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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Combinatorics.MatchingPSD.CellDecoder
namespace OAI
namespace MatchingInterpolation
open MatchingSmoothing PerfectMatchingPSD Matrix
open scoped BigOperators MatrixOrder
lemma interpolation_trace_error_bound (d : ℕ) {c L r η m : ℝ}
(hc : 0 ≤ c) (hL : 1 ≤ L)
(hw : weightMass d L ≤ c/L^2) (hnorm : r*η*m ≤ 1/L) :
weightMass d L*(2*(d:ℝ)*L)+r*(6+weightMass d L)*η*m ≤
((2*(d:ℝ)+1)*c+6)/L := by
have hLp : 0 < L := by linarith
have hwc : weightMass d L ≤ c := by
apply hw.trans
exact (div_le_self hc (by nlinarith : 1 ≤ L^2))
have h1 := mul_le_mul_of_nonneg_right hw (show 0 ≤ 2*(d:ℝ)*L by positivity)
have h2 := mul_le_mul_of_nonneg_left hnorm (show 0 ≤ 6+weightMass d L by linarith [weightMass_nonneg d L])
have h3 := mul_le_mul_of_nonneg_right (show 6+weightMass d L ≤ 6+c by linarith) (show 0 ≤ 1/L by positivity)
calc
_ = weightMass d L*(2*(d:ℝ)*L)+(6+weightMass d L)*(r*η*m) := by ring
_ ≤ (c/L^2)*(2*(d:ℝ)*L)+(6+c)*(1/L) := add_le_add h1 (h2.trans h3)
_ = _ := by field_simp; ring
lemma sqrt_error_small {q D L ε : ℝ} (hq : 0 ≤ q) (hL : 0 < L) (hε : 0 ≤ ε)
(he : ε ≤ (q*D+1)/L) (hn : 16*q^2*(q*D+1) ≤ L) :
q*Real.sqrt ε ≤ 1/4 := by
have heL : ε*L ≤ q*D+1 := (le_div_iff₀ hL).mp he
have hmult := mul_le_mul_of_nonneg_left heL (show 0 ≤ 16*q^2 by positivity)
have h16 : 16*q^2*ε ≤ 1 := by
apply (mul_le_mul_iff_of_pos_right hL).mp
nlinarith [hmult.trans hn]
have hsq : (q*Real.sqrt ε)^2 = q^2*ε := by rw [mul_pow, Real.sq_sqrt hε]
have hn0 : 0 ≤ q*Real.sqrt ε := mul_nonneg hq (Real.sqrt_nonneg _)
nlinarith
lemma normalized_eta {r m L : ℝ} (hr : 0 < r) (hm : 0 < m) (hL : 0 < L) :
r*(1/(r*m*L))*m = 1/L := by
field_simp [hr.ne', hm.ne', hL.ne']
lemma conditional_cell_lower_bound (K : ℕ) :
∃ d H : ℕ, 0 < d ∧ ∃ D : ℝ, 0 < D ∧
∀ s q l r : ℕ, H ≤ 3*s+1 → 1 ≤ s → 1 ≤ q → 2*l*d ≤ s → 1 ≤ l → 0 < r →
∀ m : ℝ, 0 < m →
∀ (Q : Equiv.Perm (GlobalVertices s q) → (Fin q → Bool) → Matrix (Fin r) (Fin r) ℝ)
(B : Pairing (GlobalVertices s q) → Matrix (Fin r) (Fin r) ℝ),
(∀ a y, (Q a y).PosSemidef) → (∀ M, (B M).PosSemidef) →
(∀ a y, (Q a y).trace ≤ m) → (∀ M, (B M).trace ≤ (r:ℝ)) →
(∀ a y M, (Q a y*B (Pairing.relabel a M)).trace =
tableCoordinate (globalCutColor y) (true,false) M-1) →
(q:ℝ)*r^4*(Fintype.card (SampleIndex d))*((2*(3*s+1):ℝ)^(2*K))⁻¹ ≤
(1/((r:ℝ)*m*(2*l)))^2 →
16*(q:ℝ)^2*((q:ℝ)*D+1) ≤ 2*l →
(2:ℝ)^q/2 ≤ r := by
obtain ⟨d,H,hd,hgood⟩ := good_cell_polynomials K
obtain ⟨c,hc,hw⟩ := weightMass_bound d
let D := (2*(d:ℝ)+1)*c+6
have hD : 0 < D := by dsimp [D]; positivity
refine ⟨d,H,hd,D,hD,?_⟩
intro s q l r hH hs hq hsl hl hr m hm Q B hQ hB htrQ htrB hpair hnum hn
let L : ℝ := 2*l
have hL : 1 ≤ L := by dsimp [L]; exact_mod_cast (by omega : 1 ≤ 2*l)
have hLp : 0 < L := by linarith
have hrp : 0 < (r:ℝ) := by exact_mod_cast hr
let η := 1/((r:ℝ)*m*L)
let δ := 1/(L*m)
have hη : 0 ≤ η := by dsimp [η]; positivity
have hδ : 0 < δ := by dsimp [δ]; positivity
obtain ⟨a,P,hP,happrox⟩ := hgood s q l r hH hs hsl B hB htrB η hη hnum
let e := weightMass d L*(2*(d:ℝ)*L)+(r:ℝ)*(6+weightMass d L)*η*m
have hw0 := weightMass_nonneg d L
have he0 : 0 ≤ e := by dsimp [e]; positivity
have he : e ≤ D/L := interpolation_trace_error_bound d hc.le hL
(hw L hL) (le_of_eq (normalized_eta hrp hm hLp))
have hδm : δ*m=1/L := by dsimp [δ]; field_simp [hLp.ne', hm.ne']
have hε : 0 ≤ (q:ℝ)*e+δ*m := by positivity
have heps : (q:ℝ)*e+δ*m ≤ ((q:ℝ)*D+1)/L := by
have heq := mul_le_mul_of_nonneg_left he (Nat.cast_nonneg q)
rw [hδm]
calc
_ ≤ (q:ℝ)*(D/L)+1/L := add_le_add heq (le_refl (1/L))
_ = _ := by ring
have hsmall := sqrt_error_small (Nat.cast_nonneg q) hLp hε heps hn
exact cell_decoder hs hq hsl (Q a) (fun M => B (Pairing.relabel a M))
(hQ a) (fun M => hB _) (hpair a) P hP hη hδ (by omega)
(htrQ a) happrox hε hsmall
end MatchingInterpolation
end OAI
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