| import OAI.Probability.InvariantIsing.Haar.HaarKernelMass |
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| namespace OAI |
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| /-! Uniform approximation by positive polynomial Haar convolution. -/ |
| noncomputable section |
| open Matrix MvPolynomial MeasureTheory Set Filter |
| open scoped Topology |
| namespace InvariantIsing |
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| 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 |
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| 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 |
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| end InvariantIsing |
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| end |
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| end OAI |
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