openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianGramExcessBound.lean
| import OAI.Probability.InvariantIsing.Gaussian.GaussianGramLowerBound | |
| import OAI.Probability.InvariantIsing.Gaussian.GaussianGramNormBound | |
| namespace OAI | |
| /-! Deterministic transport of the two singular-value edge errors to Gram spectral excess. -/ | |
| noncomputable section | |
| open scoped Topology | |
| namespace InvariantIsing | |
| lemma gaussianGram_singular_bounds {n m : ℕ} | |
| (z : EuclideanSpace ℝ (Fin (n+1) × Fin m)) (i : Fin (n+1)) : | |
| (gaussianPatternSingularMin (gaussianPatternTranspose (n+1) m z)/Real.sqrt (n+1))^2 ≤ | |
| gaussianPatternEigenvalues z i ∧ | |
| gaussianPatternEigenvalues z i ≤ (gaussianPatternSingularMax z/Real.sqrt (n+1))^2 := by | |
| have hs : (Real.sqrt (n+1))^2 = (n+1 : ℝ) := Real.sq_sqrt (by positivity) | |
| constructor | |
| · simpa only [div_pow,hs,Nat.cast_succ,Nat.cast_add,Nat.cast_one] using | |
| gaussianPatternEigenvalues_lower_singular (Nat.succ_pos n) z i | |
| · have h := (le_abs_self (gaussianPatternEigenvalues z i)).trans | |
| (gaussianPatternEigenvalues_abs_le (Nat.succ_pos n) z i) | |
| simpa only [div_pow,hs,Nat.cast_succ,Nat.cast_add,Nat.cast_one] using h | |
| lemma squared_lower_edge_bound (L b : ℝ) (hL : 0 ≤ L) (hb : 0 ≤ b) : | |
| L^2-2*L*max (L-b) 0 ≤ b^2 := by | |
| by_cases h : L ≤ b | |
| · rw [max_eq_right (sub_nonpos.mpr h)] | |
| nlinarith | |
| · rw [max_eq_left (by linarith : 0 ≤ L-b)] | |
| nlinarith [sq_nonneg (L-b)] | |
| lemma gaussianGram_spectralExcess_bound {n m : ℕ} | |
| (z : EuclideanSpace ℝ (Fin (n+1) × Fin m)) (L U : ℝ) (hL : 0 ≤ L) (hU : 0 ≤ U) : | |
| spectralExcess (gaussianPatternEigenvalues z) (L^2) (U^2) ≤ | |
| (U+max (gaussianPatternSingularMax z/Real.sqrt (n+1)-U) 0)^2-U^2 + | |
| 2*L*max (L-gaussianPatternSingularMin (gaussianPatternTranspose (n+1) m z)/Real.sqrt (n+1)) 0 := by | |
| let a := gaussianPatternSingularMax z/Real.sqrt (n+1) | |
| let b := gaussianPatternSingularMin (gaussianPatternTranspose (n+1) m z)/Real.sqrt (n+1) | |
| let du := max (a-U) 0 | |
| let dl := max (L-b) 0 | |
| have ha : 0 ≤ a := div_nonneg (norm_nonneg _) (Real.sqrt_nonneg _) | |
| have hb : 0 ≤ b := div_nonneg (gaussianPatternSingularMin_nonneg (Nat.succ_pos n) _) | |
| (Real.sqrt_nonneg _) | |
| have hdu : 0 ≤ du := le_max_right _ _ | |
| have hdl : 0 ≤ dl := le_max_right _ _ | |
| have hu : 0 ≤ (U+du)^2-U^2 := by nlinarith | |
| have hl : 0 ≤ 2*L*dl := by positivity | |
| apply spectralExcess_le _ _ _ _ (add_nonneg hu hl) | |
| intro i | |
| have hbounds := gaussianGram_singular_bounds z i | |
| change b^2 ≤ gaussianPatternEigenvalues z i ∧ gaussianPatternEigenvalues z i ≤ a^2 at hbounds | |
| have hblo := squared_lower_edge_bound L b hL hb | |
| have hau : a ≤ U+du := by dsimp only [du]; linarith [le_max_left (a-U) 0] | |
| constructor | |
| · change L^2-2*L*dl ≤ b^2 at hblo | |
| linarith [hbounds.1] | |
| · nlinarith [hbounds.2] | |
| end InvariantIsing | |
| end | |
| end OAI | |