| import OAI.Probability.InvariantIsing.Arrays.TensorMinimumFluctuations |
| import OAI.Probability.InvariantIsing.Arrays.TensorGGMinimum |
| import OAI.Probability.InvariantIsing.Arrays.FullPerturbationVariance |
|
|
| namespace OAI |
|
|
| /-! For the complete perturbation, both coordinate minima force the |
| Ghirlanda–Guerra errors to vanish. -/ |
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| noncomputable section |
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|
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators |
|
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| namespace InvariantIsing |
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|
| def tensorPerturbationPressureMean {N m : ℕ} |
| (μ : Measure (SpecialOrthogonal N)) (eig c : Fin N → ℝ) |
| (I : Fin m → Finset (Fin N)) (u : Fin N → ℝ) (v : Fin m → ℝ) (t : ℝ) |
| (n : ℕ) (b h : ℕ → ℝ) : ℝ := |
| let degree := fun j : Fin N => enumeratedSpectralDegree m j |
| let treeDegree := fun j : Fin N => enumeratedTreeDegree m j |
| ∫ p, tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree |
| (tensorPerturbationAmplitude N u) n p |
| ∂(μ.prod (tensorRootTreeLaw I degree n b |
| (fun i => tensorPathProfile I degree n treeDegree h (i + 1)) |
| (tensorPathProfile I degree n treeDegree h 0))) |
|
|
| def diagonalMinimumCenter (N : ℕ) (w : ℝ) : ℝ := |
| let a := N * perturbationScale N |
| 2 * (N / a ^ 2) * (a * w - a * (3 / 2)) |
|
|
| /-- For the manuscript's full finite perturbation, genuine pressure-coordinate |
| minima give a GG residual bounded by two explicit rates tending to zero. -/ |
| theorem fullPerturbationPressure_minimumGG (hhaar : HaarConcentrationInput) |
| (hgauss : GaussianLipschitzVarianceInput) : |
| ∃ C : ℝ, 0 < C ∧ |
| ∀ N : ℕ, 3 ≤ N → |
| ∀ μ : Measure (SpecialOrthogonal N), IsProbabilityMeasure μ → μ.IsMulLeftInvariant → |
| ∀ m : ℕ, ∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) → |
| ∀ I : Fin m → Finset (Fin N), ∀ u : Fin N → ℝ, (∀ j, u j ∈ Set.Icc (1 : ℝ) 2) → |
| ∀ v : Fin m → ℝ, (∀ a, v a ∈ Set.Icc (1 : ℝ) 2) → ∀ t : ℝ, |t| ≤ 1 → |
| ∀ n : ℕ, ∀ b : ℕ → ℝ, CascadeExponents n b → |
| ∀ h : ℕ → ℝ, Monotone h → 0 ≤ h 0 → ∀ H : ℝ, h n ≤ H → |
| let M := tensorPerturbationPressureMean μ eig c I (n := n) (b := b) (h := h) |
| let degree := fun j : Fin N => enumeratedSpectralDegree m j |
| let treeDegree := fun j : Fin N => enumeratedTreeDegree m j |
| let L := 4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2 |
| ∂(rawCascadeLaw n b : Measure (RawTree n))) + H + 4 + C * (K + 4 * m + 8) ^ 2 |
| perturbationScale N ≤ 1 / 32 → contactStep N ≤ 1 / 4 → |
| (∀ (j : Fin N) (w : ℝ), w ∈ Set.Icc (1 : ℝ) 2 → |
| -M u v t + perturbationWeight j * (u j - 3 / 2) ^ 2 ≤ |
| -M (Function.update u j w) v t + perturbationWeight j * (w - 3 / 2) ^ 2) → |
| (∀ (a : Fin m) (w : ℝ), w ∈ Set.Icc (1 : ℝ) 2 → |
| -M u v t + (v a - 3 / 2) ^ 2 ≤ -M u (Function.update v a w) t + (w - 3 / 2) ^ 2) → |
| ∀ j : Fin N, ∀ q : ℕ, ∀ B : ℝ, 0 ≤ B → |
| ∀ D : SpecialOrthogonal N → (Fin (q + 1) → Spin N × LabeledLeaf n) → ℝ, |
| Measurable (Function.uncurry D) → (∀ U σ, |D U σ| ≤ B) → |
| |tensorSpectralGGResidual μ (diagonalPerturbedEigenvalues eig I v t) c I degree |
| (tensorPerturbationAmplitude N u) n b treeDegree h j D| ≤ |
| contactGGRate N j L B + |
| 2 * B * (∑ a, (degree j a : ℝ)) * diagonalContactRate N L := by |
| obtain ⟨C, hC, hvar⟩ := fullPerturbationPressure_variance hhaar hgauss |
| refine ⟨C, hC, ?_⟩ |
| intro N hN μ hμ hμinv m eig c K hK heig I u hu v hv t ht n b hb h hh h0 H hH |
| M degree treeDegree L he hs hminu hminv j q B hB D hD hDb |
| let : IsProbabilityMeasure μ := hμ |
| have hNp : 0 < N := by omega |
| have hua : ∀ j, |u j| ≤ 2 := fun j => abs_le.mpr ⟨by linarith [(hu j).1], (hu j).2⟩ |
| have hva : ∀ a, |v a| ≤ 2 := fun a => abs_le.mpr ⟨by linarith [(hv a).1], (hv a).2⟩ |
| let amplitude := tensorPerturbationAmplitude N u |
| 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)) |
| have hgu (w : ℝ) (hw : |w| ≤ 2) : |
| MemLp (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree |
| (Function.update amplitude j (perturbationAmplitude N j * w)) n) 2 R ∧ |
| variance (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree |
| (Function.update amplitude j (perturbationAmplitude N j * w)) n) R ≤ L / N := by |
| simpa only [tensorPerturbationAmplitude_update] using |
| hvar N hN μ hμ hμinv m eig c K hK heig I (Function.update u j w) |
| (update_abs_le_two hua j hw) v hva t ht n b hb h hh h0 H hH |
| have hgv (a : Fin m) (w : ℝ) (hw : |w| ≤ 2) : |
| MemLp (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I (Function.update v a w) t) |
| c I degree amplitude n) 2 R ∧ |
| variance (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I (Function.update v a w) t) |
| c I degree amplitude n) R ≤ L / N := by |
| exact hvar N hN μ hμ hμinv m eig c K hK heig I u hua (Function.update v a w) |
| (update_abs_le_two hva a hw) t ht n b hb h hh h0 H hH |
| have hdiag (a : Fin m) : tensorNamespacedObservableAverage μ (diagonalPerturbedEigenvalues eig I v t) |
| c I degree amplitude n b treeDegree h |
| (fun U x => |projectedOverlap (specialRotation U) (I a) x.1 x.1 - diagonalMinimumCenter N (v a)|) ≤ |
| diagonalContactRate N L := by |
| have hd := tensorDiagonal_fluctuation_at_minimum hNp μ eig c I degree amplitude n b treeDegree h |
| hh h0 hb v t a (v a) L (hv a) he hs (fun w hw => (hgv a w hw).1) |
| (fun w hw => (hgv a w hw).2) (by |
| dsimp only |
| intro w hw |
| simpa only [M, tensorPerturbationPressureMean, Function.update_eq_self] using hminv a w hw) |
| simpa only [Function.update_eq_self, diagonalMinimumCenter] using hd |
| have hgm : ∀ w ∈ Set.Icc (1 : ℝ) 2, |
| -tensorGaussianCoordinatePressure μ (diagonalPerturbedEigenvalues eig I v t) c I degree amplitude |
| n b treeDegree h j (u j) + perturbationWeight j * (u j - 3 / 2) ^ 2 ≤ |
| -tensorGaussianCoordinatePressure μ (diagonalPerturbedEigenvalues eig I v t) c I degree amplitude |
| n b treeDegree h j w + perturbationWeight j * (w - 3 / 2) ^ 2 := by |
| intro w hw |
| simpa only [amplitude, M, tensorPerturbationPressureMean, tensorGaussianCoordinatePressure, ← tensorPerturbationAmplitude_update, |
| Function.update_eq_self] using hminu j w hw |
| have hg := tensorGaussianGG_at_minimum hNp μ (diagonalPerturbedEigenvalues eig I v t) c I degree |
| amplitude n b treeDegree h hh h0 hb j (u j) L (hu j) (by linarith) hs rfl |
| (fun w hw => (hgu w hw).1) (fun w hw => (hgu w hw).2) hgm B hB D hD hDb |
| (fun a => diagonalMinimumCenter N (v a)) (fun _ => diagonalContactRate N L) hdiag |
| simpa only [← Finset.sum_mul, mul_assoc] using hg |
|
|
| end InvariantIsing |
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| end |
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| end OAI |
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