| import Mathlib |
| import OAI.Probability.SKGap.Matrix.RegularizedLogDet |
| import OAI.Probability.SKGap.Brownian.LogDetPath |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators Matrix.Norms.Frobenius Topology |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι] |
|
|
| def coordAverage (a : ι → ℝ) : ℝ := (∑ i,a i)/(Fintype.card ι:ℝ) |
|
|
| omit [DecidableEq ι] in |
| lemma coordAverage_bounds {a : ι → ℝ} (ha : ∀ i, 0 ≤ a i) (ha1 : ∀ i, a i ≤ 1) : |
| 0 ≤ coordAverage a ∧ coordAverage a ≤ 1 := by |
| have hn : (0:ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos |
| constructor |
| · exact div_nonneg (Finset.sum_nonneg (fun i _ => ha i)) hn.le |
| · apply (div_le_iff₀ hn).mpr |
| simpa using Finset.sum_le_sum (s := Finset.univ) (fun i _ => ha1 i) |
|
|
| omit [DecidableEq ι] in |
| lemma coordAverage_sub_le {a b : ι → ℝ} {δ : ℝ} (hab : ∀ i, |a i-b i| ≤ δ) : |
| |coordAverage a-coordAverage b| ≤ δ := by |
| have hn : (0:ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos |
| unfold coordAverage |
| rw [← sub_div,← Finset.sum_sub_distrib,abs_div,abs_of_pos hn] |
| apply (div_le_iff₀ hn).mpr |
| exact ((Finset.abs_sum_le_sum_abs _ _).trans |
| (Finset.sum_le_sum (fun i _ => hab i))).trans_eq (by simp [mul_comm]) |
|
|
| omit [DecidableEq ι] in |
| lemma coordAverage_sq_sub_le {j δ : ℝ} (hj : 0 ≤ j) {a b : ι → ℝ} |
| (ha : ∀ i, 0 ≤ a i) (ha1 : ∀ i,a i ≤ 1) (hb : ∀ i,0 ≤ b i) (hb1 : ∀ i,b i ≤ 1) |
| (hab : ∀ i, |a i-b i| ≤ δ) : |
| |j*(coordAverage a)^2-j*(coordAverage b)^2| ≤ 2*j*δ := by |
| have haa := coordAverage_bounds ha ha1 |
| have hbb := coordAverage_bounds hb hb1 |
| have hd := coordAverage_sub_le hab |
| have hh := mul_le_mul_of_nonneg_left hd hj |
| have he : j*(coordAverage a)^2-j*(coordAverage b)^2= |
| (j*(coordAverage a-coordAverage b))*(coordAverage a+coordAverage b) := by ring |
| rw [he,abs_mul,abs_mul,abs_of_nonneg hj,abs_of_nonneg (by linarith : 0 ≤ coordAverage a+coordAverage b)] |
| calc |
| _ ≤ (j*δ)*2 := mul_le_mul hh (by linarith) (by linarith : 0 ≤ coordAverage a+coordAverage b) (by nlinarith [abs_nonneg (coordAverage a-coordAverage b)]) |
| _ = _ := by ring |
|
|
| lemma logPath_cube_difference {j δ R : ℝ} (hj : 0 ≤ j) (hδ : 0 ≤ δ) (_hR : 0 ≤ R) |
| {a b : ι → ℝ} (ha : ∀ i,0 ≤ a i) (ha1 : ∀ i,a i ≤ 1) |
| (hb : ∀ i,0 ≤ b i) (hb1 : ∀ i,b i ≤ 1) (hab : ∀ i,|a i-b i| ≤ δ) |
| {M : Matrix ι ι ℝ} (hM : opNorm M ≤ R) : |
| ‖logPath (j*coordAverage a) a M 1-logPath (j*coordAverage b) b M 1‖ ≤ |
| sqrt (Fintype.card ι:ℝ)*(2*j+R)*δ := by |
| have haa := coordAverage_bounds ha ha1 |
| have hd := coordAverage_sub_le hab |
| have hDa : opNorm (diagonal a-diagonal b) ≤ δ := by |
| rw [diagonal_sub] |
| exact opNorm_diagonal_le hδ hab |
| have hDb : opNorm (diagonal b) ≤ 1 := opNorm_diagonal_le (by norm_num) |
| (fun i => by rw [abs_of_nonneg (hb i)]; exact hb1 i) |
| have hF₁ := (LogDet.frobenius_le_sqrt_card_opNorm (diagonal a-diagonal b)).trans |
| (mul_le_mul_of_nonneg_left hDa (sqrt_nonneg _)) |
| have hF₂ := (LogDet.frobenius_le_sqrt_card_opNorm (diagonal b)).trans |
| (mul_le_mul_of_nonneg_left hDb (sqrt_nonneg _)) |
| simp only [mul_one] at hF₂ |
| have hq : |j*coordAverage a| ≤ j := by rw [abs_of_nonneg (mul_nonneg hj haa.1)]; nlinarith |
| have hq' : |j*coordAverage a-j*coordAverage b| ≤ j*δ := by |
| rw [← mul_sub,abs_mul,abs_of_nonneg hj]; exact mul_le_mul_of_nonneg_left hd hj |
| have he : logPath (j*coordAverage a) a M 1-logPath (j*coordAverage b) b M 1= |
| (j*coordAverage a) • (diagonal a-diagonal b)+ |
| (j*coordAverage a-j*coordAverage b) • diagonal b-(diagonal a-diagonal b)*M := by |
| simp only [logPath,one_pow,one_mul,one_smul,smul_sub,sub_smul,Matrix.sub_mul] |
| abel |
| rw [he] |
| apply (norm_sub_le _ _).trans |
| apply (add_le_add (norm_add_le _ _) (frobenius_mul_le_opNorm_right _ _)).trans |
| rw [norm_smul,norm_smul,Real.norm_eq_abs,Real.norm_eq_abs] |
| have hf₁ := mul_le_mul hq hF₁ (norm_nonneg _) hj |
| have hf₂ := mul_le_mul hq' hF₂ (norm_nonneg _) (mul_nonneg hj hδ) |
| have hf₃ := mul_le_mul hF₁ hM (norm_nonneg _) (mul_nonneg (sqrt_nonneg _) hδ) |
| nlinarith |
|
|
| lemma scalar_cube_net {δ : ℝ} (hδ : 0 < δ) : |
| ∃ s : Finset ℝ, (∀ x ∈ s, x ∈ Icc (0:ℝ) 1) ∧ |
| ∀ x ∈ Icc (0:ℝ) 1, ∃ y ∈ s, |x-y| ≤ δ := by |
| classical |
| obtain ⟨s,hs,hsfin,hcover⟩ := (isCompact_Icc : IsCompact (Icc (0:ℝ) 1)).finite_cover_balls hδ |
| refine ⟨hsfin.toFinset,fun x hx => hs (hsfin.mem_toFinset.mp hx),?_⟩ |
| intro x hx |
| obtain ⟨y,hy,hxy⟩ := mem_iUnion₂.mp (hcover hx) |
| exact ⟨y,hsfin.mem_toFinset.mpr hy,(show |x-y| < δ from Metric.mem_ball.mp hxy).le⟩ |
| end SKGap |
| end |
| end |
|
|
| end OAI |
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