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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 | import OAI.Probability.InvariantIsing.Fields.FieldProductSpinAveraging
import OAI.Probability.InvariantIsing.Fields.FieldCanonicalVectorMoments
namespace OAI
/-! The actual averaged constrained terminal loss, with constants
independent of the number of Gaussian steps. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace InvariantIsing
lemma magnetic_projection_distance_mean {N : 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)
(hcount : ∀ a, (k a : ℝ) = (spinGroupSize group a : ℝ) * ((1 + m a) / 2))
(h : FieldStep) :
(∫ z, fieldProjectionDistance (spinGroupProjection group k hk) z
∂fieldCanonicalVectorLaw N h (fun i => magneticBias h (m (group i)))) ≤
∑ a, Real.sqrt (spinGroupSize group a) := by
have he := field_product_spin_average
(fun i : Fin N => fieldCanonicalEndpointLaw h (magneticBias h (m (group i))))
(fun σ => (hammingDist σ (spinGroupProjection group k hk σ) : ℝ))
simp only [← fieldCanonicalSpinLaw_endpoint] at he
exact he.trans_le (magneticCanonical_projection_distance group k hk m hm hcount h)
theorem magnetic_block_terminal_loss {N : 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)
(hcount : ∀ a, (k a : ℝ) = (spinGroupSize group a : ℝ) * ((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) :
(∫ z, (∑ i, Real.log (Real.cosh (z i))) - restrictedFieldTerminal (spinGroupSlice group k) z
∂fieldCanonicalVectorLaw N h (fun i => magneticBias h (m (group i)))) ≤
ε * (N * (2 * Real.exp (2 * B + 4 * H))) +
(ε⁻¹ + t) * (∑ a, Real.sqrt (spinGroupSize group a)) +
N * Real.log (1 + Real.exp (-t)) := by
have hl := field_projection_mean_loss (spinGroupSlice group k)
(spinGroupSlice_nonempty group k hk) (spinGroupProjection group k hk)
(spinGroupProjection_mem group k hk)
(fieldCanonicalVectorLaw N h (fun i => magneticBias h (m (group i))))
(fieldCanonicalVectorLaw_square_integrable h (fun i => magneticBias h (m (group i)))) hε t
have hM := fieldCanonicalVectorLaw_second_moment h
(fun i => magneticBias h (m (group i))) hB hH
have hD := magnetic_projection_distance_mean group k hk m hm hcount h
exact hl.trans (add_le_add (add_le_add
(mul_le_mul_of_nonneg_left hM hε.le)
(mul_le_mul_of_nonneg_left hD (by positivity))) le_rfl)
lemma sum_sqrt_spinGroupSize_le {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A]
(group : Fin N → A) :
(∑ a, Real.sqrt (spinGroupSize group a)) ≤ Fintype.card A * Real.sqrt N := by
calc
_ ≤ ∑ _a : A, Real.sqrt (N : ℝ) := by
apply Finset.sum_le_sum
intro a _
apply Real.sqrt_le_sqrt
exact_mod_cast (show spinGroupSize group a ≤ N from by
simpa only [spinGroupSize, Finset.card_univ, Fintype.card_fin] using
Finset.card_filter_le (s := (Finset.univ : Finset (Fin N))) (fun i => group i = a))
_ = _ := by simp
theorem magnetic_block_terminal_loss_size {N : 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)
(hcount : ∀ a, (k a : ℝ) = (spinGroupSize group a : ℝ) * ((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) :
(∫ z, (∑ i, Real.log (Real.cosh (z i))) - restrictedFieldTerminal (spinGroupSlice group k) z
∂fieldCanonicalVectorLaw N h (fun i => magneticBias h (m (group i)))) ≤
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)) :=
(magnetic_block_terminal_loss group k hk m hm hcount h hB hH hε ht).trans
(add_le_add (add_le_add le_rfl
(mul_le_mul_of_nonneg_left (sum_sqrt_spinGroupSize_le group) (by positivity))) le_rfl)
end InvariantIsing
end
end OAI
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