| import OAI.Combinatorics.MatchingPSD.DominationDecoder |
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| namespace OAI |
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| namespace MatchingInterpolation |
| open MatchingSmoothing PerfectMatchingPSD Matrix |
| open scoped BigOperators MatrixOrder |
| attribute [local instance] Classical.propDecidable |
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| noncomputable def cutValue (b : Bool) (x : Point) : ℝ := x 0+x (if b then 2 else 1)-1 |
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| lemma cell_trace_value {s q r : ℕ} |
| (Q : (Fin q → Bool) → Matrix (Fin r) (Fin r) ℝ) |
| (B : Pairing (GlobalVertices s q) → Matrix (Fin r) (Fin r) ℝ) |
| (hpair : ∀ y M, (Q y*B M).trace = tableCoordinate (globalCutColor y) (true,false) M-1) |
| (y : Fin q → Bool) (i : Fin q) (x : Point) |
| (hx : ∃ M : Pairing (TripleVertices s), crossVector threeColor M=x) : |
| (Q y*cellMatrix i B x).trace = cutValue (y i) x := by |
| have : Nonempty (Pairing (Fin s × Bool)) := ⟨standardPairing⟩ |
| rw [trace_cellMatrix] |
| apply cellAverage_const i _ x _ hx |
| intro M hM N |
| rw [hpair, extendLocal_cut, hM] |
| rfl |
|
|
| lemma cutValue_p (b : Bool) : cutValue b (origin+dirP) = 0 := by |
| cases b < |
| lemma cutValue_star (b : Bool) : cutValue b (origin+dirStar) = 2 := by |
| cases b < |
| lemma cutValue_bit (b : Bool) : cutValue b (if b then origin+dirB else origin+dirA) = 2 := by |
| cases b < |
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| lemma cutValue_grid {d : ℕ} {L : ℝ} (hL : 0 ≤ L) (g : GridIndex d) (b : Bool) : |
| cutValue b (scaledNode L g) ≤ 2*(d:ℝ)*L := by |
| have hk (a : Fin 3) : (g a : ℝ) ≤ (d:ℝ) := by exact_mod_cast (Nat.le_of_lt_succ (g a).isLt) |
| have h0 := mul_le_mul_of_nonneg_left (hk 0) hL |
| have h1 := mul_le_mul_of_nonneg_left (hk 1) hL |
| have h2 := mul_le_mul_of_nonneg_left (hk 2) hL |
| cases b < |
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| lemma origin_star_eq : origin+dirStar = (![3,0,0] : Point) := by |
| ext a |
| fin_cases a < |
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| lemma weightMass_bound (d : ℕ) : ∃ c : ℝ, 0 < c ∧ ∀ L : ℝ, 1 ≤ L → |
| weightMass d L ≤ c/L^2 := by |
| obtain ⟨c,hc,hbound⟩ := weights_bound d |
| refine ⟨(Fintype.card (GridIndex d) : ℝ)*c, mul_pos (by positivity) hc, ?_⟩ |
| intro L hL |
| have he := Finset.sum_le_sum (s := Finset.univ) (fun g _ => hbound L hL g) |
| simpa only [weightMass, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_div_assoc] using he |
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| lemma cell_decoder {s q l r d : ℕ} (hs : 1 ≤ s) (hq : 1 ≤ q) (hsl : 2*l*d ≤ s) |
| (Q : (Fin q → Bool) → Matrix (Fin r) (Fin r) ℝ) |
| (B : Pairing (GlobalVertices s q) → Matrix (Fin r) (Fin r) ℝ) |
| (hQ : ∀ y, (Q y).PosSemidef) (hB : ∀ M, (B M).PosSemidef) |
| (hpair : ∀ y M, (Q y*B M).trace = tableCoordinate (globalCutColor y) (true,false) M-1) |
| (P : Fin q → Fin r → Fin r → Point → ℝ) |
| (hP : ∀ i u v, P i u v ∈ lowDegree coord d) |
| {η m δ : ℝ} (hη : 0 ≤ η) (hδ : 0 < δ) (hl : 0 < l) |
| (hm : ∀ y, (Q y).trace ≤ m) |
| (happrox : ∀ i u v t, |cellMatrix i B (samplePoint d (2*l) t) u v- |
| P i u v (samplePoint d (2*l) t)| ≤ η) |
| (hε : 0 ≤ (q:ℝ)*(weightMass d (2*l)*(2*d*(2*l))+ |
| (r:ℝ)*(6+weightMass d (2*l))*η*m)+δ*m) |
| (hsmall : (q:ℝ)*Real.sqrt ((q:ℝ)*(weightMass d (2*l)*(2*d*(2*l))+ |
| (r:ℝ)*(6+weightMass d (2*l))*η*m)+δ*m) ≤ 1/4) : |
| (2:ℝ)^q/2 ≤ r := by |
| let L : ℝ := 2*l |
| have hL : 0 < L := by dsimp [L] |
| let C := fun i => cellMatrix i B |
| have hC (i : Fin q) (x : Point) : (C i x).PosSemidef := cellMatrix_psd i B hB x |
| have hsamp (y : Fin q → Bool) (i : Fin q) (t : SampleIndex d) : |
| (Q y*C i (samplePoint d L t)).trace = cutValue (y i) (samplePoint d L t) := |
| cell_trace_value Q B hpair y i _ (sample_witness hs hsl t) |
| have hfive (y : Fin q → Bool) (i : Fin q) (j : Fin 5) : |
| (Q y*C i (specialPoint j)).trace = cutValue (y i) (specialPoint j) := hsamp y i (.inl j) |
| have hp (y : Fin q → Bool) (i : Fin q) : (Q y*C i (origin+dirP)).trace=0 := |
| (hfive y i 1).trans (cutValue_p _) |
| have hstar (y : Fin q → Bool) (i : Fin q) : (Q y*C i (origin+dirStar)).trace=2 := |
| (hfive y i 2).trans (cutValue_star _) |
| have hgrid (y : Fin q → Bool) (i : Fin q) (g : GridIndex d) : |
| (Q y*C i (scaledNode L g)).trace ≤ 2*(d:ℝ)*L := |
| (hsamp y i (.inr g)).le.trans (cutValue_grid hL.le g (y i)) |
| have hEex (i : Fin q) := interpolation_matrix_domination d (C i) (hC i) (P i) (hP i) |
| hL.ne' hη (fun j u v => happrox i u v (.inl j)) (fun g u v => happrox i u v (.inr g)) |
| choose E hE hdom hEb using hEex |
| let i₀ : Fin q := ⟨0,by omega⟩ |
| let S := C i₀ (origin+dirStar) |
| have hcommon (i : Fin q) : C i (origin+dirStar)=S := by |
| dsimp only [S,C] |
| rw [origin_star_eq] |
| exact cellMatrix_star i i₀ B 3 |
| let T := fun i (b : Bool) => if b then C i (origin+dirB) else C i (origin+dirA) |
| have hT (i : Fin q) (b : Bool) : (T i b).PosSemidef := by |
| cases b <;> exact hC i _ |
| have hTdom (i : Fin q) : T i false+T i true ≤ S+(2:ℝ) • C i (origin+dirP)+E i := by |
| change C i (origin+dirA)+C i (origin+dirB) ≤ S+(2:ℝ) • C i (origin+dirP)+E i |
| rw [← hcommon i] |
| exact hdom i |
| have hbit (y : Fin q → Bool) (i : Fin q) : (Q y*T i (y i)).trace=2 := by |
| cases hy : y i |
| · exact (hfive y i 3).trans (by rw [hy]; exact cutValue_bit false) |
| · exact (hfive y i 4).trans (by rw [hy]; exact cutValue_bit true) |
| have he (y : Fin q → Bool) (i : Fin q) : (Q y*E i).trace ≤ |
| weightMass d L*(2*(d:ℝ)*L)+(r:ℝ)*(6+weightMass d L)*η*m := by |
| simpa only [Fintype.card_fin] using hEb i (Q y) m (2*(d:ℝ)*L) (hm y) (hgrid y i) |
| simpa only [Fintype.card_fin] using domination_decoder Q T (fun i => C i (origin+dirP)) E S hQ hT |
| (fun i => hC i _) hE (hC i₀ _) hTdom hp (fun y => hstar y i₀) hbit hδ hm he hε hsmall |
| |
| end MatchingInterpolation |
| |
| end OAI |
| |
