File size: 3,941 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 72 73 74 75 76 77 78 79 80 81 | import OAI.Probability.InvariantIsing.Magnetic.RestrictedPairTies
namespace OAI
/-! The canonical site-average overlap path and its uniform comparison
with the original constrained-block overlap path. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
open scoped BigOperators NNReal
namespace InvariantIsing
def magneticBlockLevel {N : ℕ} (h : FieldStep) (m : Fin N → ℝ)
(i : Fin (h.depth + 1)) : ℝ := (N : ℝ)⁻¹ * ∑ j, magneticFieldLevel h (m j) i
lemma magneticBlockLevel_mem_unit {N : ℕ} (hN : 0 < N) (h : FieldStep)
(m : Fin N → ℝ) (i : Fin (h.depth + 1)) : magneticBlockLevel h m i ∈ Icc (0 : ℝ) 1 := by
have hn : (0 : ℝ) < N := by exact_mod_cast hN
constructor
· exact mul_nonneg (inv_nonneg.mpr hn.le)
(Finset.sum_nonneg (fun j _ => (magneticFieldLevel_mem_unit h (m j) i).1))
· calc
_ ≤ (N : ℝ)⁻¹ * ∑ _j : Fin N, (1 : ℝ) :=
mul_le_mul_of_nonneg_left
(Finset.sum_le_sum (fun j _ => (magneticFieldLevel_mem_unit h (m j) i).2))
(inv_nonneg.mpr hn.le)
_ = 1 := by simp [hn.ne']
lemma magneticBlockLevel_monotone {N : ℕ} (h : FieldStep) (m : Fin N → ℝ) :
Monotone (magneticBlockLevel h m) := by
intro i j hij
exact mul_le_mul_of_nonneg_left
(Finset.sum_le_sum (fun k _ => magneticFieldLevel_monotone h (m k) hij)) (by positivity)
def magneticBlockPath {N : ℕ} (hN : 0 < N) (h : FieldStep) (m : Fin N → ℝ) : OverlapPath :=
fieldLevelPath h (magneticBlockLevel h m) (magneticBlockLevel_monotone h m)
(magneticBlockLevel_mem_unit hN h m)
lemma magneticBlockPairMean_comparison {N : ℕ} (hN : 0 < N)
{A : Type*} [Fintype A] [DecidableEq A] (group : Fin N → A) (k : A → ℕ)
(hk : ∀ a, k a ≤ spinGroupSize group a) (m : A → ℝ) (hm : ∀ a, |m a| < 1)
(hc : ∀ a, (k a : ℝ) = spinGroupSize group a * ((1 + m a) / 2))
(h : FieldStep) (i : Fin (h.depth + 1)) {t : ℝ} (ht : 0 < t) :
|restrictedBlockPairMean hN (spinGroupSlice group k) (spinGroupSlice_nonempty group k hk)
h (fun _ => 0) i - magneticBlockLevel h (fun j => m (group j)) i| ≤
2 * ((N : ℝ)⁻¹ * (∑ j, constrainedFieldValue h (m (group j))) -
constrainedBlockValue (spinGroupSlice group k) h) / t + t / 2 := by
have he := restrictedBlockPairMean_comparison hN (spinGroupSlice group k)
(spinGroupSlice_nonempty group k hk) h (fun j => magneticBias h (m (group j))) i ht
rw [← magneticBlockGap_eq_biasedGap hN group k hk m hm hc h,
restrictedBlockPairMean_group_bias hN group k hk (fun a => magneticBias h (m a)) h i] at he
exact he
lemma magneticBlockPath_l1_comparison {N : ℕ} (hN : 0 < N)
{A : Type*} [Fintype A] [DecidableEq A] (group : Fin N → A) (k : A → ℕ)
(hk : ∀ a, k a ≤ spinGroupSize group a) (m : A → ℝ) (hm : ∀ a, |m a| < 1)
(hc : ∀ a, (k a : ℝ) = spinGroupSize group a * ((1 + m a) / 2))
(h : FieldStep) {t : ℝ} (ht : 0 < t) :
(∫ s, |restrictedBlockOverlapPath hN (spinGroupSlice group k)
(spinGroupSlice_nonempty group k hk) h s - magneticBlockPath hN h (fun j => m (group j)) s|
∂pathMeasure) ≤
2 * ((N : ℝ)⁻¹ * (∑ j, constrainedFieldValue h (m (group j))) -
constrainedBlockValue (spinGroupSlice group k) h) / t + t / 2 := by
let p := restrictedBlockOverlapPath hN (spinGroupSlice group k)
(spinGroupSlice_nonempty group k hk) h
let q := magneticBlockPath hN h (fun j => m (group j))
calc
_ ≤ ∫ _s, (2 * ((N : ℝ)⁻¹ * (∑ j, constrainedFieldValue h (m (group j))) -
constrainedBlockValue (spinGroupSlice group k) h) / t + t / 2) ∂pathMeasure := by
apply integral_mono (p.integrable.sub q.integrable).abs (integrable_const _)
intro s
exact magneticBlockPairMean_comparison hN group k hk m hm hc h (fieldLevelIndex h s) ht
_ = _ := by simp
end InvariantIsing
end
end OAI
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