| import Mathlib |
| import OAI.Probability.SKGap.Model |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace SKGap.LogDet |
| open Matrix |
| open scoped BigOperators |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| |
| private def detMultilinear : ContinuousMultilinearMap ℝ (fun _ : ι => ι → ℝ) ℝ := |
| { (Matrix.detRowAlternating : (ι → ℝ) [⋀^ι]→ₗ[ℝ] ℝ).toMultilinearMap with |
| cont := by change Continuous (fun M : Matrix ι ι ℝ => M.det); fun_prop } |
| |
| lemma hasDerivAt_det_update {Q : ℝ → Matrix ι ι ℝ} {Q' : Matrix ι ι ℝ} {t : ℝ} |
| (hQ : ∀ i k, HasDerivAt (fun x => Q x i k) (Q' i k) t) : |
| HasDerivAt (fun x => (Q x).det) (∑ i, ((Q t).updateRow i (Q' i)).det) t := by |
| have hrow (i : ι) : HasDerivAt (fun x => Q x i) (Q' i) t := hasDerivAt_pi.mpr (hQ i) |
| have h := (HasFDerivAt.multilinear_comp (f := detMultilinear (ι := ι)) |
| (fun i => (hrow i).hasFDerivAt)).hasDerivAt |
| convert h using 1 <;> try rfl |
| simp only [_root_.sum_apply,ContinuousLinearMap.comp_apply, |
| ContinuousLinearMap.toSpanSingleton_apply,one_smul, |
| ContinuousMultilinearMap.toContinuousLinearMap_apply] |
| rfl |
| |
| lemma det_updateRow_trace (M H : Matrix ι ι ℝ) : |
| (∑ i, (M.updateRow i (H i)).det)= (H*M.adjugate).trace := by |
| simp only [Matrix.trace,Matrix.diag,Matrix.mul_apply] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [← Matrix.cramer_transpose_apply,Matrix.cramer_eq_adjugate_mulVec, |
| ← Matrix.adjugate_transpose] |
| simp only [Matrix.mulVec,Matrix.transpose_apply,dotProduct] |
| apply Finset.sum_congr rfl |
| intro k _ |
| ring |
|
|
| lemma hasDerivAt_det_adjugate {Q : ℝ → Matrix ι ι ℝ} {Q' : Matrix ι ι ℝ} {t : ℝ} |
| (hQ : ∀ i k, HasDerivAt (fun x => Q x i k) (Q' i k) t) : |
| HasDerivAt (fun x => (Q x).det) (Q'*(Q t).adjugate).trace t := by |
| simpa only [det_updateRow_trace] using hasDerivAt_det_update hQ |
|
|
| lemma adjugate_eq_det_smul_inv (M : Matrix ι ι ℝ) (hM : IsUnit M) : |
| M.adjugate=M.det • M⁻¹ := by |
| have hd : M.det ≠ 0 := (Matrix.isUnit_iff_isUnit_det M).mp hM |>.ne_zero |
| rw [Matrix.inv_def,Ring.inverse_eq_inv,smul_smul,mul_inv_cancel₀ hd,one_smul] |
|
|
| lemma hasDerivAt_det {Q : ℝ → Matrix ι ι ℝ} {Q' : Matrix ι ι ℝ} {t : ℝ} |
| (hQ : ∀ i k, HasDerivAt (fun x => Q x i k) (Q' i k) t) (hunit : IsUnit (Q t)) : |
| HasDerivAt (fun x => (Q x).det) ((Q t).det*(Q'*(Q t)⁻¹).trace) t := by |
| have h := hasDerivAt_det_adjugate hQ |
| rw [adjugate_eq_det_smul_inv _ hunit,Matrix.mul_smul,Matrix.trace_smul,smul_eq_mul] at h |
| exact h |
|
|
| lemma hasDerivAt_logdet {Q : ℝ → Matrix ι ι ℝ} {Q' : Matrix ι ι ℝ} {t : ℝ} |
| (hQ : ∀ i k, HasDerivAt (fun x => Q x i k) (Q' i k) t) (hunit : IsUnit (Q t)) : |
| HasDerivAt (fun x => Real.log (Q x).det) (Q'*(Q t)⁻¹).trace t := by |
| have hd : (Q t).det ≠ 0 := (Matrix.isUnit_iff_isUnit_det _).mp hunit |>.ne_zero |
| have h := (hasDerivAt_det hQ hunit).log hd |
| convert h using 1 |
| field_simp |
| end SKGap.LogDet |
| end |
| end |
|
|
| end OAI |
|
|
