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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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramResolvent
namespace OAI
/-! Exact column decomposition and the positive leave-one-column Gram matrix. -/
noncomputable section
open Matrix
open scoped BigOperators
namespace InvariantIsing
def gaussianGramRawColumn {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m)) (j : Fin m) : Fin N → ℝ :=
fun i => z (i,j)
def gaussianGramLeaveOneOut {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m))
(j : Fin m) : Matrix (Fin N) (Fin N) ℝ :=
(1/(N : ℝ)) • ∑ l ∈ Finset.univ.erase j, vecMulVec (gaussianGramRawColumn z l) (gaussianGramRawColumn z l)
lemma gaussianPatternCoupling_column_sum {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m)) :
gaussianPatternCoupling 1 z = (1/(N : ℝ)) •
∑ j : Fin m, vecMulVec (gaussianGramRawColumn z j) (gaussianGramRawColumn z j) := by
ext i k
simp only [gaussianPatternCoupling,Matrix.smul_apply,smul_eq_mul,Matrix.mul_apply,
Matrix.transpose_apply,gaussianPatternArray,Matrix.sum_apply,Matrix.vecMulVec_apply,
gaussianGramRawColumn]
lemma gaussianGramLeaveOneOut_posSemidef {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m))
(j : Fin m) : (gaussianGramLeaveOneOut z j).PosSemidef := by
apply Matrix.PosSemidef.smul _ (by positivity : (0 : ℝ) ≤ 1/N)
apply Matrix.posSemidef_sum
intro l _
simpa only [star_trivial] using Matrix.posSemidef_vecMulVec_self_star (gaussianGramRawColumn z l)
lemma gaussianGramLeaveOneOut_decomposition {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m))
(j : Fin m) : gaussianPatternCoupling 1 z = gaussianGramLeaveOneOut z j+
(1/(N : ℝ)) • vecMulVec (gaussianGramRawColumn z j) (gaussianGramRawColumn z j) := by
rw [gaussianPatternCoupling_column_sum,gaussianGramLeaveOneOut,← smul_add]
congr 1
exact (Finset.sum_erase_add Finset.univ
(fun l => vecMulVec (gaussianGramRawColumn z l) (gaussianGramRawColumn z l))
(Finset.mem_univ j)).symm
end InvariantIsing
end
end OAI
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