| import OAI.Probability.InvariantIsing.Gaussian.GaussianGramResolvent |
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| namespace OAI |
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| /-! Exact column decomposition and the positive leave-one-column Gram matrix. -/ |
| noncomputable section |
| open Matrix |
| open scoped BigOperators |
| namespace InvariantIsing |
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| def gaussianGramRawColumn {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m)) (j : Fin m) : Fin N → ℝ := |
| fun i => z (i,j) |
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| 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) |
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| 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] |
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| 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) |
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| 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 |
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| end InvariantIsing |
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| end |
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| end OAI |
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