| import Mathlib |
| import OAI.Probability.SKGap.Model |
|
|
| namespace OAI |
|
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| noncomputable section |
|
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| open MeasureTheory ProbabilityTheory InformationTheory Real Set |
| open scoped NNReal ENNReal |
| open Filter |
| open scoped Topology |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal |
| namespace SKGap |
| abbrev MatrixCoordinates (ι : Type*) := (ι × ι) ⊕ ι |
|
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| variable {ι : Type*} |
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| def goeMatrix (r : ℝ) (g : MatrixCoordinates ι → ℝ) : Matrix ι ι ℝ := |
| fun i k => sqrt (2*r)/2 * (g (.inl (i,k)) + g (.inl (k,i))) |
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| variable [Fintype ι] |
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| def rawCoordinateMatrix (g : MatrixCoordinates ι → ℝ) : Matrix ι ι ℝ := |
| fun i k => g (.inl (i,k)) |
|
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| lemma rawCoordinateMatrix_norm (g : EuclideanSpace ℝ (MatrixCoordinates ι)) : |
| ‖rawCoordinateMatrix g‖ ≤ ‖g‖ := by |
| apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _) |
| simp only [← sq] |
| rw [Matrix.frobenius_norm_def,← Real.sqrt_eq_rpow,Real.sq_sqrt (by positivity), |
| EuclideanSpace.norm_sq_eq] |
| simp only [Real.rpow_two,Real.norm_eq_abs,sq_abs,Fintype.sum_sum_type,Fintype.sum_prod_type, |
| rawCoordinateMatrix] |
| exact le_add_of_nonneg_right (Finset.sum_nonneg (fun _ _ => sq_nonneg _)) |
|
|
| lemma goeMatrix_norm (r : ℝ) (g : EuclideanSpace ℝ (MatrixCoordinates ι)) : |
| ‖goeMatrix r g‖ ≤ sqrt (2*r)*‖g‖ := by |
| have he : goeMatrix r g = (sqrt (2*r)/2) • |
| (rawCoordinateMatrix g+(rawCoordinateMatrix g)ᵀ) := by ext; rfl |
| rw [he,norm_smul,Real.norm_eq_abs,abs_of_nonneg (by positivity)] |
| calc |
| _ ≤ (sqrt (2*r)/2)*(‖rawCoordinateMatrix g‖+‖(rawCoordinateMatrix g)ᵀ‖) := |
| mul_le_mul_of_nonneg_left (norm_add_le _ _) (by positivity) |
| _ = sqrt (2*r)*‖rawCoordinateMatrix g‖ := by rw [Matrix.frobenius_norm_transpose]; ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left (rawCoordinateMatrix_norm g) (sqrt_nonneg _) |
|
|
| omit [Fintype ι] in |
| lemma goeMatrix_sub (r : ℝ) (g h : MatrixCoordinates ι → ℝ) : |
| goeMatrix r (g-h) = goeMatrix r g-goeMatrix r h := by |
| ext i k; simp only [goeMatrix,Pi.sub_apply,Matrix.sub_apply]; ring |
|
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| lemma goeMatrix_L2_lipschitz (r : ℝ) : |
| LipschitzWith ⟨sqrt (2*r),sqrt_nonneg _⟩ |
| (fun g : EuclideanSpace ℝ (MatrixCoordinates ι) => goeMatrix r g) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro g h |
| simpa only [dist_eq_norm,← goeMatrix_sub] using! goeMatrix_norm r (g-h) |
|
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| lemma goeMatrix_pi_lipschitz (r : ℝ) : |
| LipschitzWith (⟨sqrt (2*r),sqrt_nonneg _⟩ * |
| (Fintype.card (MatrixCoordinates ι) : ℝ≥0) ^ (1/(2:ℝ≥0∞)).toReal) |
| (goeMatrix (ι := ι) r) := by |
| exact (goeMatrix_L2_lipschitz (ι := ι) r).comp |
| (PiLp.lipschitzWith_toLp 2 (fun _ : MatrixCoordinates ι => ℝ)) |
|
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| variable [DecidableEq ι] |
|
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| def symmetricElementary (a b : ι) : Matrix ι ι ℝ := |
| Matrix.single a b 1+Matrix.single b a 1 |
|
|
| omit [Fintype ι] in |
| lemma goeMatrix_update (r : ℝ) (g : MatrixCoordinates ι → ℝ) (a b : ι) (t : ℝ) : |
| goeMatrix r (Function.update g (.inl (a,b)) t) = |
| goeMatrix r g + ((sqrt (2*r)/2)*(t-g (.inl (a,b)))) • symmetricElementary a b := by |
| ext i k |
| simp only [goeMatrix,Matrix.add_apply,Matrix.smul_apply,smul_eq_mul,symmetricElementary, |
| Matrix.single_apply] |
| by_cases hi : i=a <;> by_cases hk : k=b <;> by_cases hi' : i=b <;> by_cases hk' : k=a <;> |
| simp_all [Function.update_apply,Prod.ext_iff,eq_comm] <;> ring |
|
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| lemma symmetricElementary_norm_le (a b : ι) : ‖symmetricElementary a b‖ ≤ 2 := by |
| unfold symmetricElementary |
| exact (norm_add_le _ _).trans (by simpa [Matrix.frobenius_norm_def,Matrix.single_apply,ite_and] using (show (1:ℝ)+1 ≤ 2 by norm_num)) |
|
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| lemma goeMatrix_updated_hasDerivAt (r : ℝ) (g : MatrixCoordinates ι → ℝ) (a b : ι) : |
| HasDerivAt (fun t => goeMatrix r (Function.update g (.inl (a,b)) t)) |
| ((sqrt (2*r)/2) • symmetricElementary a b) (g (.inl (a,b))) := by |
| simp_rw [goeMatrix_update] |
| have ha := (hasDerivAt_id (g (.inl (a,b)))).sub_const (g (.inl (a,b))) |
| have hb := ha.const_mul (sqrt (2*r)/2) |
| have hh := (hb.smul_const (symmetricElementary a b)).const_add (goeMatrix r g) |
| simpa only [mul_one,id_eq] using! hh |
|
|
| end SKGap |
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| end |
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| end OAI |
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