| import Mathlib |
| import OAI.Probability.SKGap.Localization.EmpiricalConditionalData |
| import OAI.Probability.SKGap.Gaussian.ConditionalUniformTail |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Real Matrix |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| def augmentedToleranceConstant (L : ℝ) := |
| 4*(1+(L+1)*(1+L)+L*(1+2*L+L^2)) |
|
|
| lemma augmentedToleranceConstant_pos {L : ℝ} (hL : 0 ≤ L) : |
| 0 < augmentedToleranceConstant L := by unfold augmentedToleranceConstant; positivity |
|
|
| lemma augmentedGramTolerance_bound {B L ε : ℝ} (hB : 0 < B) (hL : 1 ≤ L) |
| (hε : 0 ≤ ε) (hb : B ≤ L) (hi : 1/B ≤ L) : |
| augmentedGramTolerance B ε ≤ augmentedToleranceConstant L*ε := by |
| have hL0 : 0 ≤ L := by linarith |
| have hir : (sqrt B)⁻¹ ≤ L+1 := by |
| have hs : (sqrt B)⁻¹=sqrt B⁻¹ := (sqrt_inv B).symm |
| rw [hs] |
| have hh := sq_sqrt (inv_pos.mpr hB).le |
| have hh' : B⁻¹ ≤ L := by simpa only [one_div] using hi |
| nlinarith [sq_nonneg (sqrt B⁻¹-1)] |
| have hB2 : B^2 ≤ L^2 := pow_le_pow_left₀ hB.le hb 2 |
| have ht1 : (sqrt B)⁻¹*(1+B) ≤ (L+1)*(1+L) := |
| mul_le_mul hir (by linarith) (by positivity) (by linarith) |
| have ht2 : B⁻¹*(1+2*B+B^2) ≤ L*(1+2*L+L^2) := |
| mul_le_mul (by simpa only [one_div] using hi) (by linarith) (by positivity) hL0 |
| unfold augmentedGramTolerance augmentedToleranceConstant |
| have h1 := mul_le_mul_of_nonneg_right ht1 hε |
| have h2 := mul_le_mul_of_nonneg_right ht2 hε |
| nlinarith |
|
|
| lemma conditionalMeanData_hermitian (j : ℝ) (p : ScalarPoint) (r : ℝ) : |
| (conditionalMeanData j p r).IsHermitian := by |
| exact conditionalCoefficient_hermitian _ _ _ _ |
| lemma conditionalScalarData_hermitian (j : ℝ) (p : ScalarPoint) : |
| (conditionalScalarData j p).IsHermitian := by |
| exact conditionalCoefficient_hermitian _ _ _ _ |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Real Matrix MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| lemma diagonal_mass_margin {n : ℕ} (H : Matrix (Fin n) (Fin n) ℝ) |
| (a : Fin n → ℝ) {m r A : ℝ} (ha : ∀ i,0 ≤ a i) (hA : ∀ i,a i ≤ A) |
| (hr : |r| * A ≤ m) (hH : (H-(2*m) • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef) : |
| (H-r • diagonal a-m • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef := by |
| have hd : (diagonal (fun i=>m-r*a i)).PosSemidef := Matrix.PosSemidef.diagonal (fun i=>by |
| change 0 ≤ m-r*a i |
| have hh : r*a i ≤ |r| * A := (mul_le_mul_of_nonneg_right (le_abs_self r) (ha i)).trans |
| (mul_le_mul_of_nonneg_left (hA i) (abs_nonneg r)) |
| linarith) |
| convert hH.add_posSemidef hd using 1 |
| ext i k |
| by_cases hik : i=k |
| · subst k; simp; ring |
| · simp [hik] |
|
|
| lemma empirical_moment_vector_bound {n : ℕ} [NeZero n] {R : ℝ} |
| (hR : 0 ≤ R) (y : Fin n → ℝ) (hy : empiricalLaw y ∈ momentBall R) : |
| ∑ i,y i^2 ≤ (R+1)^2*(n:ℝ) := by |
| have hh := momentBall_integral_le hR hy |
| rw [integral_empiricalLaw] at hh |
| simp only [Fintype.card_fin] at hh |
| have hn : 0 < (n:ℝ) := by exact_mod_cast NeZero.pos n |
| have hp := mul_le_mul_of_nonneg_left hh hn.le |
| have hp' : ∑ i,y i^2 ≤ R*(n:ℝ) := by |
| simpa only [← mul_assoc,mul_inv_cancel₀ hn.ne',one_mul,mul_comm (n:ℝ) R] using hp |
| have hR' : R ≤ (R+1)^2 := by nlinarith [sq_nonneg R] |
| exact hp'.trans (mul_le_mul_of_nonneg_right hR' hn.le) |
|
|
| lemma empirical_cutoff_gram_small {n : ℕ} [NeZero n] (a y : Fin n → ℝ) |
| {A L ε ν R1 ρ d s : ℝ} |
| (ha : ∀ i,0 ≤ a i) (haA : ∀ i,a i ≤ A) (hA : 0 ≤ A) |
| (hs : 0 < s) (hL : 1 ≤ L) (hd : (1+|d|)/s ≤ L) |
| (hR1 : 1 ≤ R1) (hε : 0 ≤ ε) (hν : 0 ≤ ν) |
| (haclose : ∑ i,(a i-(1-tanh (y i)^2))^2 ≤ ε^2*(n:ℝ)) |
| (hy : (∑ i,if R1 < |y i| then y i^2 else 0) ≤ ν*(n:ℝ)) |
| (hεgram : 8*L^2*(1+R1^2)*ε ≤ ρ/8) |
| (hνgram : 16*L^2*(A+1)*ν ≤ ρ/8) : |
| ‖diagonalAugmentedGram a (empiricalGramColumns y d s)- |
| augmentedGram (ν := Unit) (scalarMomentGram (empiricalLaw y) d s)‖ ≤ ρ/4 := by |
| have hdiff (i : Fin n) : |a i-(1-tanh (y i)^2)| ≤ A+1 := by |
| have hv0 := tanh_sq_lt_one (y i) |
| have hv1 := sq_nonneg (tanh (y i)) |
| exact abs_le.mpr ⟨by linarith [ha i],by linarith [haA i]⟩ |
| have hg := empirical_augmented_gram_comparison a y hs hL hd |
| (by positivity : 0 ≤ A+1) hR1 hε hν hdiff haclose hy |
| apply hg.trans |
| nlinarith |
|
|
| lemma gram_close_by_quarters {α : Type*} [SeminormedAddCommGroup α] (a b c : α) |
| {ρ : ℝ} (hab : ‖a-b‖ ≤ ρ/4) (hbc : ‖b-c‖ < ρ/4) : ‖a-c‖ < ρ/2 := by |
| exact (norm_sub_le_norm_sub_add_norm_sub a b c).trans_lt (by linarith) |
|
|
| def conditionalEmpiricalCore {n : ℕ} [NeZero n] (j t σ : ℝ) (a y : Fin n → ℝ) |
| (g : MatrixCoordinates (Fin n) → ℝ) : Matrix (Fin n) (Fin n) ℝ := |
| let p : ScalarPoint := (empiricalLaw y,t,σ) |
| let B := j*scalarBMoment p.1 |
| let BA := j/(n:ℝ)*∑ i,a i |
| let u := empiricalUnit y (scalarQMoment p.1) |
| let V := conditionalUpdateColumns j a (empiricalGramColumns y (scalarD j p) (scalarS j p)) u g |
| conditionalBase j a g-pathDiagonal a 1*V* |
| (conditionalMeanData j p (BA-B)+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • conditionalScalarData j p)* |
| Vᵀ*pathDiagonal a 1+(B-BA) • diagonal a |
|
|
| theorem conditional_empirical_fixed_tail {j A R t0 T : ℝ} |
| (hj : 0 < j) (hj1 : j < 1) (hA : 1 ≤ A) (hsub : sqrt j*A < 1) |
| (hR : 0 ≤ R) (ht0 : 0 < t0) : |
| ∃ m ν σ0 : ℝ, 0 < m ∧ 0 < ν ∧ 0 < σ0 ∧ |
| ∃ U : Set (ScalarTimeDomain R t0 T), IsOpen U ∧ scalarEqualitySet j R t0 T ⊆ U ∧ |
| ∀ R1 : ℝ, 1 ≤ R1 → ∃ ε c : ℝ, ∃ N : ℕ, 0 < ε ∧ 0 < c ∧ 0 < N ∧ |
| ∀ (n : ℕ) [NeZero n], N ≤ n → ∀ x ∈ U, ∀ y : Fin n → ℝ, |
| empiricalLaw y=(x.1 : ProbabilityMeasure ℝ) → |
| (∑ i,if R1 < |y i| then y i^2 else 0) ≤ ν*(n:ℝ) → |
| ∀ σ : ℝ, |σ| ≤ σ0 → ∀ a : Fin n → ℝ, (∀ i,0 ≤ a i) → (∀ i,a i ≤ A) → |
| (∑ i,(a i-(1-tanh (y i)^2))^2) ≤ ε^2*(n:ℝ) → |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n)=>gaussianReal 0 1)).real |
| {g | ¬(conditionalEmpiricalCore j x.2 σ a y g-m • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef} ≤ |
| 101*exp (-c*(n:ℝ)) := by |
| have hA0 : 0 < A := by linarith |
| obtain ⟨hs,hq,hb,he,hκ,_⟩ := continuous_curve_statistics (T := T) hj hj1 ht0 |
| obtain ⟨δ,ρ,hδ,_,hρ,htail⟩ := compact_conditional_uniform_tail hj hj1 hA0 hsub |
| (fun t : Icc t0 T=>scalarBMoment (scalarCurve j t)) |
| (fun t : Icc t0 T=>scalarEMoment (scalarCurve j t)) |
| (fun t : Icc t0 T=>scalarFixedPoint j t) (fun t : Icc t0 T=>scalarQ j t) |
| (fun t : Icc t0 T=>curveKappa j t) hb he hs hq hκ |
| (fun t=>scalarBMoment_pos _) (fun t=>scalarBMoment_le_one _) (fun t=>scalarEMoment_nonneg _) |
| (fun t=>(scalar_fixed_pos hj hj1 (ht0.trans_le t.2.1)).1) |
| (fun t=>(scalar_fixed_pos hj hj1 (ht0.trans_le t.2.1)).2) |
| (fun t=>curveKappa_sq hj hj1 (ht0.trans_le t.2.1)) |
| obtain ⟨L,hL,U,hU,hKU,σ0,hσ0,hnb⟩ := scalar_conditional_neighborhood (R := R) (T := T) hj hj1 hR ht0 (by positivity : 0 < ρ/4) |
| let m := δ*((1-sqrt j*A)^2/4)/2 |
| have hm : 0 < m := by dsimp [m]; positivity |
| let terr := 16*L^2*(A+1) |
| have hterr : 0 < terr := by dsimp [terr]; positivity |
| let ν := ρ/(8*terr) |
| have hν : 0 < ν := by dsimp [ν]; positivity |
| let Vbound := 2*L*(R+2)+1 |
| have hV : 0 ≤ Vbound := by dsimp [Vbound]; positivity |
| let K := augmentedToleranceConstant L |
| have hK : 0 < K := augmentedToleranceConstant_pos (by linarith) |
| let εg := ρ/(4*K) |
| let τ := ρ/(4*(L+1)) |
| have hεg : 0 < εg := by dsimp [εg]; positivity |
| have hτ : 0 < τ := by dsimp [τ]; positivity |
| obtain ⟨c,N,hc,hN,hfixed⟩ := htail Vbound hV εg τ hεg hτ |
| refine ⟨m,ν,σ0,hm,hν,hσ0,U,hU,hKU,?_⟩ |
| intro R1 hR1 |
| let berr := 8*L^2*(1+R1^2) |
| have hberr : 0 < berr := by dsimp [berr]; positivity |
| let ε := min (σ0/(j+1)) (min (ρ/(8*berr)) (m/(j*A+1))) |
| have hε : 0 < ε := by dsimp [ε]; positivity |
| have hεσ : ε*(j+1) ≤ σ0 := (le_div_iff₀ (by positivity)).mp (min_le_left _ _) |
| have hεgram : berr*ε ≤ ρ/8 := by |
| have hh : ε ≤ ρ/(8*berr) := (min_le_right _ _).trans (min_le_left _ _) |
| have hh' := (le_div_iff₀ (by positivity : 0 < 8*berr)).mp hh |
| nlinarith only [hh'] |
| have hεmass : ε*(j*A+1) ≤ m := (le_div_iff₀ (by positivity)).mp ((min_le_right _ _).trans (min_le_right _ _)) |
| refine ⟨ε,c,N,hε,hc,hN,?_⟩ |
| intro n _ hn x hx y hP hy σ hσ a ha haA haclose |
| let p : ScalarPoint := (empiricalLaw y,(x.2:ℝ),σ) |
| let BA := j/(n:ℝ)*∑ i,a i |
| let B := j*scalarBMoment (empiricalLaw y) |
| let r := BA-B |
| have hr : |r| ≤ j*ε := empirical_mass_close a y hj.le hε.le haclose |
| have hrσ : |r| ≤ σ0 := hr.trans (by nlinarith only [hεσ,hε.le]) |
| have hnbe := hnb x hx σ r hσ hrσ |
| rw [← hP] at hnbe |
| change 0 < scalarQMoment p.1 ∧ 0 < scalarS j p ∧ 0 < B+r ∧ |
| conditionalScalarSize j p r < L ∧ _ ∧ _ at hnbe |
| have hBr : B+r=BA := by dsimp [r]; ring |
| have hBA : 0 < BA := by simpa only [hBr] using hnbe.2.2.1 |
| have hd := conditionalSize_bounds hnbe.1 hnbe.2.1 hnbe.2.2.1 hnbe.2.2.2.1 |
| have hBAle : BA < L := by |
| have hh := hd.2.2.2.2.1 |
| change B+r<L at hh |
| simpa only [hBr] using hh |
| have hBAinv : 1/BA < L := by |
| have hh := hd.2.2.2.1 |
| change 1/(B+r)<L at hh |
| simpa only [hBr] using hh |
| have htol : augmentedGramTolerance BA εg < ρ/2 := by |
| apply (augmentedGramTolerance_bound hBA hL hεg.le hBAle.le hBAinv.le).trans_lt |
| have heq : augmentedToleranceConstant L*εg=ρ/4 := by |
| dsimp [εg] |
| change K*(ρ/(4*K))=ρ/4 |
| field_simp |
| rw [heq]; linarith |
| have hEbound : τ*‖conditionalScalarData j p‖ < ρ/2 := by |
| apply (mul_lt_mul_of_pos_left hd.2.2.2.2.2 hτ).trans |
| have hL0 : 0 < L+1 := by linarith |
| have hh : τ*(L+1)=ρ/4 := by dsimp [τ]; field_simp |
| nlinarith only [hh,hτ,hρ] |
| have hvecmom := empirical_moment_vector_bound hR y (by rw [hP]; exact x.1.2) |
| let cols := empiricalGramColumns y (scalarD j p) (scalarS j p) |
| let u := empiricalUnit y (scalarQMoment p.1) |
| have hpbound : ∀ k,‖cols k‖ ≤ Vbound := fun k=>(empiricalGramColumns_norm (NeZero.pos n) y hnbe.2.1 hL (by linarith : 0 ≤ R+1) hd.1.le hvecmom k).trans (by dsimp [Vbound]; linarith) |
| have hubound : ‖u‖ ≤ Vbound := by |
| rw [empiricalUnit_norm y hnbe.1] |
| dsimp [Vbound] |
| have hp : 0 ≤ 2*L*(R+2) := by positivity |
| linarith |
| have hunit : u.ofLp⬝ᵥu.ofLp=1 := empiricalUnit_unit y hnbe.1 |
| have hνgram : 16*L^2*(A+1)*ν ≤ ρ/8 := by |
| have ht : terr*ν=ρ/8 := by dsimp [ν]; field_simp |
| exact ht.le |
| have hgsmall := empirical_cutoff_gram_small a y ha haA hA0.le hnbe.2.1 hL hd.1.le |
| hR1 hε.le hν.le haclose hy hεgram hνgram |
| have hGram : ‖diagonalAugmentedGram a cols-augmentedGram (ν := Unit) |
| (limitingGram (scalarBMoment (scalarCurve j x.2)) (scalarEMoment (scalarCurve j x.2)) |
| (scalarFixedPoint j x.2))‖ < ρ/2 := by |
| exact gram_close_by_quarters _ _ _ hgsmall hnbe.2.2.2.2.1 |
| have hCoef : ‖conditionalMeanData j p r-curveCoefficient j x.2‖ < ρ/2 := |
| hnbe.2.2.2.2.2.trans (by linarith) |
| have hprob := hfixed n hn a ha haA hBA htol cols u hpbound hubound hunit x.2 |
| (conditionalMeanData j p r) (conditionalScalarData j p) |
| (conditionalMeanData_hermitian j p r) (conditionalScalarData_hermitian j p) hEbound hGram hCoef |
| apply le_trans (measureReal_mono (s₂ := {g | ¬(conditionalBase j a g- |
| pathDiagonal a 1*conditionalUpdateColumns j a cols u g* |
| (conditionalMeanData j p r+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • conditionalScalarData j p)* |
| (conditionalUpdateColumns j a cols u g)ᵀ*pathDiagonal a 1- |
| (δ*((1-sqrt j*A)^2/4)) • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef}) ?_ (measure_ne_top _ _)) hprob |
| intro g hg |
| contrapose! hg |
| simp only [Set.mem_ofPred_eq,not_not] at hg ⊢ |
| have hrm : |r| * A ≤ m := by |
| apply (mul_le_mul_of_nonneg_right hr hA0.le).trans |
| nlinarith only [hεmass,hε.le] |
| have hm2 : 2*m=δ*((1-sqrt j*A)^2/4) := by dsimp [m]; ring |
| have hh := diagonal_mass_margin _ a ha haA hrm (by rw [hm2]; exact hg) |
| convert hh using 1 |
| dsimp [conditionalEmpiricalCore,p,B,BA,r,cols,u] |
| congr 1 |
| module |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
