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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 | import OAI.Probability.FreeIsing.Ising.CopyLinearLongBound
import OAI.Probability.FreeIsing.Ising.CopySpatialEndpoints
import OAI.Probability.FreeIsing.Ising.CriticalSmallLogLoss
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
theorem path_decay_mono {c c' ω : ℝ} (hcc : c ≤ c') (hω : 0 ≤ ω) (n : ℕ) :
Real.exp (-(c' * ω) * n) ≤ Real.exp (-(c * ω) * n) := by
apply Real.exp_le_exp.mpr
have h := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_right hcc hω) (Nat.cast_nonneg n : (0 : ℝ) ≤ _)
nlinarith
theorem path_double_decay_mono {a a' q q' : ℝ}
(ha : 0 ≤ a) (haa : a ≤ a') (hq : 0 ≤ q) (hqq : q ≤ q') (n : ℕ) :
Real.exp (-a' * q' ^ n) ≤ Real.exp (-a * q ^ n) := by
apply Real.exp_le_exp.mpr
have h := mul_le_mul haa (pow_le_pow_left₀ hq hqq n)
(pow_nonneg hq n) (ha.trans haa)
nlinarith
namespace IsingCopyResponseRowPaths
open IsingObservationFactor IsingConditionalCopyBounds ConnectedHull
open IsingCopySmallLogLoss IsingCriticalSmallLogLoss
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
theorem exists_actual_spatial_endpoint_of_noise_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hb : 0 ≤ beta)
(hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) (T : ℝ≥0)
{B E c a q C₀ : ℝ} (hB : 0 ≤ B) (hE : 0 ≤ E)
(hc : 0 < c) (ha : 0 < a) (hq : 1 < q)
(hnoise : ∀ (t : ℝ≥0), 0 < t → t ≤ T → ∀ (u w : TreeVertex d), treeAdj u w →
∀ v : TreeVertex d, C₀ / min (Real.sqrt (t : ℝ)) 1 ≤ (v.1.length : ℝ) →
noiseEnergy (μ := μ) hb t v ≤
B * (logVariance hb μ u w t) *
((v.1.length : ℝ) * Real.exp (-(c * min (Real.sqrt (t : ℝ)) 1) * v.1.length)) +
E * ((v.1.length : ℝ) ^ 2 * Real.exp (-a * q ^ v.1.length))) :
∃ M : ℝ, 1 ≤ M ∧ ∀ (t : ℝ≥0), 0 < t → t ≤ T →
∀ (u w : TreeVertex d), treeAdj u w → 0 < siteVariance beta μ (root d) t →
CriticalSpatialEndpointBounds
(∫ z, IsingCopyEnergy.density beta (root d) t z ∂copies beta μ)
(∑' n, noiseShellEnergy (μ := μ) hb t n)
(∑' n, linearShellEnergy (μ := μ) hb t n)
(∑' n, Real.sqrt (noiseShellEnergy (μ := μ) hb t n))
(logVariance hb μ u w t) (siteVariance beta μ (root d) t)
M (varianceScale d beta T) (min (Real.sqrt (t : ℝ)) 1) := by
obtain ⟨cL, aL, qL, CL, hcL, haL, hqL, _, hlinear⟩ :=
exists_linear_energy_long_constants hμ hb hd hcritical
let BC := max B 1
let EC := max E ((2 * (T : ℝ) + 2 * (d : ℝ) * beta) ^ 2)
let cc := min c cL
let aa := min a aL
let qq := min q qL
let CC := max C₀ CL
have hBC : 0 ≤ BC := hB.trans (le_max_left _ _)
have hEC : 0 ≤ EC := hE.trans (le_max_left _ _)
have hcc : 0 < cc := lt_min hc hcL
have haa : 0 < aa := lt_min ha haL
have hqq : 1 < qq := lt_min hq hqL
have htheta : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1) := by
apply (tanh_threshold_iff_branching_sq hd hb).mpr
simpa only [Nat.cast_sub (by omega : 1 ≤ d), Nat.cast_one] using hcritical.le
obtain ⟨M, hM, hendpoint⟩ := exists_actual_spatial_endpoint_constant hμ hd hb htheta
(C₀ := CC) hBC hEC hcc haa hqq
refine ⟨M, hM, ?_⟩
intro t ht htT u w huw hV
have hω : 0 < min (Real.sqrt (t : ℝ)) 1 :=
lt_min (Real.sqrt_pos.2 (show 0 < (t : ℝ) from ht)) zero_lt_one
apply hendpoint t u w huw ht (varianceScale d beta T) (min (Real.sqrt (t : ℝ)) 1)
hV (siteVariance_le_varianceScale hμ hb (root d) htT)
(one_le_varianceScale d beta T) hω (min_le_right _ _) ?_ ?_
· intro v hv
have hvQ : C₀ / min (Real.sqrt (t : ℝ)) 1 ≤ (v.1.length : ℝ) :=
(div_le_div_of_nonneg_right (le_max_left _ _) hω.le).trans hv
have h := hnoise t ht htT u w huw v hvQ
apply h.trans
apply add_le_add
· apply mul_le_mul
· exact mul_le_mul_of_nonneg_right (le_max_left _ _) (logVariance_nonneg hb u w t)
· exact mul_le_mul_of_nonneg_left (path_decay_mono (min_le_left _ _) hω.le _)
(Nat.cast_nonneg v.1.length)
· positivity
· exact mul_nonneg hBC (logVariance_nonneg hb u w t)
· apply mul_le_mul
· exact le_max_left _ _
· exact mul_le_mul_of_nonneg_left
(path_double_decay_mono haa.le (min_le_left _ _)
(zero_lt_one.trans hqq).le (min_le_left _ _) _) (sq_nonneg _)
· positivity
· exact hEC
· intro v hv
have hvL : CL / min (Real.sqrt (t : ℝ)) 1 ≤ (v.1.length : ℝ) :=
(div_le_div_of_nonneg_right (le_max_right _ _) hω.le).trans hv
have h := hlinear T t ht htT v hvL
apply h.trans
apply add_le_add
· apply mul_le_mul
· exact le_mul_of_one_le_left (siteVariance_nonneg (root d) t) (le_max_right _ _)
· exact path_decay_mono (min_le_right _ _) hω.le _
· exact (Real.exp_pos _).le
· exact mul_nonneg hBC (siteVariance_nonneg (root d) t)
· apply mul_le_mul
· exact le_max_right _ _
· exact path_double_decay_mono haa.le (min_le_right _ _)
(zero_lt_one.trans hqq).le (min_le_right _ _) _
· exact (Real.exp_pos _).le
· exact hEC
end IsingCopyResponseRowPaths
end Problem367
end
end OAI
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