| import Mathlib |
| import OAI.Probability.SKRatio.Matrices.GaussianSquareTails |
| import OAI.Probability.SKRatio.Variational.CompactField |
| import OAI.Probability.SKRatio.Quantization.IntervalBins |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Real Set |
| namespace SKRatio.Bins |
| open Planted Scalar SKRatioClock.Regression |
| attribute [local instance] Classical.propDecidable |
|
|
| def gridCut (R : ℝ) (N : ℕ) (j : Fin (N+1)) : ℝ := -R+(2*R/N)*(j.val:ℝ) |
|
|
| def gridRep (R : ℝ) (N : ℕ) (j : Fin (N+2)) : ℝ := -R+(2*R/N)*((min j.val N:ℕ):ℝ) |
|
|
| def gridQuant (R : ℝ) (N : ℕ) (x : ℝ) : ℝ := gridRep R N (intervalBin (gridCut R N) x) |
|
|
| lemma gridCut_strictMono {R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) : |
| StrictMono (gridCut R N) := by |
| intro i j hij |
| have hc : (i.val:ℝ)<(j.val:ℝ) := by exact_mod_cast hij |
| exact add_lt_add_right (mul_lt_mul_of_pos_left hc (by positivity : 0<2*R/(N:ℝ))) _ |
|
|
| lemma gridCut_zero (R : ℝ) (N : ℕ) : gridCut R N 0= -R := by simp [gridCut] |
| lemma gridCut_last (R : ℝ) {N : ℕ} (hN : 0<N) : gridCut R N (Fin.last N)=R := by |
| simp only [gridCut,Fin.val_last,div_mul_cancel₀ _ ((Nat.cast_ne_zero.mpr hN.ne' : (N:ℝ)≠0))] |
| ring |
| lemma gridRep_zero (R : ℝ) (N : ℕ) : gridRep R N 0= -R := by simp [gridRep] |
| lemma gridRep_last (R : ℝ) {N : ℕ} (hN : 0<N) : gridRep R N (Fin.last (N+1))=R := by |
| simp only [gridRep,Fin.val_last,min_eq_right (show N≤N+1 from by omega),div_mul_cancel₀ _ ((Nat.cast_ne_zero.mpr hN.ne' : (N:ℝ)≠0))] |
| ring |
| lemma gridRep_middle (R : ℝ) {N : ℕ} (j : Fin (N+2)) (hj : j.val<N+1) : |
| gridRep R N j=gridCut R N ⟨j.val,hj⟩ := by |
| simp only [gridRep,gridCut,min_eq_left (show j.val≤N from by omega)] |
|
|
| lemma gridRep_abs_le {R : ℝ} (hR : 0≤R) {N : ℕ} (hN : 0<N) (j : Fin (N+2)) : |
| |gridRep R N j|≤R := by |
| have h0 : (0:ℝ)≤(min j.val N:ℕ) := Nat.cast_nonneg _ |
| have h1 : ((min j.val N:ℕ):ℝ)≤(N:ℝ) := by exact_mod_cast min_le_right j.val N |
| have hδ : 0≤2*R/(N:ℝ) := by positivity |
| have hh := mul_le_mul_of_nonneg_left h1 hδ |
| rw [div_mul_cancel₀ _ ((Nat.cast_ne_zero.mpr hN.ne' : (N:ℝ)≠0))] at hh |
| unfold gridRep |
| exact abs_le.mpr ⟨by nlinarith [mul_nonneg hδ h0],by linarith⟩ |
|
|
| lemma gridQuant_left {R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) {x : ℝ} (hx : x≤-R) : |
| gridQuant R N x= -R := by |
| unfold gridQuant |
| have he : intervalBin (gridCut R N) x=0 := (intervalBin_eq_zero (by omega) |
| (gridCut_strictMono hR hN) x).mpr (by simpa [gridCut] using hx) |
| rw [he,gridRep_zero] |
|
|
| lemma gridQuant_right {R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) {x : ℝ} (hx : R<x) : |
| gridQuant R N x=R := by |
| unfold gridQuant |
| have he : intervalBin (gridCut R N) x=Fin.last (N+1) := (intervalBin_eq_last (by omega) |
| (gridCut_strictMono hR hN) x).mpr (by simpa only [Nat.add_sub_cancel,gridCut,div_mul_cancel₀ _ ((Nat.cast_ne_zero.mpr hN.ne' : (N:ℝ)≠0)),show -R+2*R=R by ring] using hx) |
| rw [he,gridRep_last R hN] |
|
|
| lemma gridQuant_middle {R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) {x : ℝ} |
| (hxL : -R<x) (hxU : x≤R) : |x-gridQuant R N x|≤2*R/N := by |
| let j := intervalBin (gridCut R N) x |
| have hj0 : j≠0 := fun hh => not_le.mpr hxL (by |
| simpa only [gridCut,Fin.val_mk,Nat.cast_zero,mul_zero,add_zero] using |
| (intervalBin_eq_zero (by omega) (gridCut_strictMono hR hN) x).mp hh) |
| have hjLast : j≠Fin.last (N+1) := fun hh => not_lt.mpr hxU (by |
| simpa only [Nat.add_sub_cancel,gridCut,Fin.val_mk,div_mul_cancel₀ _ ((Nat.cast_ne_zero.mpr hN.ne' : (N:ℝ)≠0)), |
| show -R+2*R=R by ring] using |
| (intervalBin_eq_last (by omega) (gridCut_strictMono hR hN) x).mp hh) |
| have hj0' : 0<j.val := by have hh : j.val≠0 := fun he => hj0 (Fin.ext he); omega |
| have hjN : j.val<N+1 := by |
| have hb := j.isLt |
| have hh : j.val≠N+1 := fun he => hjLast (Fin.ext he) |
| omega |
| have hbounds := (intervalBin_eq_middle (gridCut_strictMono hR hN) x j hj0' hjN).mp rfl |
| change |x-gridRep R N j|≤_ |
| rw [gridRep_middle R j hjN] |
| simp only [gridCut] at hbounds ⊢ |
| have hcast : ((j.val-1:ℕ):ℝ)=(j.val:ℝ)-1 := by rw [Nat.cast_sub (by omega),Nat.cast_one] |
| rw [hcast] at hbounds |
| rw [abs_of_nonpos (sub_nonpos.mpr hbounds.2)] |
| linarith only [hbounds.1] |
|
|
| lemma gridQuant_square_error {R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) (x : ℝ) : |
| (x-gridQuant R N x)^2≤(2*R/N)^2 + squareTail R x := by |
| by_cases hxL : x≤-R |
| · rw [gridQuant_left hR hN hxL] |
| by_cases he : x= -R |
| · simp only [he,sub_self,zero_pow (by decide : (2:ℕ)≠0)] |
| exact add_nonneg (sq_nonneg _) (squareTail_nonneg _ _) |
| · have hlt : x < -R := lt_of_le_of_ne hxL he |
| have hx : R < |x| := by rw [abs_of_neg (by linarith)]; linarith |
| rw [squareTail,ite_eq_left hx] |
| nlinarith [sq_nonneg (2*R/(N:ℝ))] |
| · by_cases hxU : x≤R |
| · have hb := gridQuant_middle hR hN (lt_of_not_ge hxL) hxU |
| have hs : (x-gridQuant R N x)^2≤(2*R/N)^2 := by |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) (by positivity : 0≤2*R/(N:ℝ))).mpr hb |
| linarith [squareTail_nonneg R x] |
| · have hx := lt_of_not_ge hxU |
| rw [gridQuant_right hR hN hx,squareTail,ite_eq_left (by rw [abs_of_pos (hR.trans hx)]; exact hx)] |
| nlinarith [sq_nonneg (2*R/(N:ℝ))] |
|
|
| lemma measurable_gridQuant {R : ℝ} (hR : 0<R) {N : ℕ} (hN : 0<N) : |
| Measurable (gridQuant R N) := |
| (measurable_of_countable (gridRep R N)).comp (measurable_intervalBin (by omega) (gridCut_strictMono hR hN)) |
|
|
| lemma gridQuant_compact_error {δ R : ℝ} (_ : 0<δ) (hR : 0<R) {N : ℕ} (hN : 0<N) |
| (hmesh : 2*R/(N:ℝ)<δ) |
| (htail : ∀ x : ℝ, R≤|x| → |tanh x-tanh (if 0≤x then R else -R)|<δ) (x : ℝ) : |
| dist (compactWeight x) (compactWeight (gridQuant R N x))<δ := by |
| change |w x-w (gridQuant R N x)|<δ |
| rw [w,w,sub_sub_sub_cancel_left,abs_sub_comm] |
| change |tanh x-tanh (gridQuant R N x)|<δ |
| by_cases hxL : x≤-R |
| · rw [gridQuant_left hR hN hxL] |
| have hh := htail x (by rw [abs_of_neg (by linarith)]; linarith) |
| simpa only [ite_eq_right (show ¬0≤x by linarith)] using hh |
| · by_cases hxU : x≤R |
| · exact ((Calculus.tanh_lipschitz_abs _ _).trans |
| (gridQuant_middle hR hN (lt_of_not_ge hxL) hxU)).trans_lt hmesh |
| · have hx := lt_of_not_ge hxU |
| rw [gridQuant_right hR hN hx] |
| have hh := htail x (by rw [abs_of_pos (hR.trans hx)]; exact hx.le) |
| simpa only [ite_eq_left (hR.trans hx).le] using hh |
|
|
| end SKRatio.Bins |
|
|
| end |
|
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| end OAI |
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