| import Mathlib |
| import OAI.Probability.SKGap.Gaussian.GOEGaussian |
| import OAI.Probability.SKGap.Matrix.WordBiasCutoff |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| lemma gaussian_goe_centered {ι : Type*} [Fintype ι] [DecidableEq ι] (r : ℝ) |
| (x : EuclideanSpace ℝ (MatrixCoordinates ι)) : |
| matrixCentered (fun y : EuclideanSpace ℝ (MatrixCoordinates ι)=>goeMatrix r y) x=goeMatrix r x := by |
| have hm : matrixEntryMean (fun y : EuclideanSpace ℝ (MatrixCoordinates ι)=>goeMatrix r y)=0 := by |
| ext i k |
| exact goe_mean_zero r i k |
| simp only [matrixCentered,hm,sub_zero] |
|
|
| theorem goe_diagonal_frobenius_tail {j : ℝ} (hj : 0<j) {n : ℕ} (hn : 1≤n) : |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n)=>gaussianReal 0 1)) |
| {g | matrixNormConstant*sqrt (2*j)+1< |
| ‖Matrix.diagonal (fun i=>goeMatrix (j/n) g i i)‖}≤ |
| ENNReal.ofReal (exp (-2*(n:ℝ)/(π^2*(sqrt (2*j))^2))) := by |
| have hs : 0<sqrt (2*j) := sqrt_pos.mpr (by positivity) |
| have hL : LipschitzWith (Real.toNNReal (sqrt (2*j)*1/sqrt n)) |
| (fun x : EuclideanSpace ℝ (MatrixCoordinates (Fin n))=>goeMatrix (j/n) x) := by |
| apply (goeMatrix_L2_lipschitz (j/n)).weaken |
| change sqrt (2*(j/n))≤(Real.toNNReal (sqrt (2*j)*1/sqrt n):ℝ) |
| rw [Real.coe_toNNReal _ (by positivity),mul_one,show 2*(j/n)=(2*j)/n by ring,Real.sqrt_div (by positivity)] |
| have ht := gaussianMatrix_scaled_semnorm_tail hn hs (by norm_num : (0:ℝ)<1) |
| (by norm_num : (0:ℝ)≤1) (by norm_num : (0:ℝ)≤1) hL false |
| simpa only [one_pow,mul_one,gaussian_goe_centered,matrixWordSeminorm,Bool.false_eq_true,ite_false, |
| Matrix.frobenius_norm_diagonal,diagonalSeminorm,diagonalVector] using ht |
| end SKGap |
| end |
| end |
| |
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators Matrix.Norms.Frobenius SchwartzMap Topology |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| |
| def removeDiagonal (M : Matrix ι ι ℝ) : Matrix ι ι ℝ := M-Matrix.diagonal (fun i=>M i i) |
| lemma matrixWordSeminorm_le (p : Bool) (M : Matrix ι ι ℝ) : matrixWordSeminorm p M≤‖M‖ := by |
| cases p |
| · exact diagonalSeminorm_le M |
| · exact offDiagonalSeminorm_le M |
| |
| lemma actualWord_removeDiagonal [Nonempty ι] (f : 𝓢(ℝ,ℂ)) {R j A D : ℝ} |
| (hR : 0≤R) (hj : 0≤j) (hA : 0≤A) (hD : 0≤D) {a : ι→ℝ} |
| (ha : ∀ i,0≤a i) (haA : ∀ i,a i≤A) (L : ℕ) (F : List (WordLetter ι)) |
| (hFL : F.length≤L) (hF : ∀ l∈F,l.bounded D) (M : Matrix ι ι ℝ) : |
| ‖actualWord f R hR j a 1 F (removeDiagonal M)-actualWord f R hR j a 1 F M‖≤ |
| (L:ℝ)*(actualWordBound f R j A D)^L*‖Matrix.diagonal (fun i=>M i i)‖ := by |
| have hB : 1≤actualWordBound f R j A D := by |
| have hh := path_constants_nonneg f hR hj hA |
| unfold actualWordBound |
| linarith |
| have ht := ((actualWord_bounds f hR hj hA hD ha haA |
| (show (1:ℝ)∈Icc 0 1 from ⟨zero_le_one,le_rfl⟩) F hF).2).dist_le_mul (removeDiagonal M) M |
| have hd : ‖removeDiagonal M-M‖=‖Matrix.diagonal (fun i=>M i i)‖ := by |
| rw [show removeDiagonal M-M= -Matrix.diagonal (fun i=>M i i) by unfold removeDiagonal;abel,norm_neg] |
| simp only [dist_eq_norm,NNReal.coe_mul,NNReal.coe_natCast,hd] at ht |
| apply ht.trans |
| apply mul_le_mul_of_nonneg_right _ (norm_nonneg _) |
| exact mul_le_mul (by exact_mod_cast hFL) (pow_le_pow_right₀ hB hFL) |
| (pow_nonneg (zero_le_one.trans hB) _) (Nat.cast_nonneg _) |
| |
| theorem actual_word_simultaneous_zerodiag_of_cutoff {j A D : ℝ} (hj : 0<j) (hA : 0<A) |
| (hs : sqrt j*A<1) (hD : 1≤D) (hAD : A≤D) (L : ℕ) (f : 𝓢(ℝ,ℂ)) |
| (hf : ∀ x∈Icc ((1-sqrt j*A)^2/4) (2+A*(2*sqrt j+1+j*A)),f x=(x:ℂ)⁻¹) |
| (hR : 0≤2*sqrt j+1+1) : |
| ∃ (C : ℝ) (N : ℕ),0<C ∧ 0<N ∧ |
| ∀ n,N≤n → ∀ p : Bool, |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n)=>gaussianReal 0 1)) |
| {g | ∃ (a : Fin n→ℝ) (F : List (WordLetter (Fin n))), |
| (∀ i,a i∈Icc 0 A) ∧ F.length≤L ∧ (∀ l∈F,l.bounded D) ∧ inverseCount F≤1 ∧ |
| C< matrixWordSeminorm p |
| (actualWord f (2*sqrt j+1+1) hR j a 1 F (removeDiagonal (goeMatrix (j/n) g))- |
| Matrix.diagonal (wordPrediction j a 1 F))} ≤ |
| (wordPatternSet L L).card*ENNReal.ofReal (3*exp (-(n:ℝ)))+ |
| ENNReal.ofReal (exp (-2*(n:ℝ)/(π^2*(sqrt (2*j))^2))) := by |
| obtain ⟨C,N,hC,hN,hw⟩ := actual_word_simultaneous_prediction_of_cutoff hj hA hs hD hAD L f hf hR |
| let R := 2*sqrt j+1+1 |
| let B := actualWordBound f R j A D |
| let T := matrixNormConstant*sqrt (2*j)+1 |
| have hD0 := zero_le_one.trans hD |
| have hB : 0≤B := (actualWordBound_pos f hR hj.le hA.le hD0).le |
| have hT : 0≤T := by |
| have hc := matrixNormConstant_bounds.1 |
| dsimp [T] |
| positivity |
| let K := (L:ℝ)*B^L*T |
| have hK : 0≤K := by dsimp [K];positivity |
| refine ⟨C+K,N,by linarith,hN,?_⟩ |
| intro n hn p |
| have hn0 : 0<n := hN.trans_le hn |
| let : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn0 |
| let E := {g : MatrixCoordinates (Fin n)→ℝ | ∃ (a : Fin n→ℝ) (F : List (WordLetter (Fin n))), |
| (∀ i,a i∈Icc 0 A) ∧ F.length≤L ∧ (∀ l∈F,l.bounded D) ∧ inverseCount F≤1 ∧ |
| C < matrixWordSeminorm p (actualWord f R hR j a 1 F (goeMatrix (j/n) g)-Matrix.diagonal (wordPrediction j a 1 F))} |
| let V := {g : MatrixCoordinates (Fin n)→ℝ | T<‖Matrix.diagonal (fun i=>goeMatrix (j/n) g i i)‖} |
| have hsub : {g | ∃ (a : Fin n→ℝ) (F : List (WordLetter (Fin n))), |
| (∀ i,a i∈Icc 0 A) ∧ F.length≤L ∧ (∀ l∈F,l.bounded D) ∧ inverseCount F≤1 ∧ |
| C+K < matrixWordSeminorm p (actualWord f R hR j a 1 F (removeDiagonal (goeMatrix (j/n) g))- |
| Matrix.diagonal (wordPrediction j a 1 F))}⊆E∪V := by |
| intro g hg |
| by_contra hh |
| obtain ⟨a,F,ha,hFL,hF,hI,hbad⟩ := hg |
| have hd : ‖Matrix.diagonal (fun i=>goeMatrix (j/n) g i i)‖≤T := by |
| by_contra hd |
| exact hh (Or.inr (lt_of_not_ge hd)) |
| have he : matrixWordSeminorm p (actualWord f R hR j a 1 F (goeMatrix (j/n) g)-Matrix.diagonal (wordPrediction j a 1 F))≤C := by |
| by_contra he |
| exact hh (Or.inl ⟨a,F,ha,hFL,hF,hI,lt_of_not_ge he⟩) |
| have hp := (matrixWordSeminorm_le p _).trans (actualWord_removeDiagonal f hR hj.le hA.le hD0 |
| (fun i=>(ha i).1) (fun i=>(ha i).2) L F hFL hF (goeMatrix (j/n) g)) |
| have hp' : matrixWordSeminorm p |
| (actualWord f R hR j a 1 F (removeDiagonal (goeMatrix (j/n) g))-actualWord f R hR j a 1 F (goeMatrix (j/n) g))≤K := |
| hp.trans (mul_le_mul_of_nonneg_left hd (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hB _))) |
| have hac := matrixWordSeminorm_add p |
| (actualWord f R hR j a 1 F (removeDiagonal (goeMatrix (j/n) g))-actualWord f R hR j a 1 F (goeMatrix (j/n) g)) |
| (actualWord f R hR j a 1 F (goeMatrix (j/n) g)-Matrix.diagonal (wordPrediction j a 1 F)) |
| rw [sub_add_sub_cancel] at hac |
| linarith |
| exact (measure_mono hsub).trans ((measure_union_le E V).trans |
| (add_le_add (hw n hn p) (goe_diagonal_frobenius_tail hj (Nat.succ_le_iff.mpr hn0)))) |
| end SKGap |
| end |
| end |
| |
| end OAI |
| |
