| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.ProbabilityNodes |
| import OAI.Probability.LogConcave.Complexity.CenteringStateBudget |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory TensorEnergy Quadrature |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
|
|
| local instance identityOutputEmbeddingDecidableEqUnit : DecidableEq Unit := |
| Classical.decEq _ |
|
|
| def identityOutputEmbedding (d : ℕ) : Fin d ↪ (Unit → Fin d) := |
| ⟨fun i _ => i,fun _ _ h => congrFun h ()⟩ |
|
|
| def identityOutputProjection (d : ℕ) : EuclideanSpace ℝ (Unit → Fin d) →L[ℝ] Point d := |
| finiteCoordinateProjection (identityOutputEmbedding d) |
|
|
| lemma identityOutputProjection_vector {d : ℕ} (f : Point d → Point d) (y : Point d) : |
| identityOutputProjection d (tensorVector (vectorArray f) y)=f y := by |
| ext i |
| change inner ℝ (EuclideanSpace.basisFun (Fin d) ℝ i) (f y)=_ |
| simp only [EuclideanSpace.basisFun_inner] |
|
|
| lemma identityOutputProjection_lie {d : ℕ} (H : Point d → ℝ) |
| (A : (Unit ⊕ Unit → Fin d) → Point d → ℝ) |
| (hA : ∀c,PolySmooth (A c)) |
| (hsk : ∀i z y,A (Sum.elim (fun _ => i) (fun _ => z)) y= |
| -A (Sum.elim (fun _ => z) (fun _ => i)) y) (y : Point d) : |
| identityOutputProjection d (tensorVector |
| (iterTensorLie H A (vectorArray (id : Point d → Point d)) 1) y)=skewLieField H A y := by |
| ext i |
| change tensorLie H A (vectorArray (id : Point d → Point d)) (fun _ => i) y=_ |
| rw [tensorLie_eq_directional H A _ (fun c => (hA c).smooth) |
| (fun c => (identityArray_polySmooth d c).smooth) hsk] |
| change fderiv ℝ ((innerSL ℝ (EuclideanSpace.basisFun (Fin d) ℝ i)) : Point d → ℝ) y |
| (skewLieField H A y)=_ |
| rw [ContinuousLinearMap.fderiv] |
| simp only [innerSL_apply_apply, EuclideanSpace.basisFun_inner] |
|
|
| lemma projection_chainTaylor {E G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
| (L : E →L[ℝ] G) (J : ℕ → ℝ → E) (n : ℕ) (a b : ℝ) : |
| L (chainTaylor J n a b)=chainTaylor (fun k t => L (J k t)) n a b := by |
| simp only [chainTaylor,map_sum,map_smul] |
|
|
| lemma projection_rms {Ω I J : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| [Fintype I] [Fintype J] (e : I ↪ J) |
| {f : Ω → EuclideanSpace ℝ J} (hm : Measurable f) |
| {B : ℝ} (hi : Integrable (fun y => ‖f y‖^2) μ) (hb : (∫y,‖f y‖^2 ∂μ)≤B) : |
| Integrable (fun y => ‖finiteCoordinateProjection e (f y)‖^2) μ ∧ |
| (∫y,‖finiteCoordinateProjection e (f y)‖^2 ∂μ)≤B := by |
| have ha (y : Ω) : ‖finiteCoordinateProjection e (f y)‖^2≤‖f y‖^2 := |
| pow_le_pow_left₀ (norm_nonneg _) (finiteCoordinateProjection_norm_le _ _) 2 |
| have h := hi.mono' (((finiteCoordinateProjection e).continuous.measurable.comp hm).norm.pow_const 2).aestronglyMeasurable |
| (Filter.Eventually.of_forall (fun y => by simpa only [Real.norm_eq_abs,abs_sq,Function.comp_def] using ha y)) |
| exact ⟨h,(integral_mono h hi ha).trans hb⟩ |
| end LogConcaveSampling |
| |
| end |
| |
| end |
| |
| end OAI |
| |
