File size: 2,981 Bytes
7d9f027 | 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 | 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
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