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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import OAI.Probability.InvariantIsing.Magnetic.ConstrainedBlockBias
namespace OAI
/-! Finite constrained-block comparison with the actual scalar magnetic
functional. Every error term is independent of the cascade depth. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
lemma sum_sites_group {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A]
(group : Fin N → A) (f : A → ℝ) :
(∑ i, f (group i)) = ∑ a, spinGroupSize group a * f a := by
rw [← Finset.sum_fiberwise Finset.univ group]
apply Finset.sum_congr rfl
intro a _
calc
_ = ∑ _i ∈ Finset.univ.filter (fun i => group i = a), f a := by
apply Finset.sum_congr rfl
intro i hi
rw [(Finset.mem_filter.mp hi).2]
_ = _ := by simp [spinGroupSize]
lemma spinGroupFieldConstant_of_means {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A]
(group : Fin N → A) (k : A → ℕ) (m bias : A → ℝ)
(hc : ∀ a, (k a : ℝ) = spinGroupSize group a * ((1 + m a) / 2)) :
spinGroupFieldConstant group k bias = ∑ i, bias (group i) * m (group i) := by
rw [sum_sites_group group (fun a => bias a * m a)]
unfold spinGroupFieldConstant
apply Finset.sum_congr rfl
intro a _
rw [hc a]
ring
theorem biasedConstrainedBlockValue_gap {N : ℕ} (hN : 0 < N)
(S : Finset (Spin N)) (hS : S.Nonempty) (h : FieldStep) (b : Fin N → ℝ) :
0 ≤ biasedConstrainedBlockValue Finset.univ h b - biasedConstrainedBlockValue S h b ∧
biasedConstrainedBlockValue Finset.univ h b - biasedConstrainedBlockValue S h b ≤
(N : ℝ)⁻¹ * (∫ y, restrictedFieldTerminal Finset.univ y - restrictedFieldTerminal S y
∂fieldCanonicalVectorLaw N h b) := by
have hF := restrictedFieldRecursion_biased_root_integrable hN Finset.univ Finset.univ_nonempty h b
have hS' := restrictedFieldRecursion_biased_root_integrable hN S hS h b
have he : biasedConstrainedBlockValue Finset.univ h b - biasedConstrainedBlockValue S h b =
(N : ℝ)⁻¹ * (∫ z, restrictedFieldRecursion Finset.univ h.depth (chainExponent h.cut)
(fieldStepVariance h) z - restrictedFieldRecursion S h.depth (chainExponent h.cut)
(fieldStepVariance h) z
∂Measure.pi (fun i => gaussianReal (b i) (NNReal.mk (h.height 0) (h.nonneg 0)))) := by
rw [integral_sub hF hS']
unfold biasedConstrainedBlockValue
ring
rw [he]
constructor
· apply mul_nonneg (by positivity)
apply integral_nonneg
intro z
apply sub_nonneg.mpr
exact restrictedFieldRecursion_mono hN S Finset.univ hS Finset.univ_nonempty
(Finset.subset_univ _) _ _ _
(fun i hi => ((chainExponent_admissible h.ordered_cut h.first h.last).1 i hi).1) z
· exact mul_le_mul_of_nonneg_left (restrictedFieldRecursion_root_loss hN S hS h b) (by positivity)
lemma magneticBlockGap_eq_biasedGap {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) :
(N : ℝ)⁻¹ * (∑ i, constrainedFieldValue h (m (group i))) -
constrainedBlockValue (spinGroupSlice group k) h =
biasedConstrainedBlockValue Finset.univ h (fun i => magneticBias h (m (group i))) -
biasedConstrainedBlockValue (spinGroupSlice group k) h (fun i => magneticBias h (m (group i))) := by
rw [biasedConstrainedBlockValue_univ hN,
biasedConstrainedBlockValue_group hN group k hk (fun a => magneticBias h (m a)),
spinGroupFieldConstant_of_means group k m (fun a => magneticBias h (m a)) hc]
simp_rw [constrainedFieldValue_magneticBias h (hm _)]
rw [Finset.sum_sub_distrib]
ring
theorem magneticBlockGap_estimate {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) {B H ε t : ℝ} (hB : ∀ i, |magneticBias h (m (group i))| ≤ B)
(hH : h.height (Fin.last h.depth) ≤ H) (hε : 0 < ε) (ht : 0 ≤ t) :
0 ≤ (N : ℝ)⁻¹ * (∑ i, constrainedFieldValue h (m (group i))) -
constrainedBlockValue (spinGroupSlice group k) h ∧
(N : ℝ)⁻¹ * (∑ i, constrainedFieldValue h (m (group i))) -
constrainedBlockValue (spinGroupSlice group k) h ≤
(N : ℝ)⁻¹ * (ε * (N * (2 * Real.exp (2 * B + 4 * H))) +
(ε⁻¹ + t) * (Fintype.card A * Real.sqrt N) + N * Real.log (1 + Real.exp (-t))) := by
rw [magneticBlockGap_eq_biasedGap hN group k hk m hm hc]
have hg := biasedConstrainedBlockValue_gap hN (spinGroupSlice group k)
(spinGroupSlice_nonempty group k hk) h (fun i => magneticBias h (m (group i)))
refine ⟨hg.1, hg.2.trans ?_⟩
apply mul_le_mul_of_nonneg_left _ (by positivity)
have he := magnetic_block_terminal_loss_size group k hk m hm hc h hB hH hε ht
simpa only [restrictedFieldTerminal_univ] using he
end InvariantIsing
end
end OAI
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