| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace HilbertRegularization |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
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| def regularize (ε : ℝ) (z : E) : ℂ := (Real.sqrt (‖z‖^2+ε^2)-ε : ℝ) |
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| def differential (ε : ℝ) (z : E) : E →L[ℝ] ℂ := |
| (Real.sqrt (‖z‖^2+ε^2))⁻¹ • (Complex.ofRealCLM.comp (innerSL ℝ z)) |
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| lemma smooth {ε : ℝ} (hε : 0<ε) : |
| ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞) (regularize (E := E) ε) := by |
| apply Complex.ofRealCLM.contDiff.comp |
| apply ContDiff.sub _ contDiff_const |
| apply ContDiff.sqrt ((contDiff_norm_sq ℝ).add contDiff_const) |
| intro z |
| positivity |
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| omit [InnerProductSpace ℝ E] in |
| lemma zero {ε : ℝ} (hε : 0≤ε) : regularize (E := E) ε 0=0 := by |
| simp [regularize,Real.sqrt_sq hε] |
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| lemma hasFDerivAt {ε : ℝ} (hε : 0<ε) (z : E) : |
| HasFDerivAt (regularize ε) (differential ε z) z := by |
| have hn : ‖z‖^2+ε^2≠0 := by positivity |
| have h := Complex.ofRealCLM.hasFDerivAt.comp z |
| ((((hasStrictFDerivAt_norm_sq z).hasFDerivAt.add_const (ε^2)).sqrt hn).sub_const ε) |
| have hd : differential ε z = Complex.ofRealCLM.comp |
| ((1/(2*Real.sqrt (‖z‖^2+ε^2))) • (2 • innerSL ℝ z)) := by |
| ext w |
| simp [differential,two_smul,div_eq_mul_inv,mul_inv_rev] |
| ring |
| rw [hd] |
| exact h |
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| omit [InnerProductSpace ℝ E] in |
| lemma sqroot_ge_norm (ε : ℝ) (z : E) : ‖z‖≤Real.sqrt (‖z‖^2+ε^2) := |
| (Real.le_sqrt (norm_nonneg _) (by positivity)).mpr (by nlinarith [sq_nonneg ε]) |
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| lemma differential_apply_le (ε : ℝ) (z w : E) : ‖differential ε z w‖≤‖w‖ := by |
| by_cases hz : z=0 |
| · simp [hz,differential] |
| have hp : 0<Real.sqrt (‖z‖^2+ε^2) := |
| lt_of_lt_of_le (norm_pos_iff.mpr hz) (sqroot_ge_norm ε z) |
| simp only [differential,smul_apply,ContinuousLinearMap.comp_apply,norm_smul, |
| Complex.ofRealCLM_apply,Complex.norm_real,Real.norm_eq_abs,abs_inv,abs_of_pos hp,innerSL_apply_apply] |
| change _*‖inner ℝ z w‖≤_ |
| apply le_trans (mul_le_mul_of_nonneg_left (norm_inner_le_norm (𝕜 := ℝ) z w) (by positivity : |
| 0≤(Real.sqrt (‖z‖^2+ε^2))⁻¹)) |
| rw [← div_eq_inv_mul,div_le_iff₀ hp] |
| simpa only [mul_comm] using mul_le_mul_of_nonneg_right (sqroot_ge_norm ε z) (norm_nonneg w) |
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| lemma differential_le (ε : ℝ) (z : E) : ‖differential ε z‖≤1 := by |
| apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one |
| intro w |
| simpa only [one_mul] using differential_apply_le ε z w |
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| lemma fderiv_eq {ε : ℝ} (hε : 0<ε) (z : E) : |
| fderiv ℝ (regularize ε) z=differential ε z := (hasFDerivAt hε z).fderiv |
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| lemma lipschitz {ε : ℝ} (hε : 0<ε) : LipschitzWith 1 (regularize (E := E) ε) := |
| lipschitzWith_of_nnnorm_fderiv_le ((smooth hε).differentiable (by simp)) |
| (fun z ↦ by rw [fderiv_eq hε]; exact NNReal.coe_le_coe.mp (differential_le ε z)) |
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| lemma norm_le {ε : ℝ} (hε : 0<ε) (z : E) : ‖regularize ε z‖≤‖z‖ := by |
| simpa only [zero hε.le,sub_zero,NNReal.coe_one,one_mul] using |
| (lipschitz hε).norm_sub_le z 0 |
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| omit [InnerProductSpace ℝ E] in |
| lemma tendsto_regularize (z : E) {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0)) : |
| Tendsto (fun n ↦ regularize (e n) z) atTop (𝓝 (‖z‖ : ℂ)) := by |
| have h : Tendsto (fun n ↦ Real.sqrt (‖z‖^2+e n^2)-e n) atTop (𝓝 ‖z‖) := by |
| have h := (((tendsto_const_nhds (x := ‖z‖^2)).add (he.pow 2)).sqrt.sub he) |
| simpa only [zero_pow (by norm_num : 2≠0),add_zero,sub_zero,Real.sqrt_sq (norm_nonneg _)] using h |
| exact Complex.continuous_ofReal.continuousAt.tendsto.comp h |
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| end HilbertRegularization |
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| end |
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| end OAI |
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