| import Mathlib |
| import OAI.Probability.SKGap.Localization.FrobeniusSq |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set |
| open scoped NNReal ENNReal |
| open Filter |
| open scoped Topology |
| noncomputable section |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal |
| noncomputable section |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal |
| namespace SKGap |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| lemma matrix_entry_le_frobenius (M : Matrix ι ι ℝ) (i k : ι) : |M i k| ≤ ‖M‖ := by |
| have h₁ : (M i k)^2 ≤ ∑ j, (M i j)^2 := |
| Finset.single_le_sum (fun j _ => sq_nonneg _) (Finset.mem_univ k) |
| have h₂ : (∑ j, (M i j)^2) ≤ ∑ a,∑ j, (M a j)^2 := |
| Finset.single_le_sum (fun a _ => Finset.sum_nonneg (fun j _ => sq_nonneg _)) (Finset.mem_univ i) |
| have hh : (M i k)^2 ≤ ‖M‖^2 := by rw [frobenius_sq]; exact h₁.trans h₂ |
| nlinarith only [hh,sq_abs (M i k),abs_nonneg (M i k),norm_nonneg M] |
|
|
| noncomputable def matrixEntryCLM (i k : ι) : Matrix ι ι ℝ →L[ℝ] ℝ := |
| ({ toFun := fun M => M i k |
| map_add' := by intros; rfl |
| map_smul' := by intros; rfl } : Matrix ι ι ℝ →ₗ[ℝ] ℝ).mkContinuous 1 |
| (fun M => by simpa [Real.norm_eq_abs] using! matrix_entry_le_frobenius M i k) |
|
|
| lemma matrixEntry_lipschitz (i k : ι) : LipschitzWith 1 (fun M : Matrix ι ι ℝ => M i k) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro M N |
| simpa only [dist_eq_norm,Real.norm_eq_abs,NNReal.coe_one,one_mul,Matrix.sub_apply] using! |
| matrix_entry_le_frobenius (M-N) i k |
|
|
| omit [DecidableEq ι] in |
| lemma matrixEntry_derivative {f : ℝ → Matrix ι ι ℝ} {v : Matrix ι ι ℝ} {t : ℝ} |
| (hf : HasDerivAt f v t) (i k : ι) : HasDerivAt (fun s => f s i k) (v i k) t := by |
| classical |
| exact (matrixEntryCLM i k).hasFDerivAt.comp_hasDerivAt t hf |
|
|
| lemma matrix_inverse_diff_bound (K B E : Matrix ι ι ℝ) : |
| ‖K*B*E*K‖ ≤ opNorm K*(opNorm B*‖E‖)*opNorm K := by |
| calc |
| _ ≤ ‖K*B*E‖*opNorm K := frobenius_mul_le_opNorm_right _ _ |
| _ = ‖K*(B*E)‖*opNorm K := by rw [mul_assoc K B E] |
| _ ≤ (opNorm K*‖B*E‖)*opNorm K := |
| mul_le_mul_of_nonneg_right (frobenius_mul_le_opNorm _ _) (norm_nonneg _) |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left (frobenius_mul_le_opNorm _ _) (norm_nonneg _)) (norm_nonneg _) |
|
|
| noncomputable def diagonalVector (M : Matrix ι ι ℝ) : EuclideanSpace ℝ ι := |
| WithLp.toLp 2 (fun i => M i i) |
|
|
| lemma diagonalVector_norm_le (M : Matrix ι ι ℝ) : ‖diagonalVector M‖ ≤ ‖M‖ := by |
| apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg M) |
| simp only [← sq] |
| rw [EuclideanSpace.norm_sq_eq,frobenius_sq] |
| apply Finset.sum_le_sum |
| intro i _ |
| simpa only [diagonalVector,Real.norm_eq_abs,sq_abs] using |
| (Finset.single_le_sum (fun j _ => sq_nonneg (M i j)) (Finset.mem_univ i)) |
|
|
| lemma abs_trace_le_frobenius (M : Matrix ι ι ℝ) : |
| |Matrix.trace M| ≤ sqrt (Fintype.card ι)*‖M‖ := by |
| let v : EuclideanSpace ℝ ι := WithLp.toLp 2 (fun _ => 1) |
| have hv : ‖v‖ = sqrt (Fintype.card ι) := by |
| rw [EuclideanSpace.norm_eq]; simp [v] |
| have hi := abs_real_inner_le_norm (diagonalVector M) v |
| change |∑ i, 1*M i i| ≤ ‖diagonalVector M‖*‖v‖ at hi |
| simp only [one_mul,hv] at hi |
| change |Matrix.trace M| ≤ ‖diagonalVector M‖*sqrt (Fintype.card ι) at hi |
| exact hi.trans ((mul_le_mul_of_nonneg_right (diagonalVector_norm_le M) (sqrt_nonneg _)).trans_eq (mul_comm _ _)) |
|
|
| lemma matrixTrace_mul_lipschitz {Ω : Type*} [PseudoMetricSpace Ω] |
| {K : Ω → Matrix ι ι ℝ} {L : ℝ≥0} (hK : LipschitzWith L K) |
| (B : Matrix ι ι ℝ) : |
| LipschitzWith (⟨sqrt (Fintype.card ι)*opNorm B,by exact mul_nonneg (sqrt_nonneg _) (norm_nonneg _)⟩*L) |
| (fun x => Matrix.trace (K x*B)) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| change |Matrix.trace (K x*B)-Matrix.trace (K y*B)| ≤ |
| (sqrt (Fintype.card ι)*opNorm B*(L:ℝ))*dist x y |
| rw [← Matrix.trace_sub,← sub_mul] |
| calc |
| _ ≤ sqrt (Fintype.card ι)*‖(K x-K y)*B‖ := abs_trace_le_frobenius _ |
| _ ≤ sqrt (Fintype.card ι)*(‖K x-K y‖*opNorm B) := |
| mul_le_mul_of_nonneg_left (frobenius_mul_le_opNorm_right _ _) (sqrt_nonneg _) |
| _ ≤ sqrt (Fintype.card ι)*(((L:ℝ)*dist x y)*opNorm B) := by |
| exact mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_right (by simpa only [dist_eq_norm] using hK.dist_le_mul x y) |
| (norm_nonneg _)) (sqrt_nonneg _) |
| _ = _ := by ring |
|
|
| end SKGap |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open MeasureTheory.Measure |
| open scoped ENNReal NNReal MeasureTheory Topology |
|
|
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
