openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticBlockTerminalLoss.lean
| 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 | |