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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 53 54 55 | import OAI.Probability.InvariantIsing.Haar.HaarFrobeniusGeometry
import OAI.Probability.InvariantIsing.Haar.SpecialSkewRotation
import OAI.Probability.InvariantIsing.Haar.DerivativeComparison
import OAI.Probability.InvariantIsing.Haar.HaarFunctionalCurvature
import OAI.Probability.InvariantIsing.Haar.HaarPolynomialCurvature
namespace OAI
/-! A Frobenius-Lipschitz polynomial has dimension-free carré-du-champ control. -/
noncomputable section
open Matrix MvPolynomial
open scoped BigOperators Matrix.Norms.Frobenius
namespace InvariantIsing
theorem haarPolynomial_direction_le_lipschitz {N : ℕ} (p : MatrixPolynomial N)
(L : ℝ) (hp : ∀ U V, |haarPolynomialValue p U-haarPolynomialValue p V| ≤
L*frobeniusDistance U V) (U : SpecialOrthogonal N)
(A : Matrix (Fin N) (Fin N) ℝ) (hA : A.transpose = -A) :
|matrixPolynomialEval (U : Matrix (Fin N) (Fin N) ℝ)
(matrixPolynomialDerivation A p)| ≤ L*‖A‖ := by
have hf := hasDerivAt_specialSkewRotation_polynomial p A hA U
have hg := hasDerivAt_specialSkewRotation A hA U
have hh := norm_derivative_le_of_increment_bound hf hg (fun t => by
simpa only [specialSkewRotation_zero,one_mul,Real.norm_eq_abs,← frobeniusDistance_eq_norm,
haarPolynomialValue]
using hp (specialSkewRotation A hA t*U) U)
simpa only [Real.norm_eq_abs,frobenius_norm_mul_orthogonal] using hh
theorem haarPolynomial_gamma_le_lipschitz {N : ℕ} (p : MatrixPolynomial N)
(L : ℝ) (hL : 0 ≤ L)
(hp : ∀ U V, |haarPolynomialValue p U-haarPolynomialValue p V| ≤
L*frobeniusDistance U V) (U : SpecialOrthogonal N) :
haarPolynomialValue (haarPolynomialGamma p p) U ≤ 4*L^2 := by
let ell := matrixPolynomialDifferential p (U : Matrix (Fin N) (Fin N) ℝ)
let A := planeFunctionalMatrix ell
have hA : A.transpose = -A := planeFunctionalMatrix_skew ell
have hrepr := plane_functional_representation ell A hA
have hsq : ‖A‖^2 = haarPolynomialValue (haarPolynomialGamma p p) U := by
rw [frobenius_norm_sq,matrixFrobeniusPair_self]
simp only [A,planeFunctionalMatrix,haarPolynomialValue,haarPolynomialGamma,map_sum,map_mul,pow_two]
rfl
have he : ‖A‖^2 = 2*ell A := by
rw [frobenius_norm_sq]
exact hrepr
have hd : |ell A| ≤ L*‖A‖ := haarPolynomial_direction_le_lipschitz p L hp U A hA
have hn : ‖A‖ ≤ 2*L := by nlinarith [norm_nonneg A,le_abs_self (ell A)]
rw [← hsq]
nlinarith [norm_nonneg A]
end InvariantIsing
end
end OAI
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