| import Mathlib |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace GaussianTools |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| noncomputable def scalarRow (d : ℕ) := Measure.pi (fun _ : Fin d => gaussianReal 0 1) |
|
|
| lemma row_memLp {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) : |
| MemLp (fun a : Fin d → ℝ => ∑ j,a j*v j) 2 (scalarRow d) := by |
| apply memLp_finsetSum |
| intro j _ |
| exact ((memLp_id_gaussianReal (2 : NNReal)).comp_measurePreserving |
| (measurePreserving_eval (fun _ : Fin d => gaussianReal 0 1) j)).mul_const (v j) |
|
|
| lemma row_integral {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) : |
| (∫ a : Fin d → ℝ,∑ j,a j*v j ∂scalarRow d)=0 := by |
| have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance |
| have hi (j : Fin d) : Integrable (fun a : Fin d → ℝ => a j*v j) (scalarRow d) := |
| (((memLp_id_gaussianReal (2 : NNReal)).comp_measurePreserving |
| (measurePreserving_eval (fun _ : Fin d => gaussianReal 0 1) j)).mul_const (v j)).integrable (by norm_num) |
| rw [integral_finsetSum _ (fun j _ => hi j)] |
| simp only [integral_mul_const,scalarRow,integral_eval,integral_id_gaussianReal,zero_mul,Finset.sum_const_zero] |
|
|
| lemma row_energy {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) : |
| (∫ a : Fin d → ℝ,(∑ j,a j*v j)^2 ∂scalarRow d)=‖v‖^2 := by |
| have h := variance_sum_pi (μ := fun _ : Fin d => gaussianReal 0 1) |
| (X := fun j x => x*v j) (fun j => (memLp_id_gaussianReal (2 : NNReal)).mul_const (v j)) |
| have hm : Measurable (fun a : Fin d → ℝ => ∑ j,a j*v j) := by fun_prop |
| simp only [Finset.sum_fn] at h |
| rw [variance_eq_integral hm.aemeasurable] at h |
| change (∫ a,(∑ j,a j*v j - ∫ a,∑ j,a j*v j ∂scalarRow d)^2 ∂scalarRow d)=_ at h |
| rw [row_integral] at h |
| simp only [sub_zero,variance_mul_const] at h |
| rw [h,EuclideanSpace.norm_sq_eq] |
| apply Finset.sum_congr rfl |
| intro j _ |
| simp |
|
|
| noncomputable def labelDensity {p : ℕ} (r : ℝ) (y : Fin p → ℝ) : ℝ := |
| ∏ i,gaussianPDFReal 0 (NNReal.mk (r^2) (sq_nonneg r)) (y i) |
|
|
| lemma labelDensity_pos {p : ℕ} {r : ℝ} (hr : 0 < r) (y : Fin p → ℝ) : |
| 0 < labelDensity r y := by |
| apply Finset.prod_pos |
| intro i _ |
| exact gaussianPDFReal_pos _ _ _ (by intro hh; have he := congrArg (fun t : NNReal => (t : ℝ)) hh; exact (pow_ne_zero 2 hr.ne') he) |
| |
| lemma measurable_labelDensity {p : ℕ} (r : ℝ) : Measurable (@labelDensity p r) := by |
| unfold labelDensity |
| fun_prop |
| |
| lemma log_labelDensity {p : ℕ} {r : ℝ} (hr : 0 < r) (y : Fin p → ℝ) : |
| Real.log (labelDensity r y) = |
| -(p:ℝ)*Real.log r-(p:ℝ)/2*Real.log (2*Real.pi) - |
| (∑ i,(y i)^2)/(2*r^2) := by |
| unfold labelDensity |
| rw [Real.log_prod (fun i _ => (gaussianPDFReal_pos 0 (NNReal.mk (r^2) (sq_nonneg r)) (y i) (by intro hh; have he := congrArg (fun t : NNReal => (t : ℝ)) hh; exact (pow_ne_zero 2 hr.ne') he)).ne')] |
| simp only [gaussianPDFReal,NNReal.coe_mk,sub_zero] |
| have hs : Real.sqrt (2*Real.pi*r^2)=Real.sqrt (2*Real.pi)*r := by |
| rw [Real.sqrt_mul (by positivity),Real.sqrt_sq_eq_abs,abs_of_pos hr] |
| simp_rw [Real.log_mul (by positivity : (Real.sqrt (2*Real.pi*r^2))⁻¹ ≠ 0) (Real.exp_pos _).ne', |
| Real.log_inv,hs,Real.log_mul (by positivity : Real.sqrt (2*Real.pi) ≠ 0) hr.ne', |
| Real.log_sqrt (by positivity : 0 ≤ 2*Real.pi),Real.log_exp] |
| simp_rw [neg_div] |
| simp only [Finset.sum_add_distrib,Finset.sum_neg_distrib,← Finset.sum_div, |
| Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| ring |
| |
| lemma rows_energy {p d : ℕ} (v : EuclideanSpace ℝ (Fin d)) : |
| (∫ X : Fin p → Fin d → ℝ, ∑ i,(∑ j,X i j*v j)^2 |
| ∂Measure.pi (fun _ : Fin p => scalarRow d)) = (p:ℝ)*‖v‖^2 := by |
| have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance |
| have hi (i : Fin p) : Integrable (fun X : Fin p → Fin d → ℝ => (∑ j,X i j*v j)^2) |
| (Measure.pi (fun _ : Fin p => scalarRow d)) := |
| ((row_memLp v).comp_measurePreserving |
| (measurePreserving_eval (fun _ : Fin p => scalarRow d) i)).integrable_sq |
| rw [integral_finsetSum _ (fun i _ => hi i)] |
| have he (i : Fin p) : (∫ X : Fin p → Fin d → ℝ,(∑ j,X i j*v j)^2 |
| ∂Measure.pi (fun _ : Fin p => scalarRow d)) = ‖v‖^2 := by |
| rw [← row_energy v] |
| exact integral_comp_eval (μ := fun _ : Fin p => scalarRow d) (i := i) |
| (f := fun a : Fin d → ℝ => (∑ j,a j*v j)^2) ((by fun_prop : Measurable |
| (fun a : Fin d → ℝ => (∑ j,a j*v j)^2)).aestronglyMeasurable) |
| simp_rw [he] |
| simp |
| |
| lemma rows_shift_energy {p d : ℕ} (v : EuclideanSpace ℝ (Fin d)) (e : Fin p → ℝ) : |
| (∫ X : Fin p → Fin d → ℝ, ∑ i,((∑ j,X i j*v j)+e i)^2 |
| ∂Measure.pi (fun _ : Fin p => scalarRow d)) ≤ |
| 2*(p:ℝ)*‖v‖^2+2*∑ i,(e i)^2 := by |
| have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance |
| have hi (i : Fin p) : MemLp (fun X : Fin p → Fin d → ℝ => ∑ j,X i j*v j) 2 |
| (Measure.pi (fun _ : Fin p => scalarRow d)) := |
| (row_memLp v).comp_measurePreserving (measurePreserving_eval (fun _ : Fin p => scalarRow d) i) |
| have hil : Integrable (fun X : Fin p → Fin d → ℝ => ∑ i,((∑ j,X i j*v j)+e i)^2) |
| (Measure.pi (fun _ : Fin p => scalarRow d)) := |
| integrable_finsetSum _ (fun i _ => ((hi i).add (memLp_const (e i))).integrable_sq) |
| have hir : Integrable (fun X : Fin p → Fin d → ℝ => 2*(∑ i,(∑ j,X i j*v j)^2) + 2*∑ i,(e i)^2) |
| (Measure.pi (fun _ : Fin p => scalarRow d)) := |
| ((integrable_finsetSum _ (fun i _ => (hi i).integrable_sq)).const_mul 2).add (integrable_const _) |
| calc |
| _ ≤ ∫ X : Fin p → Fin d → ℝ, 2*(∑ i,(∑ j,X i j*v j)^2) + 2*∑ i,(e i)^2 |
| ∂Measure.pi (fun _ : Fin p => scalarRow d) := by |
| apply integral_mono hil hir |
| intro X |
| dsimp only |
| simp only [Finset.mul_sum,← Finset.sum_add_distrib] |
| apply Finset.sum_le_sum |
| intro i _ |
| nlinarith [sq_nonneg ((∑ j,X i j*v j)-e i)] |
| _ = _ := by |
| rw [integral_add ((integrable_finsetSum _ (fun i _ => (hi i).integrable_sq)).const_mul 2) |
| (integrable_const _),integral_const_mul, rows_energy] |
| simp |
| ring |
| |
| end GaussianTools |
| |
| end OAI |
| |
