| import Mathlib | |
| import OAI.Probability.SKGap.Localization.DiagonalTMemLp | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set | |
| open scoped NNReal ENNReal | |
| open Filter | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set Filter | |
| open MeasureTheory.Measure | |
| open scoped ENNReal NNReal MeasureTheory Topology | |
| open MeasureTheory | |
| noncomputable section | |
| noncomputable section | |
| open MeasureTheory Set NormedSpace | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Set Real | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Set Filter | |
| open scoped Topology Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Matrix NormedSpace ContinuousLinearMap | |
| open scoped Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Matrix | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set | |
| open scoped ENNReal NNReal | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set | |
| open scoped NNReal ENNReal | |
| noncomputable section | |
| open scoped BigOperators | |
| open MeasureTheory ProbabilityTheory | |
| open Real | |
| noncomputable section | |
| open scoped BigOperators Topology | |
| open Filter Real | |
| namespace SKGap | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| def matrixQuad (M : Matrix ι ι ℝ) (p : EuclideanSpace ℝ ι) : ℝ := | |
| ∑ i, ∑ k, p i * M i k * p k | |
| def matrixOperator (M : Matrix ι ι ℝ) : | |
| EuclideanSpace ℝ ι →L[ℝ] EuclideanSpace ℝ ι := | |
| M.toEuclideanLin.toContinuousLinearMap | |
| lemma matrixOperator_apply (M : Matrix ι ι ℝ) (p : EuclideanSpace ℝ ι) (i : ι) : | |
| matrixOperator M p i = ∑ k, M i k * p k := by | |
| rfl | |
| lemma matrixQuad_inner (M : Matrix ι ι ℝ) (p : EuclideanSpace ℝ ι) : | |
| matrixQuad M p = inner ℝ p (matrixOperator M p) := by | |
| simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial, matrixOperator_apply, | |
| matrixQuad, Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| ring | |
| omit [DecidableEq ι] in | |
| lemma matrixQuad_smul (M : Matrix ι ι ℝ) (p : EuclideanSpace ℝ ι) (c : ℝ) : | |
| matrixQuad M (c • p) = c^2 * matrixQuad M p := by | |
| simp only [matrixQuad, PiLp.smul_apply, smul_eq_mul, Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| ring | |
| omit [Fintype ι] [DecidableEq ι] in | |
| lemma goeMatrix_symm (r : ℝ) (g : MatrixCoordinates ι → ℝ) : | |
| (goeMatrix r g).IsHermitian := by | |
| ext i k | |
| simp [goeMatrix, add_comm] | |
| omit [Fintype ι] [DecidableEq ι] in | |
| lemma goeMatrix_neg (r : ℝ) (g : MatrixCoordinates ι → ℝ) : | |
| goeMatrix r (-g) = -goeMatrix r g := by | |
| ext i k | |
| simp only [goeMatrix, Pi.neg_apply, Matrix.neg_apply] | |
| ring | |
| omit [DecidableEq ι] in | |
| lemma goeMatrix_quad (r : ℝ) (g : MatrixCoordinates ι → ℝ) (p : EuclideanSpace ℝ ι) : | |
| matrixQuad (goeMatrix r g) p = | |
| ∑ i, ∑ k, sqrt (2*r) * p i * p k * g (.inl (i,k)) := by | |
| have hs : (∑ i, ∑ k, p i * g (.inl (k,i)) * p k) = | |
| ∑ i, ∑ k, p i * g (.inl (i,k)) * p k := by | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| ring | |
| calc | |
| _ = sqrt (2*r)/2 * ((∑ i, ∑ k, p i * g (.inl (i,k)) * p k) + | |
| ∑ i, ∑ k, p i * g (.inl (k,i)) * p k) := by | |
| simp only [matrixQuad, goeMatrix, Finset.mul_sum, ← Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| ring | |
| _ = _ := by | |
| rw [hs] | |
| simp only [Finset.mul_sum, ← Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| ring | |
| lemma matrixOperator_norm_le_of_quad {M : Matrix ι ι ℝ} (hM : M.IsHermitian) | |
| {R : ℝ} (hR : 0 ≤ R) | |
| (hq : ∀ p : EuclideanSpace ℝ ι, ‖p‖ ≤ 1 → |matrixQuad M p| ≤ R) : | |
| ‖matrixOperator M‖ ≤ R := by | |
| have hs : (matrixOperator M).IsSymmetric := Matrix.isSymmetric_toEuclideanLin_iff.mpr hM | |
| rw [ContinuousLinearMap.norm_eq_iSup_rayleighQuotient (T := matrixOperator M) hs] | |
| apply ciSup_le | |
| intro p | |
| by_cases hp : p = 0 | |
| · simp [hp, ContinuousLinearMap.rayleighQuotient, hR] | |
| have hp0 : 0 < ‖p‖ := norm_pos_iff.mpr hp | |
| have hunit : ‖‖p‖⁻¹ • p‖ ≤ 1 := by | |
| rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hp0, inv_mul_cancel₀ hp0.ne'] | |
| have hh := hq (‖p‖⁻¹ • p) hunit | |
| rw [matrixQuad_smul, abs_mul, abs_pow, abs_inv, abs_of_pos hp0] at hh | |
| simpa only [ContinuousLinearMap.rayleighQuotient, ContinuousLinearMap.reApplyInnerSelf_apply, | |
| RCLike.re_to_real, real_inner_comm, ← matrixQuad_inner, abs_div, | |
| abs_pow, abs_of_pos hp0, inv_pow, div_eq_mul_inv, mul_comm, abs_mul, abs_inv] using hh | |
| end SKGap | |
| namespace SKGap | |
| section GOEEvents | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι] | |
| omit [DecidableEq ι] [Nonempty ι] in | |
| lemma goeMatrix_quad_coeff (r : ℝ) (g : MatrixCoordinates ι → ℝ) | |
| (p : EuclideanSpace ℝ ι) : | |
| matrixQuad (goeMatrix r g) p = ∑ k, quadraticCoeff r p k * g k := by | |
| rw [goeMatrix_quad, Fintype.sum_sum_type] | |
| simp only [quadraticCoeff, zero_mul, Finset.sum_const_zero, add_zero, | |
| Fintype.sum_prod_type] | |
| omit [DecidableEq ι] [Nonempty ι] in | |
| lemma diagonal_affine_matrix (r z q : ℝ) (a : ι → ℝ) | |
| (g : MatrixCoordinates ι → ℝ) (p : UnitBall ι) : | |
| gaussianAffine (diagonalOffset a z q) | |
| (fun p k => z * quadraticCoeff r (diagonalSqrt a p) k) g p = | |
| z * matrixQuad (goeMatrix r g) (diagonalSqrt a p) - | |
| z^2*q*‖diagonalSqrt a p‖^2 := by | |
| rw [goeMatrix_quad_coeff] | |
| simp only [gaussianAffine, diagonalOffset, Finset.mul_sum] | |
| simp_rw [mul_assoc] | |
| ring | |
| omit [Fintype ι] [DecidableEq ι] [Nonempty ι] in | |
| lemma diagonalSqrt_one (p : EuclideanSpace ℝ ι) : | |
| diagonalSqrt (fun _ : ι => 1) p = p := by | |
| ext i | |
| simp [diagonalSqrt] | |
| omit [DecidableEq ι] [Nonempty ι] in | |
| lemma goe_quad_le_top (r : ℝ) (g : MatrixCoordinates ι → ℝ) | |
| (p : EuclideanSpace ℝ ι) (hp : ‖p‖ ≤ 1) : | |
| matrixQuad (goeMatrix r g) p ≤ diagonalProcess r (fun _ : ι => 1) 1 0 g := by | |
| let p' : UnitBall ι := ⟨p, by simpa only [Metric.mem_closedBall, dist_zero_right] using hp⟩ | |
| have h := gaussianAffine_le_gaussianSup (continuous_diagonalOffset (fun _ : ι => 1) 1 0) | |
| (continuous_diagonalCoeff (continuous_quadraticCoeff r) (fun _ : ι => 1) 1) g p' | |
| rw [diagonal_affine_matrix, diagonalSqrt_one] at h | |
| simpa only [one_mul, mul_zero, zero_mul, sub_zero, diagonalProcess, p'] using h | |
| omit [DecidableEq ι] [Nonempty ι] in | |
| lemma matrixQuad_neg (M : Matrix ι ι ℝ) (p : EuclideanSpace ℝ ι) : | |
| matrixQuad (-M) p = -matrixQuad M p := by | |
| simp [matrixQuad, Finset.sum_neg_distrib] | |
| omit [Nonempty ι] in | |
| lemma goe_norm_le_of_top {r R : ℝ} (hR : 0 ≤ R) (g : MatrixCoordinates ι → ℝ) | |
| (hpos : diagonalProcess r (fun _ : ι => 1) 1 0 g ≤ R) | |
| (hneg : diagonalProcess r (fun _ : ι => 1) 1 0 (-g) ≤ R) : | |
| ‖matrixOperator (goeMatrix r g)‖ ≤ R := by | |
| apply matrixOperator_norm_le_of_quad (goeMatrix_symm r g) hR | |
| intro p hp | |
| apply abs_le.mpr | |
| constructor | |
| · have h := (goe_quad_le_top r (-g) p hp).trans hneg | |
| rw [goeMatrix_neg, matrixQuad_neg] at h | |
| linarith | |
| · exact (goe_quad_le_top r g p hp).trans hpos | |
| theorem goe_norm_tail {j ε : ℝ} (hj : 0 < j) (hε : 0 ≤ ε) : | |
| (gaussianCoordinates (MatrixCoordinates ι)).real | |
| {g | 2*sqrt j+ε < ‖matrixOperator (goeMatrix (j/(Fintype.card ι:ℝ)) g)‖} ≤ | |
| 2*Real.exp (-ε^2*(Fintype.card ι:ℝ)/(π^2*j)) := by | |
| let r := j/(Fintype.card ι:ℝ) | |
| let R := 2*sqrt j+ε | |
| let P := diagonalProcess r (fun _ : ι => 1) 1 0 | |
| let E : Set (MatrixCoordinates ι → ℝ) := {g | R < P g} | |
| have hc : Continuous P := continuous_gaussianSup | |
| (continuous_diagonalOffset (fun _ : ι => 1) 1 0) | |
| (continuous_diagonalCoeff (continuous_quadraticCoeff r) (fun _ : ι => 1) 1) | |
| have hm : MeasurableSet E := measurableSet_lt measurable_const hc.measurable | |
| have hp := goe_top_tail (ι := ι) hj hε | |
| have hn : (gaussianCoordinates (MatrixCoordinates ι)).real {g | R < P (-g)} = | |
| (gaussianCoordinates (MatrixCoordinates ι)).real E := by | |
| exact congrArg ENNReal.toReal (gaussianCoordinates_preserving_neg.measure_preimage hm.nullMeasurableSet) | |
| have hs : {g : MatrixCoordinates ι → ℝ | R < ‖matrixOperator (goeMatrix r g)‖} ⊆ E ∪ {g | R < P (-g)} := by | |
| intro g hg | |
| by_contra hh | |
| have hh' : ¬ R < P g ∧ ¬ R < P (-g) := by simpa only [Set.mem_union, E, Set.mem_ofPred_eq, not_or] using hh | |
| exact (not_lt_of_ge (goe_norm_le_of_top (by dsimp [R]; positivity) g | |
| (le_of_not_gt hh'.1) (le_of_not_gt hh'.2))) hg | |
| calc | |
| _ ≤ (gaussianCoordinates (MatrixCoordinates ι)).real (E ∪ {g | R < P (-g)}) := measureReal_mono hs | |
| _ ≤ (gaussianCoordinates (MatrixCoordinates ι)).real E + | |
| (gaussianCoordinates (MatrixCoordinates ι)).real {g | R < P (-g)} := measureReal_union_le _ _ | |
| _ = 2*(gaussianCoordinates (MatrixCoordinates ι)).real E := by rw [hn]; ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left hp (by norm_num) | |
| end GOEEvents | |
| end SKGap | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |