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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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramColumns
import OAI.Probability.InvariantIsing.Spectral.PositiveRankOneQuadratic
namespace OAI
/-! Exact leave-one-column identities at the physical Gaussian normalization. -/
noncomputable section
open Matrix
namespace InvariantIsing
def gaussianPatternScaledColumn {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m))
(j : Fin m) : Fin N → ℝ := (Real.sqrt N)⁻¹ • gaussianGramRawColumn z j
def gaussianGramLeaveResolvent {N m : ℕ} (t : ℝ) (z : EuclideanSpace ℝ (Fin N × Fin m))
(j : Fin m) : Matrix (Fin N) (Fin N) ℝ :=
(t • (1 : Matrix (Fin N) (Fin N) ℝ)+gaussianGramLeaveOneOut z j)⁻¹
lemma gaussianPatternScaledColumn_outer {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m))
(j : Fin m) :
vecMulVec (gaussianPatternScaledColumn z j) (gaussianPatternScaledColumn z j) =
(1/(N : ℝ)) • vecMulVec (gaussianGramRawColumn z j) (gaussianGramRawColumn z j) := by
have he : ((Real.sqrt (N : ℝ))⁻¹)^2 = 1/(N : ℝ) := by
rw [inv_pow,Real.sq_sqrt (Nat.cast_nonneg N),one_div]
ext i k
simp only [gaussianPatternScaledColumn,Matrix.vecMulVec_apply,Matrix.smul_apply,Pi.smul_apply,smul_eq_mul]
calc
_ = ((Real.sqrt (N : ℝ))⁻¹)^2*(gaussianGramRawColumn z j i*gaussianGramRawColumn z j k) := by ring
_ = _ := by rw [he]
lemma gaussianGramLeaveShift_posDef {N m : ℕ} {t : ℝ} (ht : 0 < t)
(z : EuclideanSpace ℝ (Fin N × Fin m)) (j : Fin m) :
(t • (1 : Matrix (Fin N) (Fin N) ℝ)+gaussianGramLeaveOneOut z j).PosDef :=
(Matrix.PosDef.one.smul ht).add_posSemidef (gaussianGramLeaveOneOut_posSemidef z j)
lemma gaussianGramLeaveResolvent_quadratic {N m : ℕ} {t : ℝ} (ht : 0 < t)
(z : EuclideanSpace ℝ (Fin N × Fin m)) (j : Fin m) :
let x := gaussianPatternScaledColumn z j
x ⬝ᵥ (gaussianGramResolvent t z *ᵥ x) =
(x ⬝ᵥ (gaussianGramLeaveResolvent t z j *ᵥ x))/(1+x ⬝ᵥ (gaussianGramLeaveResolvent t z j *ᵥ x)) := by
have h := positive_rankOne_quadratic _ (gaussianGramLeaveShift_posDef ht z j)
(gaussianPatternScaledColumn z j)
rw [gaussianPatternScaledColumn_outer,add_assoc,← gaussianGramLeaveOneOut_decomposition] at h
exact h
lemma gaussianGramLeaveResolvent_trace {N m : ℕ} {t : ℝ} (ht : 0 < t)
(z : EuclideanSpace ℝ (Fin N × Fin m)) (j : Fin m) :
let x := gaussianPatternScaledColumn z j
(gaussianGramLeaveResolvent t z j).trace-(gaussianGramResolvent t z).trace =
((gaussianGramLeaveResolvent t z j *ᵥ x) ⬝ᵥ (gaussianGramLeaveResolvent t z j *ᵥ x))/
(1+x ⬝ᵥ (gaussianGramLeaveResolvent t z j *ᵥ x)) := by
have h := positive_rankOne_trace_difference _ (gaussianGramLeaveShift_posDef ht z j)
(gaussianPatternScaledColumn z j)
rw [gaussianPatternScaledColumn_outer,add_assoc,← gaussianGramLeaveOneOut_decomposition] at h
exact h
end InvariantIsing
end
end OAI
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