| import Mathlib |
| import OAI.Probability.Perceptron.Variational.HeatGrid |
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| namespace OAI |
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| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open Filter Set Metric MeasureTheory ProbabilityTheory |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
|
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| lemma equicontinuous_of_convex_nonneg_tendsto {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| (F : ℕ → E → ℝ) (f : E → ℝ) |
| (hcv : ∀ n, ConvexOn ℝ Set.univ (F n)) (h0 : ∀ n x, 0 ≤ F n x) |
| (ht : ∀ x, Tendsto (fun n => F n x) atTop (𝓝 (f x))) : Equicontinuous F := by |
| let G : E → ℝ := fun x => sSup (Set.range (fun n => F n x)) |
| have hb (x : E) : BddAbove (Set.range (fun n => F n x)) := (ht x).bddAbove_range |
| have hle (n : ℕ) (x : E) : F n x ≤ G x := le_csSup (hb x) ⟨n,rfl⟩ |
| have hG : ConvexOn ℝ Set.univ G := by |
| refine ⟨convex_univ,?_⟩ |
| intro x _ y _ a b ha hb' hab |
| apply csSup_le (Set.range_nonempty _) |
| rintro _ ⟨n,rfl⟩ |
| exact ((hcv n).2 trivial trivial ha hb' hab).trans |
| (add_le_add (mul_le_mul_of_nonneg_left (hle n x) ha) |
| (mul_le_mul_of_nonneg_left (hle n y) hb')) |
| have hGc : Continuous G := continuousOn_univ.mp (hG.continuousOn isOpen_univ) |
| intro x |
| have hn : {y : E | G y < G x+1} ∈ 𝓝 x := |
| (hGc.continuousAt).preimage_mem_nhds (Iio_mem_nhds (by linarith)) |
| obtain ⟨r,hr,hrG⟩ := Metric.mem_nhds_iff.mp hn |
| let K : ℝ≥0 := (2*(G x+1)/(r/2)).toNNReal |
| have hLip (n : ℕ) : LipschitzOnWith K (F n) (ball x (r-r/2)) := by |
| apply ((hcv n).subset (Set.subset_univ _) (convex_ball _ _)).lipschitzOnWith_of_abs_le |
| (half_pos hr) |
| intro y hy |
| rw [abs_of_nonneg (h0 n y)] |
| exact (hle n y).trans (hrG hy).le |
| apply Metric.equicontinuousAt_of_continuity_modulus (fun y => (K:ℝ)*dist x y) |
| (by |
| have hc : Continuous (fun y : E => (K:ℝ)*dist x y) := by fun_prop |
| simpa only [dist_self,mul_zero] using hc.tendsto x) F |
| filter_upwards [ball_mem_nhds x (by linarith : 0 < r-r/2)] with y hy n |
| have hx : x ∈ ball x (r-r/2) := by rw [mem_ball,dist_self]; linarith |
| exact (hLip n).dist_le_mul x hx y hy |
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|
| lemma tendstoUniformlyOn_of_convex_nonneg_tendsto {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| (F : ℕ → E → ℝ) (f : E → ℝ) |
| (hcv : ∀ n, ConvexOn ℝ Set.univ (F n)) (h0 : ∀ n x, 0 ≤ F n x) |
| (ht : ∀ x, Tendsto (fun n => F n x) atTop (𝓝 (f x))) |
| {K : Set E} (hK : IsCompact K) : TendstoUniformlyOn F f atTop K := by |
| have he := equicontinuous_of_convex_nonneg_tendsto F f hcv h0 ht |
| have : CompactSpace K := isCompact_iff_compactSpace.mp hK |
| have he' : Equicontinuous (fun n (x : K) => F n x) := by |
| intro x |
| rw [Metric.equicontinuousAt_iff] |
| intro ε hε |
| obtain ⟨δ,hδ,hd⟩ := Metric.equicontinuousAt_iff.mp (he (x:E)) ε hε |
| exact ⟨δ,hδ,fun y hy n => hd (y:E) hy n⟩ |
| have hp : Tendsto (fun n (x : K) => F n x) atTop (𝓝 (fun x : K => f x)) := |
| tendsto_pi_nhds.mpr (fun x => ht x) |
| have hu := (he'.tendsto_uniformFun_iff_pi atTop (fun x : K => f x)).mpr hp |
| exact tendstoUniformlyOn_iff_tendstoUniformly_comp_coe.mpr |
| (UniformFun.tendsto_iff_tendstoUniformly.mp hu) |
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| lemma finiteAmplitudeSphereValue_uniform {k : ℕ} (w : Fin (k+1) → ℝ) |
| (hw : ∀ i, 0 < w i) (hw1 : ∑ i, w i=1) {K : Set (Fin (k+1) → ℝ)} |
| (hK : IsCompact K) : TendstoUniformlyOn (fun n => finiteAmplitudeSphereValue n k w) |
| (finiteAmplitudeSphereDual w (fun i => (hw i).le) hw1) atTop K := |
| tendstoUniformlyOn_of_convex_nonneg_tendsto _ _ (finiteAmplitudeSphereValue_convex w hw hw1) |
| (finiteAmplitudeSphereValue_nonneg w hw hw1) (finiteAmplitudeSphereValue_tendsto w hw hw1) hK |
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| def fieldAmplitudes (k : ℕ) (h : Fin (k+1) → ℝ) (i : Fin (k+1)) : ℝ := |
| if _hi : i.val < k then stepFieldIncrement k h i.val else Real.sqrt (2*h 0) |
|
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| lemma fieldAmplitudes_last (k : ℕ) (h : Fin (k+1) → ℝ) : |
| fieldAmplitudes k h (Fin.last k)=Real.sqrt (2*h 0) := by |
| simp [fieldAmplitudes] |
|
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| lemma fieldAmplitudes_increment (k : ℕ) (h : Fin (k+1) → ℝ) : |
| finiteAmplitudeIncrement k (fieldAmplitudes k h)=stepFieldIncrement k h := by |
| funext j |
| by_cases hj : j < k |
| · simp [finiteAmplitudeIncrement,fieldAmplitudes,hj] |
| · simp [finiteAmplitudeIncrement,stepFieldIncrement,hj] |
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| lemma fieldAmplitudes_continuous (k : ℕ) : Continuous (fieldAmplitudes k) := by |
| apply continuous_pi |
| intro i |
| unfold fieldAmplitudes |
| split_ifs |
| · unfold stepFieldIncrement |
| split_ifs |
| fun_prop |
| · fun_prop |
|
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| lemma amplitudeCovarianceLevels_fieldAmplitudes (k : ℕ) (h : Fin (k+1) → ℝ) |
| (hh : Monotone h) (hh0 : 0 ≤ h 0) : |
| amplitudeCovarianceLevels k (finiteAmplitudeIncrement k (fieldAmplitudes k h)) |
| (fieldAmplitudes k h (Fin.last k))=h := by |
| rw [fieldAmplitudes_increment,fieldAmplitudes_last] |
| funext i |
| induction i using Fin.induction with |
| | zero => |
| have hr := amplitudeCovarianceLevels_root k (stepFieldIncrement k h) (Real.sqrt (2*h 0)) |
| rw [Real.sq_sqrt (mul_nonneg (by norm_num) hh0)] at hr |
| linarith |
| | succ i ih => |
| have hi := amplitudeCovarianceLevels_increment k (stepFieldIncrement k h) (Real.sqrt (2*h 0)) i |
| rw [stepFieldIncrement_sq k h hh i,ih] at hi |
| linarith |
|
|
| lemma finiteAmplitudeSphereDual_fieldAmplitudes {k : ℕ} (w h : Fin (k+1) → ℝ) |
| (hw : ∀ i, 0 ≤ w i) (hw1 : ∑ i, w i=1) (hh : Monotone h) (hh0 : 0 ≤ h 0) : |
| finiteAmplitudeSphereDual w hw hw1 (fieldAmplitudes k h)= |
| sInf (finiteSphericalDualValues w h hw hw1)+h (Fin.last k) := by |
| unfold finiteAmplitudeSphereDual amplitudeSphereDual |
| rw [amplitudeCovarianceLevels_fieldAmplitudes k h hh hh0] |
|
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| lemma finiteAmplitudeSphereValue_fieldAmplitudes (n k : ℕ) (w h : Fin (k+1) → ℝ) |
| (hh0 : 0 ≤ h 0) : |
| finiteAmplitudeSphereValue n k w (fieldAmplitudes k h)=finiteSphericalFieldValue n k w h+h (Fin.last k) := by |
| unfold finiteAmplitudeSphereValue |
| rw [fieldAmplitudes_increment,fieldAmplitudes_last,amplitudeSphereValue_eq] |
| have hv : (⟨(Real.sqrt (2*h 0))^2,sq_nonneg _⟩ : ℝ≥0)=Real.toNNReal (2*h 0) := by |
| apply NNReal.coe_injective |
| change (Real.sqrt (2*h 0))^2=(Real.toNNReal (2*h 0) : ℝ) |
| rw [Real.sq_sqrt (mul_nonneg (by norm_num) hh0), |
| Real.coe_toNNReal _ (mul_nonneg (by norm_num) hh0)] |
| rw [hv,finiteSphericalFieldValue,sub_add_cancel] |
|
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| theorem finiteSphericalFieldValue_compact_uniform {k : ℕ} (w : Fin (k+1) → ℝ) |
| (hw : ∀ i, 0 < w i) (hw1 : ∑ i, w i=1) {K : Set (Fin (k+1) → ℝ)} |
| (hK : IsCompact K) (hfield : ∀ h ∈ K, Monotone h ∧ 0 ≤ h 0) : |
| TendstoUniformlyOn (fun n h => finiteSphericalFieldValue n k w h) |
| (fun h => sInf (finiteSphericalDualValues w h (fun i => (hw i).le) hw1)) atTop K := by |
| have hu := finiteAmplitudeSphereValue_uniform w hw hw1 (hK.image (fieldAmplitudes_continuous k)) |
| rw [Metric.tendstoUniformlyOn_iff] at hu ⊢ |
| intro ε hε |
| filter_upwards [hu ε hε] with n hn h hh |
| have ht := hn (fieldAmplitudes k h) ⟨h,hh,rfl⟩ |
| rw [finiteAmplitudeSphereValue_fieldAmplitudes n k w h (hfield h hh).2, |
| finiteAmplitudeSphereDual_fieldAmplitudes w h (fun i => (hw i).le) hw1 |
| (hfield h hh).1 (hfield h hh).2,dist_add_right] at ht |
| exact ht |
|
|
| end SphericalPerceptronFreeEnergy |
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| end |
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| end OAI |
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