| import OAI.Probability.InvariantIsing.Core.Model |
| import OAI.Probability.InvariantIsing.Core.Ward |
|
|
| namespace OAI |
|
|
| /-! Exact finite-volume algebra for the coupled cavity comparison. The |
| spectral coupling and its Gaussian limit are separate source obligations. -/ |
|
|
| noncomputable section |
| open IsingPerceptron |
| open scoped BigOperators Matrix |
|
|
| namespace InvariantIsing |
|
|
| def cavityQuadratic {d : ℕ} (A : Matrix (Fin d) (Fin d) ℝ) (y : Fin d → ℝ) : ℝ := |
| (1 / 2 : ℝ) * ∑ i, ∑ j, y i * A i j * y j |
|
|
| def cavityLogFactor {d n : ℕ} (K : Matrix (Fin d) (Fin d) ℝ) |
| (L : Matrix (Fin d) (Fin n) ℝ) (C : Matrix (Fin n) (Fin n) ℝ) |
| (y : Fin d → ℝ) (ε : Spin n) : ℝ := |
| cavityQuadratic K y + (∑ i, ∑ j, y i * L i j * spinValue (ε j)) + |
| cavityQuadratic C (fun j => spinValue (ε j)) |
|
|
| lemma cavityQuadratic_sub {d : ℕ} (A B : Matrix (Fin d) (Fin d) ℝ) (y : Fin d → ℝ) : |
| cavityQuadratic (A - B) y = cavityQuadratic A y - cavityQuadratic B y := by |
| simp only [cavityQuadratic, Matrix.sub_apply, mul_sub, sub_mul, Finset.sum_sub_distrib] |
|
|
| / |
| assumption on the two symmetric special-coordinate blocks. -/ |
| theorem cavity_block_factor {d n : ℕ} (A A0 : Matrix (Fin d) (Fin d) ℝ) |
| (L : Matrix (Fin d) (Fin n) ℝ) (C : Matrix (Fin n) (Fin n) ℝ) |
| (y : Fin d → ℝ) (ε : Spin n) : |
| (1 / 2 : ℝ) * (∑ i : Fin d ⊕ Fin n, ∑ j : Fin d ⊕ Fin n, |
| Sum.elim y (fun a => spinValue (ε a)) i * |
| (Matrix.fromBlocks A L L.transpose C) i j * |
| Sum.elim y (fun a => spinValue (ε a)) j) - cavityQuadratic A0 y = |
| cavityLogFactor (A - A0) L C y ε := by |
| have hcross : (∑ j : Fin n, ∑ i : Fin d, spinValue (ε j) * L i j * y i) = |
| ∑ i : Fin d, ∑ j : Fin n, y i * L i j * spinValue (ε j) := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring_nf |
| simp only [Fintype.sum_sum_type, Sum.elim_inl, Sum.elim_inr, |
| Matrix.fromBlocks_apply₁₁, Matrix.fromBlocks_apply₁₂, Matrix.fromBlocks_apply₂₁, |
| Matrix.fromBlocks_apply₂₂, Matrix.transpose_apply, Finset.sum_add_distrib] |
| rw [hcross] |
| unfold cavityLogFactor |
| rw [cavityQuadratic_sub] |
| unfold cavityQuadratic |
| ring_nf |
|
|
| / |
| cavity cube, evaluated on the positive cavity factor. -/ |
| def cavityFactorMean {N n : ℕ} (H : Spin N → ℝ) (V : Spin N → Spin n → ℝ) : ℝ := |
| (Fintype.card (Spin n) : ℝ)⁻¹ * |
| ∑ σ, (Real.exp (H σ) / ∑ τ, Real.exp (H τ)) * ∑ ε, Real.exp (V σ ε) |
|
|
| lemma cavityFactorMean_eq_gibbsAverage {N n : ℕ} (H : Spin N → ℝ) |
| (V : Spin N → Spin n → ℝ) : |
| cavityFactorMean H V = gibbsAverage (fun _ => 1) H |
| (fun σ => (Fintype.card (Spin n) : ℝ)⁻¹ * ∑ ε, Real.exp (V σ ε)) := by |
| simp only [cavityFactorMean, gibbsAverage, finiteGibbs, finitePartition, one_mul, |
| Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro σ _ |
| ring_nf |
|
|
| lemma cavityFactorMean_eq_ratio {N n : ℕ} (H : Spin N → ℝ) |
| (V : Spin N → Spin n → ℝ) : |
| cavityFactorMean H V = (Fintype.card (Spin n) : ℝ)⁻¹ * |
| ((∑ x : Spin N × Spin n, Real.exp (H x.1 + V x.1 x.2)) / |
| ∑ σ, Real.exp (H σ)) := by |
| unfold cavityFactorMean |
| simp only [div_mul_eq_mul_div, ← Finset.sum_div] |
| congr 1 |
| congr 1 |
| simp only [Fintype.sum_prod_type, Real.exp_add, Finset.mul_sum] |
|
|
| lemma cavityFactorMean_pos {N n : ℕ} (H : Spin N → ℝ) (V : Spin N → Spin n → ℝ) : |
| 0 < cavityFactorMean H V := by |
| rw [cavityFactorMean_eq_ratio] |
| have hc : (0 : ℝ) < Fintype.card (Spin n) := by exact_mod_cast Fintype.card_pos |
| exact mul_pos (inv_pos.mpr hc) (div_pos (sum_exp_pos _) (sum_exp_pos H)) |
|
|
| / |
| partition functions, at the base endpoint of the cavity interpolation. -/ |
| theorem logPartition_cavity_increment {N n : ℕ} (H : Spin N → ℝ) |
| (V : Spin N → Spin n → ℝ) : |
| logPartition (fun x : Spin N × Spin n => H x.1 + V x.1 x.2) - logPartition H = |
| Real.log (cavityFactorMean H V) := by |
| have hN : (Fintype.card (Spin N) : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| have hn : (Fintype.card (Spin n) : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| have hfull := (sum_exp_pos (fun x : Spin N × Spin n => H x.1 + V x.1 x.2)).ne' |
| have hbase := (sum_exp_pos H).ne' |
| rw [logPartition_eq, logPartition_eq, cavityFactorMean_eq_ratio, |
| Real.log_mul (inv_ne_zero hn) (div_ne_zero hfull hbase), Real.log_inv, |
| Real.log_div hfull hbase] |
| simp only [Fintype.card_prod, Nat.cast_mul] |
| rw [Real.log_mul hN hn] |
| ring_nf |
|
|
| theorem logPartition_cavityBlock_increment {N d n : ℕ} (H : Spin N → ℝ) |
| (K : Matrix (Fin d) (Fin d) ℝ) (L : Matrix (Fin d) (Fin n) ℝ) |
| (C : Matrix (Fin n) (Fin n) ℝ) (y : Spin N → Fin d → ℝ) : |
| logPartition (fun x : Spin N × Spin n => H x.1 + cavityLogFactor K L C (y x.1) x.2) - |
| logPartition H = Real.log (cavityFactorMean H (fun σ ε => cavityLogFactor K L C (y σ) ε)) := |
| logPartition_cavity_increment (N := N) (n := n) H |
| (fun σ ε => cavityLogFactor K L C (y σ) ε) |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
