| import Mathlib |
| import OAI.Probability.SKGap.Matrix.ConditionalMiddleNorm |
| import OAI.Probability.SKGap.Localization.ConditionalMiddleSize |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| lemma outer_symmetric {ι : Type*} (x : ι → ℝ) : (vecMulVec x x).IsHermitian := by |
| ext i k |
| simp only [conjTranspose_apply,star_trivial,vecMulVec_apply] |
| ring |
|
|
| lemma outer_sum_symmetric {ι : Type*} (x y : ι → ℝ) : (vecMulVec x y+vecMulVec y x).IsHermitian := by |
| ext i k |
| simp only [conjTranspose_apply,star_trivial,Matrix.add_apply,vecMulVec_apply] |
| ring |
|
|
| lemma conditionalBlockMatrix_hermitian {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (r j q s S a B κ ell : ℝ) (u Z : ι → ℝ) (g : MatrixCoordinates ι → ℝ) : |
| (conditionalBlockMatrix r j q s S a B κ ell u Z g).IsHermitian := by |
| unfold conditionalBlockMatrix |
| rw [goeBlockResidual_expand] |
| have hh := (goeMatrix_symm r g).add ((outer_sum_symmetric u (goeMatrix r g*ᵥu)).smul |
| (R := ℝ) (isSelfAdjoint_iff.mpr (star_trivial (κ-1)))) |
| have hh' := hh.add ((outer_symmetric u).smul (R := ℝ) |
| (isSelfAdjoint_iff.mpr (star_trivial ((ell-2*κ+1)*goeBilinear r u u g)))) |
| convert (hh'.add (outer_sum_symmetric u ((j*sqrt (q/s)) • Z-(j*a/s) • u))).add |
| ((outer_symmetric u).smul (R := ℝ) (isSelfAdjoint_iff.mpr (star_trivial (2*j*(a+B*q)/S)))) using 1 |
| congr 1 |
| abel |
|
|
| lemma empiricalConditionalMiddle_hermitian {n : ℕ} [NeZero n] (j t σ : ℝ) |
| (y : Fin n → ℝ) (g : MatrixCoordinates (Fin n) → ℝ) : |
| (empiricalConditionalMiddle j t σ y g).IsHermitian := by |
| unfold empiricalConditionalMiddle |
| exact (((conditionalBlockMatrix_hermitian _ _ _ _ _ _ _ _ _ _ _ _).add |
| ((outer_symmetric _).smul (R := ℝ) (isSelfAdjoint_iff.mpr (star_trivial _)))).add |
| ((outer_symmetric _).smul (R := ℝ) (isSelfAdjoint_iff.mpr (star_trivial _)))).sub |
| (Matrix.isHermitian_one.smul (R := ℝ) (isSelfAdjoint_iff.mpr (star_trivial _))) |
|
|
| lemma empiricalConditionalMiddle_norm {n : ℕ} [NeZero n] (j t σ : ℝ) |
| (y : Fin n → ℝ) {V W : ℝ} (hV : 0 ≤ V) |
| (g : MatrixCoordinates (Fin n) → ℝ) |
| (hq : 0 < scalarQMoment (empiricalLaw y)) |
| (hcols : ∀ k, ‖empiricalGramColumns y (scalarD j (empiricalLaw y,t,σ)) |
| (scalarS j (empiricalLaw y,t,σ)) k‖ ≤ V) |
| (hW : opNorm (goeMatrix (j/(n:ℝ)) g) ≤ W) : |
| opNorm (empiricalConditionalMiddle j t σ y g) ≤ |
| conditionalMiddleSize j V W (empiricalLaw y,t,σ) := by |
| let p : ScalarPoint := (empiricalLaw y,t,σ) |
| let q := scalarQMoment p.1 |
| let s := scalarS j p |
| let B := j*scalarBMoment p.1 |
| let u := empiricalUnit y q |
| let X := empiricalGramColumns y (scalarD j p) s 0 |
| let Z := sqrt s • empiricalGramColumns y (scalarD j p) s 2 |
| have hu : ‖u‖=1 := empiricalUnit_norm y hq |
| have hZ : ‖Z‖ ≤ sqrt s*V := by |
| dsimp [Z] |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (sqrt_nonneg _)] |
| exact mul_le_mul_of_nonneg_left (hcols 2) (sqrt_nonneg _) |
| have hX : opNorm (vecMulVec X.ofLp X.ofLp) ≤ V^2 := by |
| apply (operator_vecMulVec_le X X).trans |
| simpa only [pow_two] using mul_le_mul (hcols 0) (hcols 0) (norm_nonneg _) hV |
| have hu' : opNorm (vecMulVec u.ofLp u.ofLp) ≤ 1 := by |
| simpa only [hu,mul_one] using operator_vecMulVec_le u u |
| have hI : opNorm (1 : Matrix (Fin n) (Fin n) ℝ) ≤ 1 := by |
| simpa only [Matrix.diagonal_one] using opNorm_diagonal_le (ι := Fin n) zero_le_one (d := fun _=>1) |
| (fun _=>by norm_num) |
| have hb := conditionalBlockMatrix_norm (r := j/(n:ℝ)) (j := j) (q := q) (s := s) |
| (S := s+j*q) (a := scalarA j p) (B := B) (κ := conditionalKappa j p) |
| (ell := conditionalEll j p) u Z g hu |
| have hc : opNorm (conditionalBlockMatrix (j/(n:ℝ)) j q s (s+j*q) (scalarA j p) B |
| (conditionalKappa j p) (conditionalEll j p) u.ofLp Z.ofLp g) ≤ |
| (1+2*|conditionalKappa j p-1|+|conditionalEll j p-2*conditionalKappa j p+1|)*W+ |
| 2*(|j*sqrt (q/s)| * (sqrt s*V)+|j*scalarA j p/s|)+|2*j*(scalarA j p+B*q)/(s+j*q)| := by |
| apply hb.trans |
| have hn := mul_le_mul_of_nonneg_left hW (by positivity : 0 ≤ 1+2*|conditionalKappa j p-1|+|conditionalEll j p-2*conditionalKappa j p+1|) |
| have hz := mul_le_mul_of_nonneg_left hZ (abs_nonneg (j*sqrt (q/s))) |
| linarith |
| change opNorm (_+j • vecMulVec X.ofLp X.ofLp+(2*j*q) • vecMulVec u.ofLp u.ofLp-B • 1) ≤ _ |
| rw [sub_eq_add_neg,← neg_smul] |
| apply (real_opNorm_add _ _).trans |
| have h1 := real_opNorm_add (conditionalBlockMatrix (j/(n:ℝ)) j q s (s+j*q) (scalarA j p) B |
| (conditionalKappa j p) (conditionalEll j p) u.ofLp Z.ofLp g+j • vecMulVec X.ofLp X.ofLp) |
| ((2*j*q) • vecMulVec u.ofLp u.ofLp) |
| have h2 := real_opNorm_add (conditionalBlockMatrix (j/(n:ℝ)) j q s (s+j*q) (scalarA j p) B |
| (conditionalKappa j p) (conditionalEll j p) u.ofLp Z.ofLp g) |
| (j • vecMulVec X.ofLp X.ofLp) |
| rw [real_opNorm_smul] at h1 h2 |
| rw [real_opNorm_smul,abs_neg] |
| have h3 := mul_le_mul_of_nonneg_left hX (abs_nonneg j) |
| have h4 := mul_le_mul_of_nonneg_left hu' (abs_nonneg (2*j*q)) |
| have h5 := mul_le_mul_of_nonneg_left hI (abs_nonneg B) |
| dsimp [conditionalMiddleSize,p,q,s,B,u,X,Z] at hc h1 h2 h3 h4 h5 ⊢ |
| nlinarith only [hc,h1,h2,h3,h4,h5] |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| lemma scalarS_pos_of_time {j t σ : ℝ} (hj : 0 ≤ j) (ht : 0 < t) (P : ProbabilityMeasure ℝ) : |
| 0 < scalarS j (P,t,σ) := |
| add_pos_of_pos_of_nonneg (add_pos_of_pos_of_nonneg ht (sq_nonneg _)) |
| (mul_nonneg hj (scalarQMoment_bounds _).1) |
|
|
| theorem empiricalConditionalMiddle_uniform_norm {j R t0 T : ℝ} |
| (hj : 0 ≤ j) (hR : 0 ≤ R) (ht0 : 0 < t0) (W : ℝ) : |
| ∃ L : ℝ, 1 ≤ L ∧ ∀ (n : ℕ) [NeZero n] (y : Fin n → ℝ), |
| empiricalLaw y ∈ momentBall R → 0 < scalarQMoment (empiricalLaw y) → |
| ∀ t ∈ Icc t0 T,∀ σ : ℝ, |σ| ≤ 1 → |
| ∀ g : MatrixCoordinates (Fin n) → ℝ, opNorm (goeMatrix (j/(n:ℝ)) g) ≤ W → |
| opNorm (empiricalConditionalMiddle j t σ y g) ≤ L := by |
| let K := max 1 ((1+|T|+j)/t0) |
| have hK : 1 ≤ K := le_max_left _ _ |
| let V := 2*K*(R+2) |
| have hV : 0 ≤ V := by dsimp [V]; positivity |
| obtain ⟨L,hL,hbd⟩ := conditionalMiddleSize_uniform (T := T) hj hR ht0 V W |
| refine ⟨L,hL,?_⟩ |
| intro n _ y hy hq t ht σ hσ g hg |
| have hs := scalarS_pos_of_time (σ := σ) hj (ht0.trans_le ht.1) (empiricalLaw y) |
| have hcol : ∀ k,‖empiricalGramColumns y (scalarD j (empiricalLaw y,t,σ)) |
| (scalarS j (empiricalLaw y,t,σ)) k‖ ≤ V := by |
| intro k |
| convert empiricalGramColumns_norm (NeZero.pos n) y hs hK (by linarith : 0 ≤ R+1) |
| (empirical_column_coefficient hj ht0 ht y) (empirical_moment_vector_bound hR y hy) k using 1 |
| dsimp [V]; ring |
| exact (empiricalConditionalMiddle_norm j t σ y hV g hq hcol hg).trans (hbd _ hy t ht σ hσ) |
| end SKGap |
| end |
| end |
| |
| end OAI |
| |
