File size: 3,261 Bytes
a89d282 | 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 | import OAI.Probability.InvariantIsing.Magnetic.ConstrainedBlockMagneticEstimate
import OAI.Probability.InvariantIsing.Magnetic.MagneticFieldParameter
import Mathlib.Analysis.SpecificLimits.Basic
namespace OAI
/-! The finite block error vanishes uniformly over field partitions with
bounded final height and magnetizations in a fixed interior interval. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped BigOperators NNReal Topology
namespace InvariantIsing
def magneticUniformBiasBound (H r : ℝ) : ℝ := (Real.log 2 + H / 2) / (1 - r)
lemma magneticBias_uniform_bound (h : FieldStep) {H r m : ℝ}
(hr : r < 1) (hm : |m| ≤ r) (hH : h.height (Fin.last h.depth) ≤ H) :
|magneticBias h m| ≤ magneticUniformBiasBound H r := by
have hi := hm.trans_lt hr
have hb := magneticBias_minimum_bound h hi hH (magneticBias_minimizes h hi)
have hH0 := (h.nonneg (Fin.last h.depth)).trans hH
exact hb.trans (div_le_div_of_nonneg_left (by positivity) (sub_pos.mpr hr) (by linarith))
lemma normalized_magneticBlockError {N : ℕ} (hN : 0 < N) (C A ε t : ℝ) :
(N : ℝ)⁻¹ * (ε * (N * C) + (ε⁻¹ + t) * (A * Real.sqrt N) +
N * Real.log (1 + Real.exp (-t))) =
ε * C + (ε⁻¹ + t) * A / Real.sqrt N + Real.log (1 + Real.exp (-t)) := by
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast hN.ne'
have hs : Real.sqrt (N : ℝ) ≠ 0 := (Real.sqrt_pos.mpr (by exact_mod_cast hN)).ne'
have hs2 := Real.sq_sqrt (Nat.cast_nonneg N : (0 : ℝ) ≤ N)
field_simp
linear_combination (ε⁻¹ + t) * A * hs2
lemma log_one_add_exp_neg_tendsto :
Tendsto (fun t : ℝ => Real.log (1 + Real.exp (-t))) atTop (𝓝 0) := by
have he := Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot
have hh := (tendsto_const_nhds.add he).log (by norm_num : (1 : ℝ) + 0 ≠ 0)
simpa only [add_zero, Real.log_one, Function.comp_apply] using hh
theorem tendsto_zero_of_magneticBlockError (N : ℕ → ℕ) (hN : Tendsto N atTop atTop)
(f : ℕ → ℝ) (hf : ∀ n, 0 ≤ f n) (C A : ℝ) (hC : 0 < C)
(hb : ∀ ε : ℝ, 0 < ε → ∀ t : ℝ, 0 ≤ t → ∀ n,
f n ≤ ε * C + (ε⁻¹ + t) * A / Real.sqrt (N n) + Real.log (1 + Real.exp (-t))) :
Tendsto f atTop (𝓝 0) := by
apply tendsto_order.2
constructor
· intro a ha
exact Eventually.of_forall (fun n => ha.trans_le (hf n))
· intro a ha
let ε := a / (4 * C)
have hε : 0 < ε := div_pos ha (by positivity)
have hεC : ε * C = a / 4 := by dsimp [ε]; field_simp
obtain ⟨t, ht, htl⟩ := ((eventually_ge_atTop (0 : ℝ)).and
(log_one_add_exp_neg_tendsto.eventually (gt_mem_nhds (by linarith : (0 : ℝ) < a / 4)))).exists
have hi : Tendsto (fun n => (Real.sqrt (N n : ℝ))⁻¹) atTop (𝓝 0) :=
tendsto_inv_atTop_zero.comp (Real.tendsto_sqrt_atTop.comp (tendsto_natCast_atTop_atTop.comp hN))
have hm : Tendsto (fun n => (ε⁻¹ + t) * A / Real.sqrt (N n)) atTop (𝓝 0) := by
simpa only [div_eq_mul_inv, mul_zero] using hi.const_mul ((ε⁻¹ + t) * A)
filter_upwards [hm.eventually (gt_mem_nhds (by linarith : (0 : ℝ) < a / 4))] with n hn
have h := hb ε hε t ht n
rw [hεC] at h
linarith
end InvariantIsing
end
end OAI
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