| import Mathlib |
| import OAI.Probability.SKRatio.Quantization.BinStability |
| import OAI.Probability.SKRatio.Quantization.TanhEnds |
| import OAI.Probability.SKRatio.Quantization.Quantizer |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Real Set |
| namespace SKRatio.Bins |
| open Planted Scalar SKRatioClock.Regression |
|
|
| lemma quantizer_integrable_square {R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) |
| {μ : Measure ℝ} [IsProbabilityMeasure μ] (hi : Integrable (fun x : ℝ => x^2) μ) : |
| Integrable (fun x => (x-gridQuant R N x)^2) μ := by |
| apply ((integrable_const ((2*R/N)^2)).add (integrable_squareTail hi R)).mono' |
| ((measurable_id.sub (measurable_gridQuant hR hN)).pow_const 2).aestronglyMeasurable |
| exact ae_of_all _ (fun x => by |
| rw [norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| exact gridQuant_square_error hR hN x) |
|
|
| theorem quantizer_exists |
| {Ω : ℕ → Type*} [∀ n, MeasurableSpace (Ω n)] |
| {ρ : ∀ n, Measure (Ω n)} {X : ∀ n, Ω n → Fin n → ℝ} |
| (hX : ExponentialSquareTails ρ X) |
| {μ : Measure ℝ} [IsProbabilityMeasure μ] (hi : Integrable (fun x : ℝ => x^2) μ) |
| {δ d : ℝ} (hδ : 0<δ) (hd : 0<d) : |
| ∃ R : ℝ, ∃ N : ℕ, 0<R ∧ 0<N ∧ |
| (∀ x, dist (compactWeight x) (compactWeight (gridQuant R N x))<δ) ∧ |
| (∫ x, (x-gridQuant R N x)^2 ∂μ)<d^2 ∧ |
| ExponentiallyRare ρ (fun n => {ω | d≤ siteNorm (fun i => X n ω i-gridQuant R N (X n ω i))}) := by |
| obtain ⟨Rt,hRt,ht⟩ := hX (d^2/2) (by positivity) |
| obtain ⟨Rc,hRc,hc⟩ := tanh_tail_uniform hδ |
| have he := (tendsto_integral_squareTail hi).eventually (gt_mem_nhds (by positivity : (0:ℝ)<d^2/4)) |
| obtain ⟨Ng,hNg⟩ := eventually_atTop.mp he |
| let R := max (max Rt Rc) (Ng:ℝ) |
| have hR : 0<R := hRc.trans_le ((le_max_right Rt Rc).trans (le_max_left _ _)) |
| have hRtR : Rt≤R := (le_max_left Rt Rc).trans (le_max_left _ _) |
| have hRcR : Rc≤R := (le_max_right Rt Rc).trans (le_max_left _ _) |
| have hRg : (Ng:ℝ)≤R := le_max_right _ _ |
| have hg : (∫ x, squareTail R x ∂μ)<d^2/4 := by |
| apply lt_of_le_of_lt (integral_mono (integrable_squareTail hi R) (integrable_squareTail hi Ng) |
| (squareTail_antitone hRg)) |
| exact hNg Ng le_rfl |
| let a := min δ (d/2) |
| have ha : 0<a := lt_min hδ (half_pos hd) |
| obtain ⟨N,hN⟩ := exists_nat_gt (2*R/a) |
| have hN0 : 0<N := by |
| have hnR : (0:ℝ)<N := (by positivity : (0:ℝ)<2*R/a).trans hN |
| exact_mod_cast hnR |
| have hmesh : 2*R/(N:ℝ)<a := by |
| rw [div_lt_iff₀ (Nat.cast_pos.mpr hN0)] |
| have hh := (div_lt_iff₀ ha).mp hN |
| nlinarith only [hh] |
| have hmeshd : (2*R/(N:ℝ))^2<d^2/4 := by |
| have hh := hmesh.trans_le (min_le_right δ (d/2)) |
| have h0 : 0≤2*R/(N:ℝ) := by positivity |
| nlinarith only [hh,hd,h0] |
| refine ⟨R,N,hR,hN0,?_,?_,?_⟩ |
| · exact gridQuant_compact_error hδ hR hN0 (hmesh.trans_le (min_le_left _ _)) (hc R hRcR) |
| · have hle := integral_mono (quantizer_integrable_square hR hN0 hi) |
| ((integrable_const ((2*R/(N:ℝ))^2)).add (integrable_squareTail hi R)) |
| (gridQuant_square_error hR hN0) |
| simp only [Pi.add_apply] at hle |
| rw [integral_add (integrable_const _) (integrable_squareTail hi R),integral_const] at hle |
| simp only [probReal_univ,one_smul] at hle |
| linarith [sq_pos_of_pos hd] |
| · apply ht.mono |
| filter_upwards [eventually_gt_atTop 0] with n hn |
| intro ω hω |
| change d≤ siteNorm (fun i => X n ω i-gridQuant R N (X n ω i)) at hω |
| have hnR : (0:ℝ)<n := Nat.cast_pos.mpr hn |
| have h1 : d^2≤(∑ i, (X n ω i-gridQuant R N (X n ω i))^2)/(n:ℝ) := by |
| rw [←siteNorm_sq] |
| exact (sq_le_sq₀ hd.le (siteNorm_nonneg _)).mpr hω |
| have h2 := Finset.sum_le_sum (s := Finset.univ) (fun i _ => gridQuant_square_error hR hN0 (X n ω i)) |
| have h3 := Finset.sum_le_sum (s := Finset.univ) (fun i _ => squareTail_antitone hRtR (X n ω i)) |
| rw [Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] at h2 |
| have hh := div_le_div_of_nonneg_right h2 hnR.le |
| rw [add_div,mul_div_cancel_left₀ _ (ne_of_gt hnR)] at hh |
| have hh' := div_le_div_of_nonneg_right h3 hnR.le |
| change d^2/2≤(∑ i, squareTail Rt (X n ω i))/(n:ℝ) |
| nlinarith only [h1,hh,hh',hmeshd,sq_nonneg d] |
|
|
| end SKRatio.Bins |
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| end |
|
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| end OAI |
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