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f61e547 | 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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Probability.FreeIsing.Ising.CopyEdgeVariance
import OAI.Probability.FreeIsing.Ising.CopyPathVector
import OAI.Probability.FreeIsing.Trees.ColoredPathCoefficients
import OAI.Probability.FreeIsing.Trees.PathPrefix
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.IsingCopySpectral
open IsingObservationInnovation IsingObservationFactor
open ColoredEdgeLifts ColoredPathAverage ConnectedHull
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
theorem inner_rooted_cavityLogVector_eq_of_length_eq
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (a b : TreeVertex d) (hlen : a.1.length = b.1.length) (n : ℕ) :
inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex a 1))
(cavityLogVector hμ hbeta t (takeVertex a n) a) =
inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex b 1))
(cavityLogVector hμ hbeta t (takeVertex b n) b) := by
obtain ⟨g, hga, hgroot⟩ := exists_rootFixing_treeAut_map a b hlen
have hgroot' : g.toEquiv (root d) = root d := hgroot
have hfirst := g.map_takeVertex_of_root_eq hgroot' a 1
have hlast := g.map_takeVertex_of_root_eq hgroot' a n
rw [hga] at hfirst hlast
have h := inner_cavityLogVector_map hμ hbeta g t (root d) (takeVertex a 1)
(takeVertex a n) a
rw [hgroot', hfirst, hlast, hga] at h
exact h.symm
theorem inner_pathValue_eq_rooted_cavityLogVector
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {n : ℕ} {i : Fin d} (p : Path d n i)
(v : TreeVertex d) (hlen : v.1.length = n + 1) :
inner ℝ (cavityLogEdgeVector hμ hbeta t i)
(pathValue (differenceReflection hμ) (cavityLogEdgeVector hμ hbeta t) p) =
inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1))
(cavityLogVector hμ hbeta t (takeVertex v n) v) := by
rw [pathValue_cavityLogEdgeVector]
change inner ℝ (cavityLogVector hμ hbeta t (root d) (prefixFlip i TreeEscape.root))
(cavityLogVector hμ hbeta t (tailVertex p) (headVertex p)) = _
have hfirst : takeVertex (headVertex p) 1 = prefixFlip i TreeEscape.root := by
cases n <;> apply Subtype.ext <;> rfl
have htail : tailVertex p = takeVertex (headVertex p) n := by
apply Subtype.ext
change (colors p).dropLast = (colors p).take n
rw [List.dropLast_eq_take, colors_length]
simp
rw [← hfirst, htail]
apply inner_rooted_cavityLogVector_eq_of_length_eq hμ hbeta t _ v _ n
simpa only [headVertex, colors_length] using hlen.symm
theorem inner_forwardOperator_eq_rooted_cavityLogVector
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 2 ≤ d) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (n : ℕ) (v : TreeVertex d) (hlen : v.1.length = n + 1) :
inner ℝ (cavityLogEdgeVector hμ hbeta t)
((forwardOperator (by omega : 0 < d) (differenceReflection hμ)
((d : ℝ) - 1)⁻¹ ^ n) (cavityLogEdgeVector hμ hbeta t)) =
(d : ℝ) * inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1))
(cavityLogVector hμ hbeta t (takeVertex v n) v) := by
apply inner_forwardOperator_pow_eq_card_mul hd
intro i p
exact inner_pathValue_eq_rooted_cavityLogVector hμ hbeta t p v hlen
theorem abs_inner_rooted_cavityLogVector_le_norm_pow
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 2 ≤ d) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (n : ℕ) (v : TreeVertex d) (hlen : v.1.length = n + 1)
{u w : TreeVertex d} (huw : treeAdj u w) :
|inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1))
(cavityLogVector hμ hbeta t (takeVertex v n) v)| ≤
‖forwardOperator (by omega : 0 < d) (differenceReflection hμ)
((d : ℝ) - 1)⁻¹ ^ n‖ *
∫ q, (cavityLogDifference hbeta t u w q) ^ 2 ∂copies beta μ := by
apply abs_pathCoefficient_le_norm_pow hd (differenceReflection hμ) n
(cavityLogEdgeVector hμ hbeta t)
· intro i p
exact inner_pathValue_eq_rooted_cavityLogVector hμ hbeta t p v hlen
· exact (norm_cavityLogEdgeVector_sq_eq hμ hbeta t huw).le
theorem exists_rooted_cavityLogVector_correlation_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hbeta : 0 ≤ beta)
(hthreshold : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1)) :
∃ K : ℝ, 0 ≤ K ∧ ∀ (t : ℝ≥0) (n : ℕ) (v : TreeVertex d),
v.1.length = n + 1 → ∀ (u w : TreeVertex d), treeAdj u w →
|inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1))
(cavityLogVector hμ hbeta t (takeVertex v n) v)| ≤
(K * ((n : ℝ) + 1) * (Real.sqrt ((d : ℝ) - 1))⁻¹ ^ n) *
∫ q, (cavityLogDifference hbeta t u w q) ^ 2 ∂copies beta μ := by
obtain ⟨K, hK, hpow⟩ := forwardOperator_pow_bound hμ hd hbeta hthreshold
refine ⟨K, hK, ?_⟩
intro t n v hlen u w huw
exact (abs_inner_rooted_cavityLogVector_le_norm_pow hμ (by omega) hbeta t n v
hlen huw).trans (mul_le_mul_of_nonneg_right (hpow n)
(integral_nonneg (fun q => sq_nonneg _)))
theorem exists_rooted_cavityLogDifference_pair_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hbeta : 0 ≤ beta)
(hthreshold : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1)) :
∃ K : ℝ, 0 ≤ K ∧ ∀ (t : ℝ≥0) (n : ℕ) (v : TreeVertex d),
v.1.length = n + 1 → ∀ (u w : TreeVertex d), treeAdj u w →
|∫ q, cavityLogDifference hbeta t (root d) (takeVertex v 1) q *
cavityLogDifference hbeta t (takeVertex v n) v q ∂copies beta μ| ≤
(K * ((n : ℝ) + 1) * (Real.sqrt ((d : ℝ) - 1))⁻¹ ^ n) *
∫ q, (cavityLogDifference hbeta t u w q) ^ 2 ∂copies beta μ := by
obtain ⟨K, hK, h⟩ := exists_rooted_cavityLogVector_correlation_bound hμ hd hbeta hthreshold
refine ⟨K, hK, ?_⟩
intro t n v hlen u w huw
simpa only [inner_cavityLogVector_eq_integral] using h t n v hlen u w huw
end Problem367.IsingCopySpectral
end
end OAI
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