| import OAI.Combinatorics.ContingencyTables.Dense.DenseRegularity |
| import OAI.Combinatorics.ContingencyTables.Dense.GridConductance |
|
|
| namespace OAI |
|
|
| /-! |
| # Conductance geometry for the actual dense penalty |
|
|
| The free table coordinates are reindexed by `Fin q`. The bin and cut-tube |
| density comparisons below follow from the actual margins and integer |
| widths. Thus the weighted cut theorem has no assumed penalty regularity |
| or bin comparison when used for the dense completion sampler. |
| -/ |
|
|
| namespace ContingencyTables.GridBoundary |
|
|
| open MeasureTheory |
| open scoped BigOperators |
|
|
| noncomputable section |
|
|
| variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] |
| variable {q : ℕ} |
|
|
| def realCorner {K : ℕ} (a : Grid (I := Fin q) K) : Fin q → ℝ := |
| fun j => (corner a j : ℝ) |
|
|
| def denseGridPenalty (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) |
| (e : I × J ≃ Fin q) (x : Fin q → ℝ) : ℝ := |
| DenseAffine.densePenalty R P d (fun p => x (e p)) |
|
|
| theorem denseGridPenalty_continuous (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (e : I × J ≃ Fin q) : Continuous (denseGridPenalty R P d e) := by |
| exact (DenseAffine.densePenalty_continuous R P d).comp |
| (continuous_pi fun p => continuous_apply (e p)) |
|
|
| theorem denseGridPenalty_convex (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (e : I × J ≃ Fin q) : |
| ConvexOn ℝ Set.univ (denseGridPenalty R P d e) := by |
| refine ⟨convex_univ, ?_⟩ |
| intro x _ y _ a b ha hb hab |
| exact (DenseAffine.densePenalty_convex R P d Set.univ convex_univ).2 |
| (Set.mem_univ _) (Set.mem_univ _) ha hb hab |
|
|
| theorem denseGridPenalty_change (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hfree : Fintype.card (I × J) ≤ d) (e : I × J ≃ Fin q) |
| (x y : Fin q → ℝ) (r : ℝ) (hr : 0 ≤ r) (hxy : ∀ j, |x j - y j| ≤ r) : |
| |(d : ℝ) ^ 4 * denseGridPenalty R P d e x - |
| (d : ℝ) ^ 4 * denseGridPenalty R P d e y| ≤ r := |
| DenseAffine.densePenalty_scaled_change R P d hd hR hP hRmax hPmax hfree |
| _ _ r hr (fun p => hxy (e p)) |
|
|
| private theorem closedBin_distance {K : ℕ} (a : Grid (I := Fin q) K) |
| (x : Fin q → ℝ) (hx : ClosedBin (corner a) x) : |
| ∀ j, |x j - realCorner a j| ≤ 1 := by |
| intro j |
| change |x j - (corner a j : ℝ)| ≤ 1 |
| apply abs_le.mpr |
| constructor <;> linarith [(hx j).1, (hx j).2] |
|
|
| private theorem tube_distances {K : ℕ} {Q : (Fin q → ℤ) → Prop} |
| (edge : CutEdge K Q) (r : ℝ) (_hr : 0 ≤ r) (x : Fin q → ℝ) |
| (hx : x ∈ edgeTube edge r) : |
| (∀ j, |x j - realCorner edge.val.1 j| ≤ 1 + r) ∧ |
| (∀ j, |x j - realCorner edge.val.2.1 j| ≤ 1 + r) := by |
| have hcoord (j : Fin q) := hx j (Set.mem_univ j) |
| constructor |
| · intro j |
| have h := hcoord j |
| dsimp [tubeLower, tubeUpper] at h |
| change |x j - (corner edge.val.1 j : ℝ)| ≤ 1 + r |
| by_cases hj : j = edge.val.2.2 |
| · subst j |
| simp only [ite_true] at h |
| apply abs_le.mpr |
| constructor <;> linarith [h.1, h.2] |
| · simp only [ite_eq_right hj] at h |
| apply abs_le.mpr |
| constructor <;> linarith [h.1, h.2] |
| · intro j |
| have h := hcoord j |
| dsimp [tubeLower, tubeUpper] at h |
| change |x j - (corner edge.val.2.1 j : ℝ)| ≤ 1 + r |
| by_cases hj : j = edge.val.2.2 |
| · have hadj : (corner edge.val.2.1 j : ℝ) = |
| (corner edge.val.1 j : ℝ) + 1 := by |
| simpa only [hj, Int.cast_add, Int.cast_one] using |
| congrArg (fun z : ℤ => (z : ℝ)) edge.property.1.1 |
| simp only [ite_eq_left hj] at h |
| rw [hadj] |
| apply abs_le.mpr |
| constructor <;> linarith [h.1, h.2] |
| · have heq : corner edge.val.2.1 j = corner edge.val.1 j := |
| congrArg Subtype.val (edge.property.1.2 j hj) |
| simp only [ite_eq_right hj] at h |
| rw [heq] |
| apply abs_le.mpr |
| constructor <;> linarith [h.1, h.2] |
|
|
| theorem denseGrid_bin_comparison (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hfree : Fintype.card (I × J) ≤ d) (e : I × J ≃ Fin q) |
| {K : ℕ} (a : Grid (I := Fin q) K) (x : Fin q → ℝ) |
| (hx : ClosedBin (corner a) x) : |
| DenseWeights.binWeight ((d : ℝ) ^ 4) (denseGridPenalty R P d e) (realCorner a) / 4 ≤ |
| DenseWeights.density ((d : ℝ) ^ 4) (denseGridPenalty R P d e) x ∧ |
| DenseWeights.density ((d : ℝ) ^ 4) (denseGridPenalty R P d e) x ≤ |
| 2 * DenseWeights.binWeight ((d : ℝ) ^ 4) (denseGridPenalty R P d e) (realCorner a) := |
| DenseWeights.bin_density_comparison _ _ _ _ |
| (denseGridPenalty_change R P d hd hR hP hRmax hPmax hfree e x _ 1 |
| (by norm_num) (closedBin_distance a x hx)) |
|
|
| theorem denseGrid_tube_comparison (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hfree : Fintype.card (I × J) ≤ d) (e : I × J ≃ Fin q) |
| {K : ℕ} {Q : (Fin q → ℤ) → Prop} (edge : CutEdge K Q) (r : ℝ) |
| (hr : 0 ≤ r) (hr₁ : r ≤ 1) (x : Fin q → ℝ) (hx : x ∈ edgeTube edge r) : |
| DenseWeights.density ((d : ℝ) ^ 4) (denseGridPenalty R P d e) x ≤ |
| 4 * min |
| (DenseWeights.binWeight ((d : ℝ) ^ 4) (denseGridPenalty R P d e) (realCorner edge.val.1)) |
| (DenseWeights.binWeight ((d : ℝ) ^ 4) (denseGridPenalty R P d e) (realCorner edge.val.2.1)) := by |
| obtain ⟨hlo, hhi⟩ := tube_distances edge r hr x hx |
| rw [mul_min_of_nonneg _ _ (by norm_num : (0 : ℝ) ≤ 4)] |
| apply le_min |
| · apply DenseWeights.tube_density_comparison |
| exact (denseGridPenalty_change R P d hd hR hP hRmax hPmax hfree e x _ (1 + r) |
| (by positivity) hlo).trans (by linarith) |
| · apply DenseWeights.tube_density_comparison |
| exact (denseGridPenalty_change R P d hd hR hP hRmax hPmax hfree e x _ (1 + r) |
| (by positivity) hhi).trans (by linarith) |
|
|
| / |
| conditionally only on the published finite-box integral inequality. -/ |
| theorem denseGrid_weighted_cut (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hfree : Fintype.card (I × J) ≤ d) (e : I × J ≃ Fin q) |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q) |
| {K : ℕ} (Q : (Fin q → ℤ) → Prop) (t : ℝ) (ht : 0 < t) (ht₁ : t < 1) |
| (hr : t * K ≤ 1 / 4) (hsize : (q - 1 : ℕ) * (2 * (t * K)) ≤ (1 / 2 : ℝ)) |
| (hSmass : 0 < ∫ x in selectedBins K Q, |
| DenseWeights.density ((d : ℝ) ^ 4) (denseGridPenalty R P d e) x) |
| (hcut : 2 * (∫ x in selectedBins K Q, |
| DenseWeights.density ((d : ℝ) ^ 4) (denseGridPenalty R P d e) x) ≤ |
| ∫ x in {x | Cube K x}, |
| DenseWeights.density ((d : ℝ) ^ 4) (denseGridPenalty R P d e) x) : |
| (((2 : ℝ) ^ t - 1) / 16) * |
| (∑ a : Selected K Q, DenseWeights.binWeight ((d : ℝ) ^ 4) |
| (denseGridPenalty R P d e) (realCorner a.val)) ≤ |
| ∑ edge : CutEdge K Q, min |
| (DenseWeights.binWeight ((d : ℝ) ^ 4) (denseGridPenalty R P d e) (realCorner edge.val.1)) |
| (DenseWeights.binWeight ((d : ℝ) ^ 4) (denseGridPenalty R P d e) (realCorner edge.val.2.1)) := by |
| apply weighted_cut_of_interpolation hPL Q t ht ht₁ hr hsize |
| (DenseWeights.density ((d : ℝ) ^ 4) (denseGridPenalty R P d e)) |
| (fun a => DenseWeights.binWeight ((d : ℝ) ^ 4) (denseGridPenalty R P d e) (realCorner a)) |
| · intro a |
| exact (DenseWeights.binWeight_pos _ _ _).le |
| · exact (DenseWeights.density_continuous _ _ (denseGridPenalty_continuous R P d e)).continuousOn |
| · intro x _ |
| exact DenseWeights.density_pos _ _ _ |
| · intro x _ y _ s hs hs₁ |
| exact DenseWeights.density_logConcave _ (by positivity) _ Set.univ |
| (denseGridPenalty_convex R P d e) x (Set.mem_univ _) y (Set.mem_univ _) s hs hs₁ |
| · intro a x hx |
| exact (denseGrid_bin_comparison R P d hd hR hP hRmax hPmax hfree e a x |
| (fun j => ⟨(hx j).1.le, (hx j).2.le⟩)).1 |
| · intro edge x hx |
| exact denseGrid_tube_comparison R P d hd hR hP hRmax hPmax hfree e edge (t * K) |
| (mul_nonneg ht.le (Nat.cast_nonneg _)) (by linarith) x hx.1 |
| · exact hSmass |
| · exact hcut |
|
|
| end |
|
|
| end ContingencyTables.GridBoundary |
|
|
| end OAI |
|
|
