| import Mathlib |
| import OAI.Probability.SKGap.Matrix.MatrixEntryCLM |
| import OAI.Probability.SKGap.Matrix.OpNormDiagonal |
|
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| namespace OAI |
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| section |
| noncomputable section |
| namespace SKGap |
| open Real Matrix MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
| variable {ι X : Type*} [Fintype ι] [DecidableEq ι] [PseudoMetricSpace X] |
|
|
| def matrixFactorProduct (F : List (X→Matrix ι ι ℝ)) (x : X) : Matrix ι ι ℝ := |
| (F.map (fun f=>f x)).prod |
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|
| omit [PseudoMetricSpace X] in |
| @[simp] lemma matrixFactorProduct_nil (x : X) : matrixFactorProduct ([] : List (X→Matrix ι ι ℝ)) x=1 := rfl |
| omit [PseudoMetricSpace X] in |
| @[simp] lemma matrixFactorProduct_cons (f : X→Matrix ι ι ℝ) (F : List (X→Matrix ι ι ℝ)) (x : X) : |
| matrixFactorProduct (f::F) x=f x*matrixFactorProduct F x := rfl |
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| omit [PseudoMetricSpace X] in |
| lemma matrixFactorProduct_opNorm (F : List (X→Matrix ι ι ℝ)) {R : NNReal} |
| (hF : ∀ f∈F,∀ x,opNorm (f x) ≤ R) (x : X) : |
| opNorm (matrixFactorProduct F x) ≤ (R:ℝ)^F.length := by |
| induction F with |
| | nil => simpa using opNorm_one_le (ι:=ι) |
| | cons f F ih => |
| rw [matrixFactorProduct_cons,List.length_cons,pow_succ'] |
| exact (opNorm_mul _ _).trans (mul_le_mul (hF f (List.mem_cons_self) x) |
| (ih (fun p hp=>hF p (List.mem_cons_of_mem f hp))) (show 0 ≤ opNorm (matrixFactorProduct F x) from norm_nonneg _) R.coe_nonneg) |
|
|
| theorem matrixFactorProduct_lipschitz (F : List (X→Matrix ι ι ℝ)) {R : NNReal} |
| (hF : ∀ f∈F,∀ x,opNorm (f x) ≤ R) |
| (hL : ∀ f∈F,LipschitzWith R f) : |
| LipschitzWith ((F.length:NNReal)*R^F.length) (matrixFactorProduct F) := by |
| induction F with |
| | nil => simp [matrixFactorProduct] |
| | cons f F ih => |
| have hFt : ∀ p∈F,∀ x,opNorm (p x) ≤ R := fun p hp=>hF p (List.mem_cons_of_mem f hp) |
| have hLt : ∀ p∈F,LipschitzWith R p := fun p hp=>hL p (List.mem_cons_of_mem f hp) |
| have ht := ih hFt hLt |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| rw [dist_eq_norm,matrixFactorProduct_cons,matrixFactorProduct_cons] |
| have he : f x*matrixFactorProduct F x-f y*matrixFactorProduct F y= |
| f x*(matrixFactorProduct F x-matrixFactorProduct F y)+(f x-f y)*matrixFactorProduct F y := by noncomm_ring |
| rw [he] |
| apply (norm_add_le _ _).trans |
| apply (add_le_add (frobenius_mul_le_opNorm _ _) (frobenius_mul_le_opNorm_right _ _)).trans |
| have hd : ‖matrixFactorProduct F x-matrixFactorProduct F y‖ ≤ ((F.length:ℝ)*(R:ℝ)^F.length)*dist x y := by |
| simpa only [dist_eq_norm,NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_pow] using ht.dist_le_mul x y |
| have hf : ‖f x-f y‖ ≤ (R:ℝ)*dist x y := by |
| simpa only [dist_eq_norm] using (hL f List.mem_cons_self).dist_le_mul x y |
| have hb := matrixFactorProduct_opNorm F hFt y |
| have hp := hF f List.mem_cons_self x |
| have h1 := mul_le_mul hp hd (norm_nonneg _) R.coe_nonneg |
| have h2 := mul_le_mul hf hb (show 0 ≤ opNorm (matrixFactorProduct F y) from norm_nonneg _) (mul_nonneg R.coe_nonneg dist_nonneg) |
| apply (add_le_add h1 h2).trans_eq |
| norm_num only [NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_pow,List.length_cons,Nat.cast_add,Nat.cast_one,NNReal.coe_add,NNReal.coe_one,pow_succ] |
| ring |
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| lemma matrixFactorProduct_entry_lipschitz (F : List (X→Matrix ι ι ℝ)) {R : NNReal} |
| (hF : ∀ f∈F,∀ x,opNorm (f x) ≤ R) (hL : ∀ f∈F,LipschitzWith R f) (i k : ι) : |
| LipschitzWith ((F.length:NNReal)*R^F.length) (fun x=>matrixFactorProduct F x i k) := by |
| simpa only [one_mul,Function.comp_def] using (matrixEntry_lipschitz i k).comp (matrixFactorProduct_lipschitz F hF hL) |
|
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| lemma matrixFactorProduct_trace_lipschitz (F : List (X→Matrix ι ι ℝ)) {R : NNReal} |
| (hF : ∀ f∈F,∀ x,opNorm (f x) ≤ R) (hL : ∀ f∈F,LipschitzWith R f) : |
| LipschitzWith (⟨sqrt (Fintype.card ι),sqrt_nonneg _⟩*((F.length:NNReal)*R^F.length)) |
| (fun x=>trace (matrixFactorProduct F x)) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| have hh := (matrixFactorProduct_lipschitz F hF hL).dist_le_mul x y |
| have ht := abs_trace_le_frobenius (matrixFactorProduct F x-matrixFactorProduct F y) |
| rw [trace_sub] at ht |
| have hm := mul_le_mul_of_nonneg_left hh (sqrt_nonneg (Fintype.card ι:ℝ)) |
| change |trace (matrixFactorProduct F x)-trace (matrixFactorProduct F y)| ≤ |
| (sqrt (Fintype.card ι)*((F.length:ℝ)*(R:ℝ)^F.length))*dist x y |
| simpa only [dist_eq_norm,NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_pow,mul_assoc] using ht.trans hm |
|
|
| lemma goe_word_entry_lipschitz (F : List (Matrix ι ι ℝ→Matrix ι ι ℝ)) {R : NNReal} |
| (hF : ∀ f∈F,∀ M,opNorm (f M) ≤ R) (hL : ∀ f∈F,LipschitzWith R f) |
| (r : ℝ) (i k : ι) : |
| LipschitzWith (((F.length:NNReal)*R^F.length)*⟨sqrt (2*r),sqrt_nonneg _⟩) |
| (fun z : EuclideanSpace ℝ (MatrixCoordinates ι)=>matrixFactorProduct F (goeMatrix r z) i k) := |
| (matrixFactorProduct_entry_lipschitz F hF hL i k).comp (goeMatrix_L2_lipschitz r) |
|
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| lemma goe_word_normalized_trace_lipschitz (F : List (Matrix ι ι ℝ→Matrix ι ι ℝ)) {R : NNReal} |
| (hF : ∀ f∈F,∀ M,opNorm (f M) ≤ R) (hL : ∀ f∈F,LipschitzWith R f) (r : ℝ) : |
| LipschitzWith (⟨(Fintype.card ι:ℝ)⁻¹,inv_nonneg.mpr (Nat.cast_nonneg _)⟩* |
| (⟨sqrt (Fintype.card ι),sqrt_nonneg _⟩*((F.length:NNReal)*R^F.length))*⟨sqrt (2*r),sqrt_nonneg _⟩) |
| (fun z : EuclideanSpace ℝ (MatrixCoordinates ι)=>trace (matrixFactorProduct F (goeMatrix r z))/(Fintype.card ι:ℝ)) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro z w |
| have hg := (goeMatrix_L2_lipschitz (ι:=ι) r).dist_le_mul z w |
| have ht := (matrixFactorProduct_trace_lipschitz F hF hL).dist_le_mul (goeMatrix r z) (goeMatrix r w) |
| change dist (goeMatrix r z) (goeMatrix r w) ≤ sqrt (2*r)*dist z w at hg |
| change dist (trace (matrixFactorProduct F (goeMatrix r z))) (trace (matrixFactorProduct F (goeMatrix r w))) ≤ |
| (sqrt (Fintype.card ι)*((F.length:ℝ)*(R:ℝ)^F.length))*dist (goeMatrix r z) (goeMatrix r w) at ht |
| have hc : 0 ≤ sqrt (Fintype.card ι)*((F.length:ℝ)*(R:ℝ)^F.length) := by positivity |
| have hi : 0 ≤ (Fintype.card ι:ℝ)⁻¹ := inv_nonneg.mpr (Nat.cast_nonneg _) |
| have hb := mul_le_mul_of_nonneg_left (ht.trans (mul_le_mul_of_nonneg_left hg hc)) hi |
| change dist (trace (matrixFactorProduct F (goeMatrix r z))/(Fintype.card ι:ℝ)) |
| (trace (matrixFactorProduct F (goeMatrix r w))/(Fintype.card ι:ℝ)) ≤ |
| ((Fintype.card ι:ℝ)⁻¹*(sqrt (Fintype.card ι)*((F.length:ℝ)*(R:ℝ)^F.length))*sqrt (2*r))*dist z w |
| rw [Real.dist_eq,← sub_div,abs_div] |
| simp only [abs_of_nonneg (show (0:ℝ) ≤ (Fintype.card ι:ℝ) by positivity)] |
| simpa only [Real.dist_eq,div_eq_mul_inv,mul_assoc,mul_comm (Fintype.card ι:ℝ)⁻¹] using hb |
| end SKGap |
| end |
| end |
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| end OAI |
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