| import OAI.Probability.InvariantIsing.Arrays.TensorPerturbationCost |
| import OAI.Probability.InvariantIsing.Spectral.SpectralDiagonalPerturbation |
|
|
| namespace OAI |
|
|
| /-! Uniform vanishing cost of both terms of the manuscript perturbation. -/ |
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| noncomputable section |
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|
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators NNReal |
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| namespace InvariantIsing |
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|
| private lemma rotatedEnergy_mul {N : ℕ} (eig : Fin N → ℝ) (U : Rotation N) (t : ℝ) (σ : Spin N) : |
| rotatedEnergy (fun i => t * eig i) U σ = t * rotatedEnergy eig U σ := by |
| simp only [rotatedEnergy, mul_assoc, ← Finset.mul_sum] |
| ring |
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| lemma abs_tensorPathPressure_sub_base_le {N m k : ℕ} (hN : 0 < N) |
| (eig eig' c : Fin N → ℝ) (U : Rotation N) |
| (I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (a : Fin k → ℝ) |
| (n : ℕ) (r : Fin k → ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (h : ℕ → ℝ) (hh : Monotone h) (h0 : 0 ≤ h 0) (C : ℝ) |
| (hE : ∀ σ, |rotatedEnergy eig U σ - rotatedEnergy eig' U σ| ≤ C) : |
| let v := tensorPathProfile I degree n r h |
| |tensorEnrichedPressure eig U c I degree a n b (fun i => v (i + 1)) (v 0) - |
| tensorEnrichedPressure eig' U c I degree a n b (fun i => v (i + 1)) (v 0)| ≤ |
| (N : ℝ)⁻¹ * C := by |
| intro v |
| have hA := tensorFlatLog_integrable eig U c I degree a n b v |
| have hB := tensorFlatLog_integrable eig' U c I degree a n b v |
| have hpoint (T : LabeledTree n) : |
| |(∫ g, tensorFlatLog eig U c I degree a n v (T, g) ∂gaussianCoordinates) - |
| ∫ g, tensorFlatLog eig' U c I degree a n v (T, g) ∂gaussianCoordinates| ≤ C := by |
| let ν := labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) T |
| have he := countable_cylinder_log_mean_base_compare ν |
| (fun x => rotatedEnergy eig U x.1 + fieldEnergy c x.1) |
| (fun x => rotatedEnergy eig' U x.1 + fieldEnergy c x.1) |
| (finite_spin_base_exp_integrable ν (fun σ => rotatedEnergy eig U σ + fieldEnergy c σ)) |
| (finite_spin_base_exp_integrable ν (fun σ => rotatedEnergy eig' U σ + fieldEnergy c σ)) |
| (tensorLeafCoefficients U I degree a n (fun i => v i)) |
| (tensorPathProfile_variance_cap U I degree a n r h hh h0) |
| (fun x => by simpa only [add_sub_add_right_eq_sub] using hE x.1) |
| exact he |
| have hi := norm_integral_le_of_norm_le_const |
| (μ := (labeledCascadeLaw n b : Measure (LabeledTree n))) |
| (f := fun T => (∫ g, tensorFlatLog eig U c I degree a n v (T, g) ∂gaussianCoordinates) - |
| ∫ g, tensorFlatLog eig' U c I degree a n v (T, g) ∂gaussianCoordinates) |
| (C := C) (ae_of_all _ fun T => by simpa only [Real.norm_eq_abs] using hpoint T) |
| rw [tensorEnrichedPressure_eq_flat_mean hN eig U c I degree a n b v hb, |
| tensorEnrichedPressure_eq_flat_mean hN eig' U c I degree a n b v hb, |
| ← mul_sub, ← integral_sub hA.integral_prod_left hB.integral_prod_left, abs_mul, |
| abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg N))] |
| apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr (Nat.cast_nonneg N)) |
| simpa only [Real.norm_eq_abs, probReal_univ, mul_one] using hi |
|
|
| / |
| `2 m e_N + 8 e_N²`, uniformly in the other deterministic parameters. -/ |
| theorem fullPerturbationPressure_cost {N m : ℕ} (hN : 0 < N) |
| (eig c : Fin N → ℝ) (U : Rotation N) (I : Fin m → Finset (Fin N)) |
| (degree : Fin N → Fin m → ℕ) (u : Fin N → ℝ) (hu : ∀ j, |u j| ≤ 2) |
| (z : Fin m → ℝ) (hz : ∀ a, |z a| ≤ 2) (t : ℝ) |
| (n : ℕ) (r : Fin N → ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (h : ℕ → ℝ) (hh : Monotone h) (h0 : 0 ≤ h 0) : |
| let v := tensorPathProfile I degree n r h |
| |tensorEnrichedPressure (diagonalPerturbedEigenvalues eig I z t) U c I degree |
| (tensorPerturbationAmplitude N u) n b (fun i => v (i + 1)) (v 0) - |
| tensorEnrichedPressure (fun i => t * eig i) U c I degree 0 n b |
| (fun i => v (i + 1)) (v 0)| ≤ |
| 2 * m * perturbationScale N + 8 * perturbationScale N ^ 2 := by |
| intro v |
| have hg := tensorPerturbationPressure_cost hN (diagonalPerturbedEigenvalues eig I z t) c U I |
| degree u hu n r b hb h hh h0 |
| have hd := abs_tensorPathPressure_sub_base_le hN (diagonalPerturbedEigenvalues eig I z t) |
| (fun i => t * eig i) c U I degree 0 n r b hb h hh h0 |
| (2 * m * N * perturbationScale N) (fun σ => by |
| rw [rotatedEnergy_mul] |
| exact abs_rotatedEnergy_diagonalPerturbation_le hN eig U I z hz t σ) |
| have hn : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN.ne' |
| have he : (N : ℝ)⁻¹ * (2 * m * N * perturbationScale N) = 2 * m * perturbationScale N := by |
| field_simp |
| rw [he] at hd |
| exact (abs_sub_le _ _ _).trans ((add_le_add hg hd).trans_eq (by ring)) |
|
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| lemma fullPerturbationCost_tendsto (m : ℕ) : |
| Filter.Tendsto (fun N => 2 * m * perturbationScale N + 8 * perturbationScale N ^ 2) |
| Filter.atTop (nhds (0 : ℝ)) := by |
| simpa only [mul_zero, zero_pow (by norm_num : 2 ≠ 0), add_zero] using |
| (perturbationScale_tendsto.const_mul (2 * (m : ℝ))).add |
| ((perturbationScale_tendsto.pow 2).const_mul 8) |
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|
| end InvariantIsing |
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| end |
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| end OAI |
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