| import OAI.Analysis.FreeEntropy.MatrixReparam |
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| namespace OAI |
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| universe u1 u2 u3 u4 |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset Filter |
| open scoped Classical Topology |
| namespace FiniteEntropySeparation |
| theorem complex_mean_square_bound {ι : Type u1} [Fintype ι] [DecidableEq ι] |
| {f : (ι → ℝ) → ℂ} (hf : IsComplexPolynomial f) (z : ℂ) : |
| (∫ x, ‖f x-z‖^2 ∂productGaussian ι) ≤ |
| 2*(∫ x, ‖f x-∫ y, f y ∂productGaussian ι‖^2 ∂productGaussian ι)+2*‖(∫ y, f y ∂productGaussian ι)-z‖^2 := by |
| let a := ∫ y, f y ∂productGaussian ι |
| have hg := (hf.sub (IsComplexPolynomial.const a)).norm_sq.integrable |
| calc |
| _ ≤ ∫ x, 2*‖f x-a‖^2+2*‖a-z‖^2 ∂productGaussian ι := by |
| apply integral_mono ((hf.sub (IsComplexPolynomial.const z)).norm_sq.integrable) ((hg.const_mul 2).add (integrable_const _)) |
| intro x |
| have ht : ‖f x-z‖ ≤ ‖f x-a‖+‖a-z‖ := by |
| simpa only [sub_add_sub_cancel] using norm_add_le (f x-a) (a-z) |
| have hp := pow_le_pow_left₀ (norm_nonneg _) ht 2 |
| change ‖f x-z‖^2 ≤ 2*‖f x-a‖^2+2*‖a-z‖^2 |
| nlinarith [sq_nonneg (‖f x-a‖-‖a-z‖)] |
| _ = _ := by rw [integral_add (hg.const_mul _) (integrable_const _),integral_const_mul]; simp [a] |
| theorem real_tendsto_zero_of_inv_sq {D : ℕ → ℕ} (hD : Tendsto D atTop atTop) (C : ℝ) : |
| Tendsto (fun k => C/(D k:ℝ)^2) atTop (𝓝 0) := by |
| have hc : Tendsto (fun k => (D k:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop.comp hD |
| have hi := tendsto_inv_atTop_zero.comp (tendsto_pow_atTop (by norm_num : (2:ℕ) ≠ 0) |>.comp hc) |
| simpa [div_eq_mul_inv] using tendsto_const_nhds.mul hi |
| theorem matrixTraceWord_mean_square_tendsto {α : Type u2} {n : ℕ} {D : ℕ → ℕ} |
| (hD0 : ∀ k,0<D k) (hD : Tendsto D atTop atTop) |
| (A : (k : ℕ) → α → Matrix (Fin (D k)) (Fin (D k)) ℂ) {R : ℝ} (hR : 0 ≤ R) |
| (hA : ∀ k a, matrixOpNorm (A k a) ≤ R) (w : List (α ⊕ Fin n)) {z : ℂ} |
| (hm : Tendsto (fun k => expectedMatrixWord A k w) atTop (𝓝 z)) : |
| Tendsto (fun k => ∫ x, ‖matrixTraceWord (jointMatrix (A k) x) w-z‖^2 ∂productGaussian (MatrixIndex n (D k))) atTop (𝓝 0) := by |
| have hvar : Tendsto (fun k => ∫ x, ‖matrixTraceWord (jointMatrix (A k) x) w-expectedMatrixWord A k w‖^2 ∂productGaussian (MatrixIndex n (D k))) atTop (𝓝 0) := by |
| apply squeeze_zero (fun k => integral_nonneg (fun _ => sq_nonneg _)) |
| (fun k => (matrixTraceWord_variance_bound (hD0 k) (A k) hR (hA k) w).trans_eq (by ring)) |
| (real_tendsto_zero_of_inv_sq hD ((w.length:ℝ)^2*jointMomentBound n R (2*(w.length-1)))) |
| have hb : Tendsto (fun k => 2*(∫ x, ‖matrixTraceWord (jointMatrix (A k) x) w-expectedMatrixWord A k w‖^2 ∂productGaussian (MatrixIndex n (D k)))+2*‖expectedMatrixWord A k w-z‖^2) atTop (𝓝 0) := by |
| have hh := (hvar.const_mul 2).add (((hm.sub_const z).norm.pow 2).const_mul 2) |
| simpa using hh |
| exact squeeze_zero (fun k => integral_nonneg (fun _ => sq_nonneg _)) |
| (fun k => complex_mean_square_bound (matrixTraceWord_polynomial (A k) w) z) hb |
| theorem probability_error_le_mean_square {Ω : Type u3} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] |
| {f : Ω → ℂ} {z : ℂ} {ε : ℝ} (hε : 0 < ε) (hi : Integrable (fun x => ‖f x-z‖^2) μ) : |
| μ.real {x | ε ≤ ‖f x-z‖} ≤ (∫ x, ‖f x-z‖^2 ∂μ)/ε^2 := by |
| have he : {x | ε ≤ ‖f x-z‖} = {x | ε^2 ≤ ‖f x-z‖^2} := by |
| ext x; simp only [Set.mem_ofPred_eq]; exact (sq_le_sq₀ hε.le (norm_nonneg _)).symm |
| rw [he] |
| exact (le_div_iff₀ (sq_pos_of_pos hε)).mpr (by |
| simpa [mul_comm] using mul_meas_ge_le_integral_of_nonneg (ae_of_all _ (fun x => sq_nonneg ‖f x-z‖)) hi (ε^2)) |
| theorem matrixTraceWord_probability_tendsto {α : Type u4} {n : ℕ} {D : ℕ → ℕ} |
| (hD0 : ∀ k,0<D k) (hD : Tendsto D atTop atTop) |
| (A : (k : ℕ) → α → Matrix (Fin (D k)) (Fin (D k)) ℂ) {R : ℝ} (hR : 0 ≤ R) |
| (hA : ∀ k a, matrixOpNorm (A k a) ≤ R) (w : List (α ⊕ Fin n)) {z : ℂ} |
| (hm : Tendsto (fun k => expectedMatrixWord A k w) atTop (𝓝 z)) {ε : ℝ} (hε : 0 < ε) : |
| Tendsto (fun k => (productGaussian (MatrixIndex n (D k))).real {x | ε ≤ ‖matrixTraceWord (jointMatrix (A k) x) w-z‖}) atTop (𝓝 0) := by |
| apply squeeze_zero (fun k => measureReal_nonneg) |
| (fun k => probability_error_le_mean_square _ hε ((matrixTraceWord_polynomial (A k) w).sub (IsComplexPolynomial.const z)).norm_sq.integrable) |
| simpa using (matrixTraceWord_mean_square_tendsto hD0 hD A hR hA w hm).div_const (ε^2) |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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