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a89d282 | 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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianColumnDeviation
import OAI.Probability.InvariantIsing.Gaussian.GaussianGramTraceGap
import OAI.Probability.InvariantIsing.Gaussian.GaussianGramResolventBounds
namespace OAI
/-! Replacing a leave-one-column quadratic form by the full normalized trace. -/
noncomputable section
open Matrix
namespace InvariantIsing
lemma gaussianGramColumnQuadratic_error {N m : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t)
(z : EuclideanSpace ℝ (Fin N × Fin m)) (j : Fin m) :
|gaussianGramColumnQuadratic t z j-gaussianGramStieltjes t z| ≤
gaussianGramColumnDeviation t z j+1/((N : ℝ)*t) := by
have hn : (0 : ℝ) < N := by exact_mod_cast hN
have hg := gaussianGramLeaveResolvent_trace_gap ht z j
have he : (gaussianGramLeaveResolvent t z j).trace/N-gaussianGramStieltjes t z =
((gaussianGramLeaveResolvent t z j).trace-(gaussianGramResolvent t z).trace)/N := by
unfold gaussianGramStieltjes
ring
have hd : |(gaussianGramLeaveResolvent t z j).trace/N-gaussianGramStieltjes t z| ≤
1/((N : ℝ)*t) := by
rw [he,abs_of_nonneg (div_nonneg hg.1 hn.le)]
calc
_ ≤ (1/t)/N := div_le_div_of_nonneg_right hg.2 hn.le
_ = _ := by ring
exact (abs_sub_le (gaussianGramColumnQuadratic t z j)
((gaussianGramLeaveResolvent t z j).trace/N) (gaussianGramStieltjes t z)).trans
(add_le_add le_rfl hd)
end InvariantIsing
end
end OAI
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