| import Mathlib |
| import OAI.Probability.SKGap.Matrix.RankNorm |
| import OAI.Probability.SKGap.Brownian.LogDetMesh |
| import OAI.Probability.SKGap.Stability.LogDetUniformMean |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators Matrix.Norms.Frobenius Topology |
|
|
| theorem actual_logdet_uniform {j ε : ℝ} (hj : 0 < j) (hj1 : j < 1) (hε : 0 < ε) : |
| ∃ γ₀ : ℝ, 0 < γ₀ ∧ ∀ γ : ℝ, 0 < γ → γ ≤ γ₀ → ∃ N : ℕ, 0 < N ∧ |
| ∀ n : ℕ, N ≤ n → |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1)).real |
| {g | ∃ a : Fin n → ℝ, (∀ i,0 ≤ a i) ∧ (∀ i,a i ≤ 1) ∧ |
| j*(coordAverage a)^2+ε < LogDet.regularizedLogDet γ |
| (logPath (j*coordAverage a) a (goeMatrix (j/(n:ℝ)) g) 1)} ≤ |
| 4*Real.exp (-min 1 (1/(π^2*j))*(n:ℝ)) := by |
| classical |
| obtain ⟨B,hB,hmean⟩ := actual_logdet_mean hj hj1 |
| refine ⟨ε/(4*(B^2+1)),by positivity,?_⟩ |
| intro γ hγ hγ₀ |
| obtain ⟨C,hC,N₀,hN₀,hm⟩ := hmean γ hγ |
| have hγB : γ*B^2 ≤ ε/4 := by |
| have hh := (le_div_iff₀ (by positivity : 0 < 4*(B^2+1))).mp hγ₀ |
| nlinarith |
| let R := 2*sqrt j+1 |
| have hR : 0 ≤ R := by dsimp [R]; positivity |
| let K := (2*j+R)/sqrt γ+2*j |
| have hK : 0 ≤ K := by dsimp [K]; positivity |
| let δ := ε/(4*(K+1)) |
| have hδ : 0 < δ := by dsimp [δ]; positivity |
| have hKδ : K*δ ≤ ε/4 := by |
| have hh : δ*(4*(K+1))=ε := div_mul_cancel₀ ε (ne_of_gt (by positivity : 0 < 4*(K+1))) |
| nlinarith |
| obtain ⟨s,hs,hcover⟩ := scalar_cube_net hδ |
| let q := γ*(ε/4)^2/(π^2*j) |
| have hq : 0 < q := by dsimp [q]; positivity |
| obtain ⟨N₁,hN₁⟩ := exists_nat_ge (4*C/ε) |
| obtain ⟨N₂,hN₂⟩ := exists_nat_ge (((s.card:ℝ)+1)/q) |
| refine ⟨max N₀ (max N₁ N₂),hN₀.trans_le (le_max_left _ _),?_⟩ |
| intro n hn |
| have hn₀ := (le_max_left N₀ (max N₁ N₂)).trans hn |
| have hn₁ := (le_max_left N₁ N₂).trans ((le_max_right N₀ (max N₁ N₂)).trans hn) |
| have hn₂ := (le_max_right N₁ N₂).trans ((le_max_right N₀ (max N₁ N₂)).trans hn) |
| have hnpos := hN₀.trans_le hn₀ |
| have hnreal : (0:ℝ) < n := Nat.cast_pos.mpr hnpos |
| let : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hnpos |
| have hCn : C/(n:ℝ) ≤ ε/4 := by |
| have hh := hN₁.trans (Nat.cast_le.mpr hn₁) |
| have hh' := (div_le_iff₀ hε).mp hh |
| apply (div_le_iff₀ hnreal).mpr |
| nlinarith |
| let μ := Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1) |
| let M := fun g : MatrixCoordinates (Fin n) → ℝ => goeMatrix (j/(n:ℝ)) g |
| let T := {g | ∃ v : Fin n → s, j*(coordAverage (fun i => (v i:ℝ)))^2+ε/2+ε/4 < |
| LogDet.regularizedLogDet γ (logPath (j*coordAverage (fun i => (v i:ℝ))) |
| (fun i => (v i:ℝ)) (M g) 1)} |
| let U := {g | R < opNorm (M g)} |
| have hT : μ.real T ≤ 2*Real.exp (-(n:ℝ)) := by |
| apply (logdet_mesh_tail hj hγ (by positivity : 0 ≤ ε/4) hnpos s hs ?_).trans |
| · have hh := product_mesh_absorption hq s.card n (hN₂.trans (Nat.cast_le.mpr hn₂)) |
| have he : -γ*(n:ℝ)^2*(ε/4)^2/(π^2*j)= -q*(n:ℝ)^2 := by dsimp only [q]; ring |
| simpa only [he] using hh |
| · intro a ha ha1 |
| have hh := hm n hn₀ a ha ha1 |
| have he : (j/(n:ℝ))*∑ i,a i=j*coordAverage a := by simp only [coordAverage,Fintype.card_fin]; ring |
| rw [he] at hh |
| change (∫ g, LogDet.regularizedLogDet γ (logPath (j*coordAverage a) a (M g) 1) ∂μ) ≤ _ |
| have heavg : (∑ i,a i)/(n:ℝ)=coordAverage a := by simp only [coordAverage,Fintype.card_fin] |
| rw [heavg] at hh |
| linarith |
| have hU : μ.real U ≤ 2*Real.exp (-(n:ℝ)/(π^2*j)) := by |
| change (gaussianCoordinates (MatrixCoordinates (Fin n))).real |
| {g | 2*sqrt j+1 < ‖matrixOperator (goeMatrix (j/(n:ℝ)) g)‖} ≤ _ |
| simpa only [one_pow,_root_.neg_one_mul,Fintype.card_fin] using |
| (goe_norm_tail (ι := Fin n) hj (by norm_num : (0:ℝ) ≤ 1)) |
| have hsub : {g | ∃ a : Fin n → ℝ, (∀ i,0 ≤ a i) ∧ (∀ i,a i ≤ 1) ∧ |
| j*(coordAverage a)^2+ε < LogDet.regularizedLogDet γ (logPath (j*coordAverage a) a (M g) 1)} ⊆ T ∪ U := by |
| rintro g ⟨a,ha,ha1,hbad⟩ |
| by_contra hg |
| have hnot := not_or.mp hg |
| have hop : opNorm (M g) ≤ R := le_of_not_gt hnot.2 |
| have hnet (v : Fin n → s) : LogDet.regularizedLogDet γ (logPath (j*coordAverage (fun i => (v i:ℝ))) |
| (fun i => (v i:ℝ)) (M g) 1) ≤ j*(coordAverage (fun i => (v i:ℝ)))^2+(ε/2+ε/4) := by |
| have hh : ¬ j*(coordAverage (fun i => (v i:ℝ)))^2+ε/2+ε/4 < |
| LogDet.regularizedLogDet γ (logPath (j*coordAverage (fun i => (v i:ℝ))) |
| (fun i => (v i:ℝ)) (M g) 1) := fun hv => hnot.1 ⟨v,hv⟩ |
| linarith [le_of_not_gt hh] |
| have hh := logdet_cube_net_transfer hj.le hδ hR hγ s hs hcover hop hnet ha ha1 |
| change LogDet.regularizedLogDet γ (logPath (j*coordAverage a) a (M g) 1) ≤ |
| j*(coordAverage a)^2+(ε/2+ε/4)+K*δ at hh |
| linarith |
| have hb := (measureReal_mono hsub (measure_ne_top _ _)).trans |
| ((measureReal_union_le T U).trans (add_le_add hT hU)) |
| apply hb.trans |
| have e₁ : Real.exp (-(n:ℝ)) ≤ Real.exp (-min 1 (1/(π^2*j))*(n:ℝ)) := by |
| apply Real.exp_le_exp.mpr |
| nlinarith [mul_nonneg (Nat.cast_nonneg n) (sub_nonneg.mpr (min_le_left 1 (1/(π^2*j))))] |
| have e₂ : Real.exp (-(n:ℝ)/(π^2*j)) ≤ Real.exp (-min 1 (1/(π^2*j))*(n:ℝ)) := by |
| apply Real.exp_le_exp.mpr |
| have hh := mul_le_mul_of_nonneg_right (min_le_right 1 (1/(π^2*j))) (Nat.cast_nonneg n) |
| simp only [div_eq_mul_inv,_root_.one_mul] at hh ⊢ |
| nlinarith |
| linarith |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators Matrix.Norms.Frobenius Topology |
|
|
| theorem actual_simultaneous_logdet {j ε : ℝ} (hj : 0 < j) (hj1 : j < 1) (hε : 0 < ε) : |
| ∃ γ₀ : ℝ, 0 < γ₀ ∧ ∀ γ : ℝ, 0 < γ → γ ≤ γ₀ → ∀ r₀ : ℕ, ∀ M₂ : ℝ, |
| ∃ N : ℕ, 0 < N ∧ ∀ n : ℕ, N ≤ n → |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1)).real |
| {g | ∃ a : Fin n → ℝ, (∀ i,0 ≤ a i) ∧ (∀ i,a i ≤ 1) ∧ |
| ∃ E : Matrix (Fin n) (Fin n) ℝ, E.rank ≤ r₀ ∧ opNorm E ≤ M₂ ∧ |
| j*(coordAverage a)^2+ε < LogDet.regularizedLogDet γ |
| (1+((j*coordAverage a) • 1-goeMatrix (j/(n:ℝ)) g)*diagonal a+E)} ≤ |
| 4*Real.exp (-min 1 (1/(π^2*j))*(n:ℝ)) := by |
| obtain ⟨γ₀,hγ₀,hbound⟩ := actual_logdet_uniform hj hj1 (half_pos hε) |
| refine ⟨γ₀,hγ₀,?_⟩ |
| intro γ hγ hγle r₀ M₂ |
| obtain ⟨N₀,hN₀,hb⟩ := hbound γ hγ hγle |
| let D := max 0 M₂ |
| have hD : 0 ≤ D := le_max_left _ _ |
| obtain ⟨N₁,hN₁⟩ := exists_nat_ge ((2*D*sqrt (r₀:ℝ)/ε)^2/γ) |
| refine ⟨max N₀ N₁,hN₀.trans_le (le_max_left _ _),?_⟩ |
| intro n hn |
| have hn₀ := (le_max_left N₀ N₁).trans hn |
| have hn₁ := (le_max_right N₀ N₁).trans hn |
| have hnpos := hN₀.trans_le hn₀ |
| have hnreal : (0:ℝ) < n := Nat.cast_pos.mpr hnpos |
| let : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hnpos |
| have hroot : 2*D*sqrt (r₀:ℝ)/ε ≤ sqrt ((n:ℝ)*γ) := by |
| apply (sq_le_sq₀ (by positivity) (sqrt_nonneg _)).mp |
| rw [sq_sqrt (by positivity)] |
| have hh := (div_le_iff₀ hγ).mp (hN₁.trans (Nat.cast_le.mpr hn₁)) |
| exact hh |
| have hpert : D*sqrt (r₀:ℝ)/sqrt ((n:ℝ)*γ) ≤ ε/2 := by |
| apply (div_le_iff₀ (sqrt_pos.mpr (mul_pos hnreal hγ))).mpr |
| have hh := (div_le_iff₀ hε).mp hroot |
| nlinarith |
| apply (measureReal_mono (s₂ := {g | ∃ a : Fin n → ℝ, (∀ i,0 ≤ a i) ∧ (∀ i,a i ≤ 1) ∧ |
| j*(coordAverage a)^2+ε/2 < LogDet.regularizedLogDet γ |
| (logPath (j*coordAverage a) a (goeMatrix (j/(n:ℝ)) g) 1)}) ?_ (measure_ne_top _ _)).trans (hb n hn₀) |
| rintro g ⟨a,ha,ha1,E,hr,hE,hbad⟩ |
| refine ⟨a,ha,ha1,?_⟩ |
| let W := goeMatrix (j/(n:ℝ)) g |
| let L := 1+((j*coordAverage a) • 1-W)*diagonal a |
| have htr : Lᵀ=logPath (j*coordAverage a) a W 1 := by |
| have hW : Wᵀ=W := goeMatrix_symm (j/(n:ℝ)) g |
| simp only [L,transpose_add,transpose_one,transpose_mul,transpose_sub,transpose_smul, |
| diagonal_transpose,hW,logPath,one_pow,one_mul,one_smul,Matrix.mul_sub,Matrix.mul_smul,Matrix.mul_one] |
| abel |
| have hf : LogDet.regularizedLogDet γ (L+E)= |
| LogDet.regularizedLogDet γ (logPath (j*coordAverage a) a W 1+Eᵀ) := by |
| rw [← regularizedLogDet_transpose (ne_of_gt hγ) (L+E),transpose_add,htr] |
| have hh := (abs_le.mp (logdet_rank_perturbation hγ hD |
| (logPath (j*coordAverage a) a W 1) Eᵀ |
| (by simpa only [Matrix.rank_transpose] using hr) |
| (by rw [opNorm_transpose]; exact hE.trans (le_max_right _ _)))).2 |
| simp only [Fintype.card_fin] at hh |
| change j*(coordAverage a)^2+ε < LogDet.regularizedLogDet γ (L+E) at hbad |
| rw [hf] at hbad |
| linarith |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
