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c4cdbbd | 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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import OAI.Probability.InvariantIsing.Cavity.CavityRetainedNorm
import OAI.Probability.InvariantIsing.Cavity.CavityConcreteFrameOrbit
namespace OAI
/-! A single physical base eigenframe identifies every retained group
overlap with the corresponding projected spin overlap. -/
noncomputable section
open scoped BigOperators Matrix
namespace InvariantIsing
lemma cavity_physical_frame_coordinates {N n : ℕ} {ι : Type*} [Fintype ι]
(e : ι ≃ Fin N) (U : SpecialOrthogonal (N + n))
(F : Matrix (Fin (N + n)) ι ℝ)
(hF : ∀ i : Fin n, ∀ j,
((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * F) (Fin.natAdd N i) j = 0)
(V : Orthogonal N)
(hV : ∀ i j, (V : Matrix (Fin N) (Fin N) ℝ) i j =
cavityReservoirRows ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * F)
i (e.symm j))
(σ : Spin (N + n)) (j : ι) :
matrixRotation V⁻¹ (spinVector (fun i => σ (Fin.castAdd n i))) (e j) =
(F.transpose *ᵥ (specialRotation U (spinVector σ)).ofLp) j := by
let P := (U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose * F
have he : F.transpose *ᵥ (specialRotation U (spinVector σ)).ofLp =
P.transpose *ᵥ (spinVector σ).ofLp := by
change F.transpose *ᵥ ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ) *ᵥ
(spinVector σ).ofLp) = _
simp only [P, Matrix.transpose_mul, Matrix.transpose_transpose, Matrix.mulVec_mulVec]
rw [he]
change (∑ i : Fin N, (V : Matrix (Fin N) (Fin N) ℝ) i (e j) *
spinValue (σ (Fin.castAdd n i))) = ∑ i : Fin (N + n), P i j * spinValue (σ i)
rw [Fin.sum_univ_add]
simp only [hF, zero_mul, Finset.sum_const_zero, add_zero, hV,
Equiv.symm_apply_apply, cavityReservoirRows, P]
def cavityBaseGroup {N m d : ℕ} (k : Fin m → ℕ)
(e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N)
(a₀ : Fin d → Fin m) (a : Fin m) : Finset (Fin N) :=
Finset.univ.filter (fun j => Sum.elim (fun v => v.1) a₀ (e.symm j) = a)
theorem cavity_base_group_overlap {N n m d : ℕ}
(g : Fin (N + n) → Fin m) (U : SpecialOrthogonal (N + n))
(B : Matrix (Fin (m * n)) (Fin d) ℝ)
(k : Fin m → ℕ) (R : (a : Fin m) → Matrix (Fin (N + n)) (Fin (k a)) ℝ)
(e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N)
(a₀ : Fin d → Fin m)
(hF : ∀ i : Fin n, ∀ j,
((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose *
Matrix.fromCols (cavityRetainedStack k R)
(cavityEigenspaceFrame (cavitySpectralImage g
(cavityColumns (cavitySpecialOrthogonal U))) * B)) (Fin.natAdd N i) j = 0)
(V : Orthogonal N)
(hV : ∀ i j, (V : Matrix (Fin N) (Fin N) ℝ) i j =
cavityReservoirRows ((U : Matrix (Fin (N + n)) (Fin (N + n)) ℝ).transpose *
Matrix.fromCols (cavityRetainedStack k R)
(cavityEigenspaceFrame (cavitySpectralImage g
(cavityColumns (cavitySpecialOrthogonal U))) * B)) i (e.symm j))
(a : Fin m) (σ τ : Spin (N + n)) :
projectedOverlap (matrixRotation V⁻¹) (cavityBaseGroup k e a₀ a)
(fun i => σ (Fin.castAdd n i)) (fun i => τ (Fin.castAdd n i)) =
cavityRetainedBaseOverlap g U B k R
(fun a => Finset.univ.filter (fun j => a₀ j = a)) a σ τ := by
let F := Matrix.fromCols (cavityRetainedStack k R)
(cavityEigenspaceFrame (cavitySpectralImage g (cavityColumns (cavitySpecialOrthogonal U))) * B)
let z := fun σ : Spin (N + n) => F.transpose *ᵥ (specialRotation U (spinVector σ)).ofLp
have hc (σ : Spin (N + n)) (j) :
matrixRotation V⁻¹ (spinVector (fun i => σ (Fin.castAdd n i))) (e j) = z σ j :=
cavity_physical_frame_coordinates e U F hF V hV σ j
have hr (σ : Spin (N + n)) (b : Fin m) (i : Fin (k b)) :
z σ (Sum.inl ⟨b, i⟩) =
((R b).transpose *ᵥ (specialRotation U (spinVector σ)).ofLp) i := rfl
have hy (σ : Spin (N + n)) (i : Fin d) :
z σ (Sum.inr i) = cavityFullSpecialCoordinates g B U σ i := rfl
rw [projectedOverlap, cavityBaseGroup, Finset.sum_filter]
rw [← Equiv.sum_comp e]
simp only [Equiv.symm_apply_apply, hc, Fintype.sum_sum_type, Fintype.sum_sigma,
Sum.elim_inl, Sum.elim_inr, hr, hy]
simp only [Finset.sum_ite_irrel]
rw [cavityRetainedBaseOverlap, Finset.sum_filter, div_eq_inv_mul]
simp only [Finset.sum_const_zero, Finset.sum_ite_eq', Finset.mem_univ, ite_true, dotProduct]
end InvariantIsing
end
end OAI
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