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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | import OAI.Probability.FreeIsing.Ising.CopyFixedPathBound
import OAI.Probability.FreeIsing.Trees.IndexedPathVariance
import OAI.Probability.FreeIsing.Trees.PathRerooting
import OAI.Probability.FreeIsing.Trees.PathReverseRerooting
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.IsingCopySpectral
open IsingObservationInnovation IsingObservationFactor
open ConnectedHull
open scoped NNReal
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
private theorem inner_path_pair_bound_of_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {K q S : ℝ}
(hroot : ∀ (n : ℕ) (w : TreeVertex d), w.1.length = n + 1 →
|inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex w 1))
(cavityLogVector hμ hbeta t (takeVertex w n) w)| ≤
K * ((n : ℝ) + 1) * q ^ n * S)
(v : TreeVertex d) {i j : ℕ} (hij : i ≤ j) (hj : j < v.1.length) :
|inner ℝ (cavityLogVector hμ hbeta t (takeVertex v i) (takeVertex v (i + 1)))
(cavityLogVector hμ hbeta t (takeVertex v j) (takeVertex v (j + 1)))| ≤
(K * S) * pathCorrelationKernel q ((j : ℤ) - (i : ℤ)) := by
obtain ⟨w, g, hlen, hi, hi1, hj0, hj1⟩ :=
exists_treeAut_reroot_edge_pair v hij hj
have hm := inner_cavityLogVector_map hμ hbeta g t
(takeVertex v i) (takeVertex v (i + 1))
(takeVertex v j) (takeVertex v (j + 1))
rw [hi, hi1, hj0, hj1] at hm
rw [← hm]
have hlag : ((j : ℤ) - (i : ℤ)).natAbs = j - i := by
rw [← Int.natCast_sub hij]
rfl
calc
_ ≤ K * (((j - i : ℕ) : ℝ) + 1) * q ^ (j - i) * S := hroot _ w hlen
_ = _ := by rw [pathCorrelationKernel, hlag]; ring
private theorem inner_reverse_path_pair_bound_of_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {K q S : ℝ}
(hroot : ∀ (n : ℕ) (w : TreeVertex d), w.1.length = n + 1 →
|inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex w 1))
(cavityLogVector hμ hbeta t (takeVertex w n) w)| ≤
K * ((n : ℝ) + 1) * q ^ n * S)
(v : TreeVertex d) {i j : ℕ} (hij : i ≤ j) (hj : j < v.1.length) :
|inner ℝ (cavityLogVector hμ hbeta t (takeVertex v (i + 1)) (takeVertex v i))
(cavityLogVector hμ hbeta t (takeVertex v (j + 1)) (takeVertex v j))| ≤
(K * S) * pathCorrelationKernel q ((j : ℤ) - (i : ℤ)) := by
obtain ⟨w, g, hlen, hi, hi1, hj0, hj1⟩ :=
exists_treeAut_reroot_reverse_edge_pair v hij hj
have hm := inner_cavityLogVector_map hμ hbeta g t
(takeVertex v (j + 1)) (takeVertex v j)
(takeVertex v (i + 1)) (takeVertex v i)
rw [hi, hi1, hj0, hj1] at hm
rw [real_inner_comm, ← hm]
have hlag : ((j : ℤ) - (i : ℤ)).natAbs = j - i := by
rw [← Int.natCast_sub hij]
rfl
calc
_ ≤ K * (((j - i : ℕ) : ℝ) + 1) * q ^ (j - i) * S := hroot _ w hlen
_ = _ := by rw [pathCorrelationKernel, hlag]; ring
private theorem integral_image_path_sq_bound_of_root_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {K q S : ℝ} (hK : 0 ≤ K) (hq0 : 0 ≤ q) (hq1 : q < 1)
(hS : 0 ≤ S)
(hroot : ∀ (n : ℕ) (w : TreeVertex d), w.1.length = n + 1 →
|inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex w 1))
(cavityLogVector hμ hbeta t (takeVertex w n) w)| ≤
K * ((n : ℝ) + 1) * q ^ n * S)
(g : TreeAut d) (v : TreeVertex d) :
(∫ ω, (∑ i : Fin v.1.length, cavityLogDifference hbeta t
(g.toEquiv (takeVertex v i)) (g.toEquiv (takeVertex v (i + 1))) ω) ^ 2
∂copies beta μ) ≤
(K * (2 / (1 - q) ^ 2 - 1)) * (v.1.length : ℝ) * S := by
have h := integral_sum_fin_sq_le_ordered_geometric_pair_bound
(fun i => cavityLogDifference hbeta t
(g.toEquiv (takeVertex v i)) (g.toEquiv (takeVertex v (i + 1))))
hq0 hq1 (mul_nonneg hK hS)
(fun i => memLp_cavityLogDifference hμ hbeta t _ _)
(fun i j hij => by
rw [← inner_cavityLogVector_eq_integral hμ hbeta t,
inner_cavityLogVector_map]
have hpair := inner_path_pair_bound_of_le hμ hbeta t hroot v hij j.isLt
simpa only [pathCorrelationKernel,
Int.natAbs_natCast_sub_natCast_of_ge hij, mul_assoc] using hpair)
calc
_ ≤ (v.1.length : ℝ) * (K * S) * (2 / (1 - q) ^ 2 - 1) := h
_ = _ := by ring
private theorem integral_image_reverse_path_sq_bound_of_root_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {K q S : ℝ} (hK : 0 ≤ K) (hq0 : 0 ≤ q) (hq1 : q < 1)
(hS : 0 ≤ S)
(hroot : ∀ (n : ℕ) (w : TreeVertex d), w.1.length = n + 1 →
|inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex w 1))
(cavityLogVector hμ hbeta t (takeVertex w n) w)| ≤
K * ((n : ℝ) + 1) * q ^ n * S)
(g : TreeAut d) (v : TreeVertex d) :
(∫ ω, (∑ i : Fin v.1.length, cavityLogDifference hbeta t
(g.toEquiv (takeVertex v (i + 1))) (g.toEquiv (takeVertex v i)) ω) ^ 2
∂copies beta μ) ≤
(K * (2 / (1 - q) ^ 2 - 1)) * (v.1.length : ℝ) * S := by
have h := integral_sum_fin_sq_le_ordered_geometric_pair_bound
(fun i => cavityLogDifference hbeta t
(g.toEquiv (takeVertex v (i + 1))) (g.toEquiv (takeVertex v i)))
hq0 hq1 (mul_nonneg hK hS)
(fun i => memLp_cavityLogDifference hμ hbeta t _ _)
(fun i j hij => by
rw [← inner_cavityLogVector_eq_integral hμ hbeta t,
inner_cavityLogVector_map]
have hpair := inner_reverse_path_pair_bound_of_le hμ hbeta t hroot v hij j.isLt
simpa only [pathCorrelationKernel,
Int.natAbs_natCast_sub_natCast_of_ge hij, mul_assoc] using hpair)
calc
_ ≤ (v.1.length : ℝ) * (K * S) * (2 / (1 - q) ^ 2 - 1) := h
_ = _ := by ring
theorem exists_cavityLog_path_variance_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hbeta : 0 ≤ beta)
(hthreshold : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1)) :
∃ C : ℝ, 0 ≤ C ∧ ∀ (t : ℝ≥0) (u w : TreeVertex d), treeAdj u w →
∀ (g : TreeAut d) (v : TreeVertex d),
((∫ ω, (∑ i : Fin v.1.length, cavityLogDifference hbeta t
(g.toEquiv (takeVertex v i)) (g.toEquiv (takeVertex v (i + 1))) ω) ^ 2
∂copies beta μ) ≤
C * (v.1.length : ℝ) *
∫ ω, (cavityLogDifference hbeta t u w ω) ^ 2 ∂copies beta μ) ∧
((∫ ω, (∑ i : Fin v.1.length, cavityLogDifference hbeta t
(g.toEquiv (takeVertex v (i + 1))) (g.toEquiv (takeVertex v i)) ω) ^ 2
∂copies beta μ) ≤
C * (v.1.length : ℝ) *
∫ ω, (cavityLogDifference hbeta t u w ω) ^ 2 ∂copies beta μ) := by
obtain ⟨K, hK, hbound⟩ :=
exists_rooted_cavityLogVector_correlation_bound hμ hd hbeta hthreshold
let q := (Real.sqrt ((d : ℝ) - 1))⁻¹
have hq0 : 0 ≤ q := (ColoredEdgeLifts.degree_decay_parameters hd).1.le
have hq1 : q < 1 := (ColoredEdgeLifts.degree_decay_parameters hd).2.1
have hmass : 0 ≤ 2 / (1 - q) ^ 2 - 1 := by
rw [← tsum_pathCorrelationKernel hq0 hq1]
exact tsum_nonneg (pathCorrelationKernel_nonneg hq0)
refine ⟨K * (2 / (1 - q) ^ 2 - 1), mul_nonneg hK hmass, ?_⟩
intro t u w huw g v
have hS : 0 ≤ ∫ ω, (cavityLogDifference hbeta t u w ω) ^ 2 ∂copies beta μ :=
integral_nonneg (fun ω => sq_nonneg _)
have hroot := fun n x hx => hbound t n x hx u w huw
exact ⟨integral_image_path_sq_bound_of_root_bound hμ hbeta t hK hq0 hq1 hS
hroot g v,
integral_image_reverse_path_sq_bound_of_root_bound hμ hbeta t hK hq0 hq1 hS
hroot g v⟩
end Problem367.IsingCopySpectral
end
end OAI
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