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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Probability.InvariantIsing.Arrays.NSpinTensorFlatMean
import OAI.Probability.InvariantIsing.Arrays.TensorGaussianMean
import OAI.Probability.InvariantIsing.Arrays.TensorProfiles
namespace OAI
/-! Rotation comparison for the actual disorder-averaged tensor cascade pressure. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
lemma measurable_tensorEnrichedPressure {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) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(root : SpinTensorIndex I degree → ℝ≥0) :
Measurable (fun U : SpecialOrthogonal N =>
tensorEnrichedPressure eig (specialRotation U) c I degree amplitude n b v root) :=
(measurable_tensorCascadeValue_joint eig c I degree amplitude n b v).stronglyMeasurable.integral_prod_right.measurable.const_mul (N : ℝ)⁻¹
/-- Gaussian covariance comparison transfers through the exact flat-model
mean identity to the backward recursion pressure itself. -/
theorem abs_tensorEnrichedPressure_sub_rotation_le {N m k : ℕ} (hN : 0 < N)
(eig c : Fin N → ℝ) (U V : SpecialOrthogonal N)
(K : ℝ) (hK : 0 ≤ K) (heig : ∀ 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) (hb : CascadeExponents n b)
(site : ℕ → ℝ≥0) (monomial : ℕ → Fin k → ℝ≥0) :
let v := fun i => tensorVarianceProfile I degree (site i) (monomial i)
|tensorEnrichedPressure eig (specialRotation U) c I degree amplitude n b (fun i => v (i + 1)) (v 0) -
tensorEnrichedPressure eig (specialRotation V) c I degree amplitude n b (fun i => v (i + 1)) (v 0)| ≤
(K + 2 * (N : ℝ)⁻¹ * (∑ i : Fin (n + 1), ∑ j, (monomial i j : ℝ) * amplitude j ^ 2 *
∑ a, (degree j a : ℝ))) * frobeniusDistance U V := by
intro v
let F := fun R T => ∫ g : ℕ → ℝ,
tensorFlatLog eig (specialRotation R) c I degree amplitude n v (T, g) ∂gaussianCoordinates
have hi (R : SpecialOrthogonal N) : Integrable (F R)
(labeledCascadeLaw n b : Measure (LabeledTree n)) :=
(tensorFlatLog_integrable eig (specialRotation R) c I degree amplitude n b v).integral_prod_left
let L := K * N + 2 * (∑ i : Fin (n + 1), ∑ j, (monomial i j : ℝ) * amplitude j ^ 2 *
∑ a, (degree j a : ℝ))
have hbound (T : LabeledTree n) : |F U T - F V T| ≤ L * frobeniusDistance U V := by
exact tensorLeafLogMean_abs_sub_le hN eig c U V K hK heig I degree amplitude n
(fun i => site i) (fun i => monomial i)
(labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) T)
have hm : |(∫ T, F U T ∂(labeledCascadeLaw n b : Measure (LabeledTree n))) -
∫ T, F V T ∂(labeledCascadeLaw n b : Measure (LabeledTree n))| ≤ L * frobeniusDistance U V := by
rw [← integral_sub (hi U) (hi V), ← Real.norm_eq_abs]
simpa only [probReal_univ, mul_one] using
norm_integral_le_of_norm_le_const (μ := (labeledCascadeLaw n b : Measure (LabeledTree n)))
(C := L * frobeniusDistance U V) (ae_of_all _ fun T => by
simpa only [Real.norm_eq_abs] using hbound T)
rw [tensorEnrichedPressure_eq_flat_mean hN eig (specialRotation U) c I degree amplitude n b v hb,
tensorEnrichedPressure_eq_flat_mean hN eig (specialRotation V) c I degree amplitude n b v hb,
← mul_sub, abs_mul, abs_of_nonneg (show 0 ≤ (N : ℝ)⁻¹ from inv_nonneg.mpr (Nat.cast_nonneg _))]
refine (mul_le_mul_of_nonneg_left hm (by positivity)).trans_eq ?_
have hn : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN.ne'
dsimp only [L]
field_simp
/-- For the manuscript's tree power increments, the rotation modulus
depends on the perturbation amplitudes and degrees, not the cascade depth. -/
theorem abs_tensorPathPressure_sub_rotation_le {N m k : ℕ} (hN : 0 < N)
(eig c : Fin N → ℝ) (U V : SpecialOrthogonal N)
(K : ℝ) (hK : 0 ≤ K) (heig : ∀ i, |eig i| ≤ K)
(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)
(b : ℕ → ℝ) (hb : CascadeExponents n b) :
let v := tensorPathProfile I degree n treeDegree h
|tensorEnrichedPressure eig (specialRotation U) c I degree amplitude n b (fun i => v (i + 1)) (v 0) -
tensorEnrichedPressure eig (specialRotation V) c I degree amplitude n b (fun i => v (i + 1)) (v 0)| ≤
(K + 2 * (N : ℝ)⁻¹ * (∑ j, amplitude j ^ 2 * ∑ a, (degree j a : ℝ))) * frobeniusDistance U V := by
intro v
have hc := abs_tensorEnrichedPressure_sub_rotation_le hN eig c U V K hK heig I degree amplitude n b hb
(varianceIncrement h) (fun i j => varianceIncrement (monomialPath n (treeDegree j)) i)
refine hc.trans ?_
apply mul_le_mul_of_nonneg_right ?_ (show 0 ≤ frobeniusDistance U V from Real.sqrt_nonneg _)
have hc := mul_le_mul_of_nonneg_left
(tensorPathProfile_rotation_cap degree amplitude n treeDegree)
(show 0 ≤ 2 * (N : ℝ)⁻¹ by positivity)
linarith
end InvariantIsing
end
end OAI
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