| 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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|
