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a2ee19a | 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 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.Probability.PosteriorReplicas.Geometry.FrameTubes
namespace OAI
open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Metric
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace PosteriorReplicas
@[fun_prop] lemma continuous_basePoint_label {d k : ℕ} (base : BaseLaw)
(A : Rows d k) (x : Vec d) : Continuous (fun y => basePoint base A y x) := by
cases base <;> unfold basePoint <;> fun_prop
lemma baseDensity_sphere_clipped {d k : ℕ} (hk : k + 2 < d) {A : Rows d k}
(hA : LinearIndependent ℝ A) (y : Labels k) :
baseDensity .spherical A y = ENNReal.ofReal
(baseProjectionConstant .spherical d k *
(max (1 - ‖liftLabel A y‖^2) 0) ^ (((d : ℝ) - k - 2) / 2) /
Real.sqrt (rowGram A).det) := by
have hβ : 0 < ((d : ℝ) - k - 2) / 2 := by
have : (k : ℝ) + 2 < d := by exact_mod_cast hk
linarith
rw [baseDensity.eq_def, ite_eq_left hA]
dsimp only [baseProjectionConstant]
by_cases hy : ‖liftLabel A y‖ < 1
· rw [ite_eq_left hy, max_eq_left (by nlinarith [norm_nonneg (liftLabel A y)])]
· rw [ite_eq_right hy, max_eq_right (by nlinarith [le_of_not_gt hy]),
Real.zero_rpow hβ.ne', mul_zero, zero_div, ENNReal.ofReal_zero]
lemma continuous_baseDensity {d k : ℕ} (hk : k + 2 < d) (base : BaseLaw)
{A : Rows d k} (hA : LinearIndependent ℝ A) : Continuous (baseDensity base A) := by
cases base
· have hβ : 0 ≤ ((d : ℝ) - k - 2) / 2 := by
have : (k : ℝ) + 2 < d := by exact_mod_cast hk
linarith
have he : baseDensity .spherical A = fun y => ENNReal.ofReal
(baseProjectionConstant .spherical d k *
(max (1 - ‖liftLabel A y‖^2) 0) ^ (((d : ℝ) - k - 2) / 2) /
Real.sqrt (rowGram A).det) := funext (baseDensity_sphere_clipped hk hA)
rw [he]
apply ENNReal.continuous_ofReal.comp
apply Continuous.div_const
apply Continuous.const_mul
exact (Real.continuous_rpow_const hβ).comp (by fun_prop)
· have he : baseDensity .gaussian A = fun y => ENNReal.ofReal
((2 * Real.pi) ^ (-(k : ℝ) / 2) *
Real.exp (-‖liftLabel A y‖^2 / 2) / Real.sqrt (rowGram A).det) := by
funext y
rw [baseDensity.eq_def, ite_eq_left hA]
rw [he]
exact ENNReal.continuous_ofReal.comp (by fun_prop)
lemma base_sample_representation {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A)
(hk : k < d) (base : BaseLaw) :
((volume.withDensity (baseDensity base A)).prod (stdGaussian (Vec d))).map
(fun z => basePoint base A z.1 z.2) = base.measure d := by
have : NeZero d := ⟨by omega⟩
have hb0 := (measurable_baseDensity (d := d) (k := k) base).comp
(show Measurable (fun y : Labels k => (A,y)) from measurable_const.prodMk measurable_id)
simp only [Function.comp_def] at hb0
have hb : Measurable (baseDensity base A) := hb0
have hm : Measurable (fun z : Labels k × Vec d => basePoint base A z.1 z.2) := by
have h := (measurable_basePoint (d := d) (k := k) base).comp
(show Measurable (fun z : Labels k × Vec d => ((A,z.1),z.2)) from
(measurable_const.prodMk measurable_fst).prodMk measurable_snd)
exact h
have : IsProbabilityMeasure (volume.withDensity (baseDensity base A)) := by
rw [← projected_base_measure hA hk]
infer_instance
apply Measure.ext_of_lintegral
intro g hg
have hgm : Measurable (fun z : Labels k × Vec d => g (basePoint base A z.1 z.2)) := hg.comp hm
rw [lintegral_map hg hm, lintegral_prod _ hgm.aemeasurable,
lintegral_withDensity_eq_lintegral_mul _ hb (by fun_prop), base_lintegral hA hk base hg]
apply lintegral_congr
intro y
simp only [Pi.mul_apply]
rw [baseFiber_eq_map, lintegral_map hg (measurable_basePoint_fixed base A y)]
end PosteriorReplicas
end
end OAI
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