monomial | |
| have hsec (U : SpecialOrthogonal N) : MemLp (fun q => F (U, q)) 2 P ∧ | |
| variance (fun q => F (U, q)) P ≤ | |
| (4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2 | |
| ∂(rawCascadeLaw n b : Measure (RawTree n))) + | |
| (site 0 : ℝ) * N + ∑ j, (monomial 0 j : ℝ) * amplitude j ^ 2) / (N : ℝ) ^ 2 := | |
| normalized_tensorCascadeLog_root_tree_variance_le hgauss hNm eig (specialRotation U) c | |
| I degree amplitude n b (fun i => v (i + 1)) hb (site 0) (monomial 0) | |
| have hc (U : SpecialOrthogonal N) : (∫ q, F (U, q) ∂P) = X U := | |
| tensorDisorderPressure_conditional_mean hgauss hNm eig c U I degree amplitude n b | |
| (fun i => v (i + 1)) hb (site 0) (monomial 0) | |
| have hi (U : SpecialOrthogonal N) : Integrable (fun q => (F (U, q) - X U) ^ 2) P := | |
| ((hsec U).1.sub (memLp_const (X U))).integrable_sq | |
| have hbnd (U : SpecialOrthogonal N) : (∫ q, (F (U, q) - X U) ^ 2 ∂P) ≤ | |
| (4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2 | |
| ∂(rawCascadeLaw n b : Measure (RawTree n))) + | |
| (site 0 : ℝ) * N + ∑ j, (monomial 0 j : ℝ) * amplitude j ^ 2) / (N : ℝ) ^ 2 := by | |
| have he := variance_eq_integral (hsec U).1.aemeasurable | |
| rw [hc U] at he | |
| rw [← he] | |
| exact (hsec U).2 | |
| have h := variance_of_conditional_center μ P | |
| (measurable_tensorDisorderPressure eig c I degree amplitude n) hX.1 hi hc hbnd | |
| exact ⟨h.1, h.2.trans (add_le_add_right hX.2 _)⟩ | |
| end InvariantIsing | |
| end | |
| end OAI | |