File size: 5,504 Bytes
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 97 98 99 100 101 102 103 104 105 106 | import OAI.Probability.InvariantIsing.Haar.HaarKernelMass
namespace OAI
/-! Uniform approximation by positive polynomial Haar convolution. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory Set Filter
open scoped Topology
namespace InvariantIsing
theorem haarKernelApprox_error {N : ℕ} (hN : 0 < N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
(f : SpecialOrthogonal N → ℝ) (hf : Continuous f) (B ε a : ℝ)
(hB : 0 ≤ B) (hfB : ∀ V, |f V| ≤ B) (hε : 0 ≤ ε) (ha : 0 ≤ a)
(hloc : ∀ U V, a < haarKernelBase 1 V → |f (U*V)-f U| ≤ ε)
(n : ℕ) (U : SpecialOrthogonal N) :
|haarPolynomialValue (haarKernelApprox μ f n) U-f U| ≤
ε+2*B*(a^n/haarKernelMass μ n) := by
let Z := haarKernelMass μ n
let k (V : SpecialOrthogonal N) := (haarKernelBase 1 V)^n
have hZ : 0 < Z := haarKernelMass_pos hN μ n
have hk : Continuous k := (continuous_haarKernelBase 1).pow n
have hi : Integrable (fun V => (f (U*V)-f U)*k V) μ :=
continuous_haar_integrable μ _ (((hf.comp (continuous_const.mul continuous_id)).sub continuous_const).mul hk)
have he : haarPolynomialValue (haarKernelApprox μ f n) U-f U =
Z⁻¹*(∫ V, (f (U*V)-f U)*k V ∂μ) := by
simp only [haarKernelApprox,haarPolynomialValue,map_smul,smul_eq_mul]
change Z⁻¹*haarPolynomialValue (haarKernelAverage μ f n) U-f U = _
rw [haarKernelAverage_convolution μ f hf]
have he' : (∫ V, (f (U*V)-f U)*k V ∂μ) =
(∫ V, f (U*V)*k V ∂μ)-f U*Z := by
simp_rw [sub_mul]
have hi1 : Integrable (fun V => f (U*V)*k V) μ :=
continuous_haar_integrable μ _ ((hf.comp (continuous_const.mul continuous_id)).mul hk)
have hi2 : Integrable (fun V => f U*k V) μ :=
(continuous_haar_integrable μ _ hk).const_mul (f U)
rw [integral_sub hi1 hi2,integral_const_mul]
rfl
rw [he']
dsimp only [k]
field_simp
have hb (V : SpecialOrthogonal N) :
|(f (U*V)-f U)*k V| ≤ ε*k V+2*B*a^n := by
rw [abs_mul,abs_of_nonneg (pow_nonneg (haarKernelBase_nonneg 1 V) n)]
by_cases hV : a < haarKernelBase 1 V
· have hh := mul_le_mul_of_nonneg_right (hloc U V hV)
(pow_nonneg (haarKernelBase_nonneg 1 V) n)
have ht : 0 ≤ 2*B*a^n := by positivity
exact hh.trans (le_add_of_nonneg_right ht)
· have hbd : |f (U*V)-f U| ≤ 2*B := (abs_sub (f (U*V)) (f U)).trans (by linarith [hfB (U*V),hfB U])
have hpow : k V ≤ a^n := pow_le_pow_left₀ (haarKernelBase_nonneg 1 V) (le_of_not_gt hV) n
have hh := mul_le_mul hbd hpow (pow_nonneg (haarKernelBase_nonneg 1 V) n) (by positivity : 0 ≤ 2*B)
have ht : 0 ≤ ε*k V := mul_nonneg hε (pow_nonneg (haarKernelBase_nonneg 1 V) n)
exact hh.trans (le_add_of_nonneg_left ht)
rw [he,abs_mul,abs_of_pos (inv_pos.mpr hZ)]
have hI : |∫ V, (f (U*V)-f U)*k V ∂μ| ≤ ε*Z+2*B*a^n := by
calc
_ ≤ ∫ V, |(f (U*V)-f U)*k V| ∂μ := abs_integral_le_integral_abs
_ ≤ ∫ V, ε*k V+2*B*a^n ∂μ :=
integral_mono hi.abs (((continuous_haar_integrable μ _ hk).const_mul ε).add (integrable_const _)) hb
_ = ε*Z+2*B*a^n := by
rw [integral_add ((continuous_haar_integrable μ _ hk).const_mul ε) (integrable_const _),integral_const_mul]
simp only [integral_const,probReal_univ,smul_eq_mul,one_mul]
rfl
calc
_ ≤ Z⁻¹*(ε*Z+2*B*a^n) := mul_le_mul_of_nonneg_left hI (inv_pos.mpr hZ).le
_ = ε+2*B*(a^n/Z) := by field_simp
theorem haarKernelApprox_uniform {N : ℕ} (hN : 0 < N)
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
(f : SpecialOrthogonal N → ℝ) (hf : Continuous f) (L : ℝ) (hL : 0 < L)
(hLip : ∀ U V, |f U-f V| ≤ L*frobeniusDistance U V) :
∀ ε > 0, ∀ᶠ n : ℕ in atTop, ∀ U,
|haarPolynomialValue (haarKernelApprox μ f n) U-f U| ≤ ε := by
intro ε hε
obtain ⟨B,hB⟩ := isCompact_univ.exists_bound_of_continuousOn hf.continuousOn
have hfB (V : SpecialOrthogonal N) : |f V| ≤ B := by simpa only [Real.norm_eq_abs] using hB V (mem_univ V)
have hB0 : 0 ≤ B := (abs_nonneg (f 1)).trans (hfB 1)
let δ := ε/(2*L)
have hδ : 0 < δ := div_pos hε (mul_pos (by norm_num) hL)
have hδeq : L*δ = ε/2 := by dsimp [δ]; field_simp
let a := max 0 (2*N-δ^2/2)
have ha : 0 ≤ a := le_max_left _ _
have haN : a < 2*N := max_lt (by positivity) (by nlinarith [sq_pos_of_pos hδ])
have hloc (U V : SpecialOrthogonal N) (hV : a < haarKernelBase 1 V) :
|f (U*V)-f U| ≤ ε/2 := by
have hda : 2*N-δ^2/2 < haarKernelBase 1 V := (le_max_right _ _).trans_lt hV
rw [haarKernelBase_dist] at hda
have hd : frobeniusDistance 1 V < δ := by nlinarith [Real.sqrt_nonneg (∑ i, ∑ j, ((1 : Matrix (Fin N) (Fin N) ℝ) i j-(V : Matrix (Fin N) (Fin N) ℝ) i j)^2)]
have hh := hLip (U*V) (U*1)
rw [frobeniusDistance_mul_left,mul_one,frobeniusDistance_comm V 1] at hh
have hh' := mul_lt_mul_of_pos_left hd hL
rw [hδeq] at hh'
exact hh.trans hh'.le
have ht : Tendsto (fun n : ℕ => 2*B*(a^n/haarKernelMass μ n)) atTop (𝓝 0) := by
simpa using (haarKernelMass_ratio_tendsto_zero hN μ a ha haN).const_mul (2*B)
filter_upwards [ht.eventually (gt_mem_nhds (by linarith : (0:ℝ) < ε/2))] with n hn U
have hh := haarKernelApprox_error hN μ f hf B (ε/2) a hB0 hfB (by positivity) ha hloc n U
linarith
end InvariantIsing
end
end OAI
|