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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 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 92 93 94 95 96 | import OAI.Probability.InvariantIsing.Haar.HaarFrobeniusGeometry
import OAI.Probability.InvariantIsing.Haar.HaarGroupTopology
namespace OAI
/-! Haar convolution by a positive polynomial kernel preserves Lipschitz constants. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory Set
open scoped BigOperators
namespace InvariantIsing
lemma continuous_haarKernelBase {N : ℕ} (U : SpecialOrthogonal N) :
Continuous (haarKernelBase U) := by
unfold haarKernelBase
apply continuous_const.add
apply continuous_finsetSum
intro i _
apply continuous_finsetSum
intro j _
exact continuous_const.mul ((continuous_apply j).comp
((continuous_apply i).comp continuous_subtype_val))
def haarKernelMass {N : ℕ} (μ : Measure (SpecialOrthogonal N)) (n : ℕ) : ℝ :=
∫ V, (haarKernelBase 1 V)^n ∂μ
def haarKernelApprox {N : ℕ} (μ : Measure (SpecialOrthogonal N))
(f : SpecialOrthogonal N → ℝ) (n : ℕ) : MatrixPolynomial N :=
(haarKernelMass μ n)⁻¹ • haarKernelAverage μ f n
lemma haarKernelMass_pos {N : ℕ} (hN : 0 < N) (μ : Measure (SpecialOrthogonal N))
[IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] (n : ℕ) :
0 < haarKernelMass μ n := by
apply integral_pos_of_integrable_nonneg_nonzero (x := (1 : SpecialOrthogonal N))
((continuous_haarKernelBase 1).pow n)
(continuous_haar_integrable μ _ ((continuous_haarKernelBase 1).pow n))
(fun V => pow_nonneg (haarKernelBase_nonneg 1 V) n)
change (haarKernelBase 1 1)^n ≠ 0
rw [haarKernelBase_self]
exact (pow_pos (mul_pos (by norm_num) (Nat.cast_pos.mpr hN)) n).ne'
lemma haarKernelAverage_convolution {N : ℕ} (μ : Measure (SpecialOrthogonal N))
[IsFiniteMeasure μ] [μ.IsMulLeftInvariant] (f : SpecialOrthogonal N → ℝ)
(hf : Continuous f) (n : ℕ) (U : SpecialOrthogonal N) :
haarPolynomialValue (haarKernelAverage μ f n) U =
∫ V, f (U*V)*(haarKernelBase 1 V)^n ∂μ := by
rw [haarKernelAverage_value μ f hf]
have h := integral_mul_left_eq_self (μ := μ)
(fun V => f V*(haarKernelBase U V)^n) U
rw [← h]
apply integral_congr_ae
exact ae_of_all μ fun V => by
dsimp only
rw [show haarKernelBase U (U*V) = haarKernelBase 1 V by
simpa only [mul_one] using haarKernelBase_mul_left U 1 V]
theorem haarKernelApprox_lipschitz {N : ℕ} (hN : 0 < N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
(f : SpecialOrthogonal N → ℝ) (hf : Continuous f) (L : ℝ)
(hL : ∀ U V, |f U-f V| ≤ L*frobeniusDistance U V) (n : ℕ)
(U V : SpecialOrthogonal N) :
|haarPolynomialValue (haarKernelApprox μ f n) U-
haarPolynomialValue (haarKernelApprox μ f n) V| ≤ L*frobeniusDistance U V := by
have hZ := haarKernelMass_pos hN μ n
let k (W : SpecialOrthogonal N) := (haarKernelBase 1 W)^n
have hk : Continuous k := (continuous_haarKernelBase 1).pow n
have hi (U : SpecialOrthogonal N) : Integrable (fun W => f (U*W)*k W) μ :=
continuous_haar_integrable μ _ ((hf.comp (continuous_const.mul continuous_id)).mul hk)
have he (U : SpecialOrthogonal N) : haarPolynomialValue (haarKernelApprox μ f n) U =
(haarKernelMass μ n)⁻¹*(∫ W, f (U*W)*k W ∂μ) := by
simp only [haarKernelApprox,haarPolynomialValue,map_smul,smul_eq_mul]
change (haarKernelMass μ n)⁻¹*haarPolynomialValue (haarKernelAverage μ f n) U = _
rw [haarKernelAverage_convolution μ f hf]
rw [he U,he V,← mul_sub,abs_mul,abs_of_pos (inv_pos.mpr hZ),← integral_sub (hi U) (hi V)]
apply (inv_mul_le_iff₀ hZ).mpr
calc
|∫ W, (f (U*W)*k W-f (V*W)*k W) ∂μ| ≤
∫ W, |f (U*W)*k W-f (V*W)*k W| ∂μ := abs_integral_le_integral_abs
_ ≤ ∫ W, (L*frobeniusDistance U V)*k W ∂μ := by
apply integral_mono
· exact ((hi U).sub (hi V)).abs
· exact (continuous_haar_integrable μ _ hk).const_mul _
· intro W
dsimp only
rw [← sub_mul,abs_mul,abs_of_nonneg (pow_nonneg (haarKernelBase_nonneg 1 W) n)]
apply mul_le_mul_of_nonneg_right _ (pow_nonneg (haarKernelBase_nonneg 1 W) n)
exact (hL (U*W) (V*W)).trans_eq (by rw [frobeniusDistance_mul_right])
_ = haarKernelMass μ n*(L*frobeniusDistance U V) := by
rw [integral_const_mul,mul_comm]
rfl
end InvariantIsing
end
end OAI
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