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a89d282 | 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 | import OAI.Probability.InvariantIsing.Haar.HaarExponentialPolynomial
import OAI.Probability.InvariantIsing.Haar.PositiveExponentialSum
namespace OAI
/-! Uniform logarithmic-gradient bounds for positive exponential truncations. -/
noncomputable section
open Matrix MvPolynomial
open scoped BigOperators
namespace InvariantIsing
lemma haarExpPolynomial_gamma {N : ℕ} (n : ℕ) (p : MatrixPolynomial N) :
haarPolynomialGamma (haarExpPolynomial (n+1) p) (haarExpPolynomial (n+1) p) =
(haarExpPolynomial n p)^2*haarPolynomialGamma p p := by
simp only [haarPolynomialGamma,haarExpPolynomial_derivation,Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
lemma haarExpPolynomial_positive {N : ℕ} (n : ℕ) (p : MatrixPolynomial N)
(M : Matrix (Fin N) (Fin N) ℝ) (hp : 0 ≤ matrixPolynomialEval M p) :
1 ≤ matrixPolynomialEval M (haarExpPolynomial n p) := by
rw [haarExpPolynomial_eval]
exact positiveExpSum_one_le n hp
lemma haarExpPolynomial_log_gradient_bound {N : ℕ} (n : ℕ) (p : MatrixPolynomial N)
(M : Matrix (Fin N) (Fin N) ℝ) (hp : 0 ≤ matrixPolynomialEval M p)
(C : ℝ) (hC : 0 ≤ C) (hG : matrixPolynomialEval M (haarPolynomialGamma p p) ≤ C) :
matrixPolynomialEval M (haarPolynomialGamma (haarExpPolynomial (n+1) p) (haarExpPolynomial (n+1) p)) ≤
C*(matrixPolynomialEval M (haarExpPolynomial (n+1) p))^2 := by
have h0 := haarExpPolynomial_positive n p M hp
have h1 := haarExpPolynomial_positive (n+1) p M hp
have hm : matrixPolynomialEval M (haarExpPolynomial n p) ≤
matrixPolynomialEval M (haarExpPolynomial (n+1) p) := by
simp only [haarExpPolynomial_eval]
exact positiveExpSum_mono n hp
rw [haarExpPolynomial_gamma,map_mul,map_pow]
calc
_ ≤ (matrixPolynomialEval M (haarExpPolynomial n p))^2*C :=
mul_le_mul_of_nonneg_left hG (sq_nonneg _)
_ ≤ C*(matrixPolynomialEval M (haarExpPolynomial (n+1) p))^2 := by
rw [mul_comm _ C]
exact mul_le_mul_of_nonneg_left ((sq_le_sq₀ (by linarith) (by linarith)).mpr hm) hC
end InvariantIsing
end
end OAI
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