| import Mathlib |
| import OAI.Probability.Perceptron.Sphere.GaussianRotation |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal BigOperators Topology RealInnerProductSpace |
| namespace SphericalPerceptronFreeEnergy |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| variable {S : Type*} [MeasurableSpace S] (ρ : Measure S) [IsProbabilityMeasure ρ] |
| {W : S → ℝ} {V : S → StrongDual ℝ E} |
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| def vectorPartition (W : S → ℝ) (V : S → StrongDual ℝ E) (x : E) : ℝ := |
| ∫ s, Real.exp (W s + V s x) ∂ρ |
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| def vectorLogPartition (W : S → ℝ) (V : S → StrongDual ℝ E) (x : E) : ℝ := |
| Real.log (vectorPartition ρ W V x) |
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| def vectorLogDerivative (W : S → ℝ) (V : S → StrongDual ℝ E) (x : E) : StrongDual ℝ E := |
| (vectorPartition ρ W V x)⁻¹ • (∫ s, Real.exp (W s + V s x) • V s ∂ρ) |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace S] in |
| lemma vectorEnergy_bound {K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) |
| (x : E) (s : S) : |W s + V s x| ≤ K+C*‖x‖ := by |
| apply (abs_add_le _ _).trans |
| apply add_le_add (hW s) |
| exact (ContinuousLinearMap.le_opNorm _ _).trans |
| (mul_le_mul_of_nonneg_right (hV s) (norm_nonneg x)) |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma vectorPartition_integrable (hWm : Measurable W) (hVm : Measurable V) |
| {K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) : |
| Integrable (fun s => Real.exp (W s + V s x)) ρ := by |
| refine Integrable.of_bound (by fun_prop) (Real.exp (K+C*‖x‖)) (ae_of_all _ fun s => ?_) |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| exact Real.exp_le_exp.mpr ((le_abs_self _).trans (vectorEnergy_bound hW hV x s)) |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma vectorPartition_pos (hWm : Measurable W) (hVm : Measurable V) |
| {K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) : |
| 0 < vectorPartition ρ W V x := by |
| apply integral_pos_iff_support_of_nonneg (fun s => (Real.exp_pos _).le) |
| (vectorPartition_integrable ρ hWm hVm hW hV x) |>.mpr |
| simp [Function.support,Real.exp_ne_zero] |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma vectorPartition_hasFDerivAt (hWm : Measurable W) (hVm : Measurable V) |
| {K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) : |
| HasFDerivAt (vectorPartition ρ W V) (∫ s, Real.exp (W s+V s x) • V s ∂ρ) x := by |
| apply hasFDerivAt_integral_of_dominated_of_fderiv_le |
| (s := Metric.ball x 1) (bound := fun _ => Real.exp (K+C*(‖x‖+1))*C) |
| (F' := fun y s => Real.exp (W s+V s y) • V s) (Metric.ball_mem_nhds x zero_lt_one) |
| (Filter.Eventually.of_forall fun y => by fun_prop) |
| (vectorPartition_integrable ρ hWm hVm hW hV x) (by fun_prop) |
| (ae_of_all _ fun s y hy => ?_) (integrable_const _) (ae_of_all _ fun s y _ => ?_) |
| · have hy' : ‖y‖ ≤ ‖x‖+1 := by |
| have hh : ‖y-x‖ < 1 := by simpa only [Metric.mem_ball,dist_eq_norm] using hy |
| have hn := norm_add_le (y-x) x |
| rw [sub_add_cancel] at hn |
| linarith |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| apply mul_le_mul _ (hV _) (norm_nonneg _) (Real.exp_pos _).le |
| apply Real.exp_le_exp.mpr |
| exact ((le_abs_self _).trans (vectorEnergy_bound hW hV y s)).trans |
| (add_le_add le_rfl (mul_le_mul_of_nonneg_left hy' hC)) |
| · exact (((V s).hasFDerivAt).const_add (W s)).exp |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma vectorLogPartition_hasFDerivAt (hWm : Measurable W) (hVm : Measurable V) |
| {K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) : |
| HasFDerivAt (vectorLogPartition ρ W V) (vectorLogDerivative ρ W V x) x := |
| (vectorPartition_hasFDerivAt ρ hWm hVm hC hW hV x).log |
| (vectorPartition_pos ρ hWm hVm hW hV x).ne' |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma vectorLogDerivative_bound (hWm : Measurable W) (hVm : Measurable V) |
| {K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) : |
| ‖vectorLogDerivative ρ W V x‖ ≤ C := by |
| have hp := vectorPartition_pos ρ hWm hVm hW hV x |
| have hn : ‖∫ s, Real.exp (W s+V s x) • V s ∂ρ‖ ≤ C*vectorPartition ρ W V x := by |
| have hh := norm_integral_le_of_norm_le (f := fun s => Real.exp (W s+V s x) • V s) |
| ((vectorPartition_integrable ρ hWm hVm hW hV x).const_mul C) |
| (ae_of_all _ fun s => by |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| exact (mul_le_mul_of_nonneg_left (hV s) (Real.exp_pos _).le).trans_eq (mul_comm ..)) |
| simpa only [integral_const_mul,vectorPartition] using hh |
| rw [vectorLogDerivative,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hp)] |
| calc |
| _ ≤ (vectorPartition ρ W V x)⁻¹*(C*vectorPartition ρ W V x) := |
| mul_le_mul_of_nonneg_left hn (inv_nonneg.mpr hp.le) |
| _ = C := by field_simp |
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| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma vectorLogDerivative_continuous (hWm : Measurable W) (hVm : Measurable V) |
| {K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) : |
| Continuous (vectorLogDerivative ρ W V) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x |
| have hp := (vectorPartition_hasFDerivAt ρ hWm hVm hC hW hV x).continuousAt |
| apply (hp.inv₀ (vectorPartition_pos ρ hWm hVm hW hV x).ne').smul |
| apply continuousAt_of_dominated (bound := fun _ => Real.exp (K+C*(‖x‖+1))*C) |
| · exact Filter.Eventually.of_forall fun y => by fun_prop |
| · filter_upwards [Metric.ball_mem_nhds x zero_lt_one] with y hy |
| exact ae_of_all _ fun s => by |
| have hy' : ‖y‖ ≤ ‖x‖+1 := by |
| have hh : ‖y-x‖ < 1 := by simpa only [Metric.mem_ball,dist_eq_norm] using hy |
| have hn := norm_add_le (y-x) x |
| rw [sub_add_cancel] at hn |
| linarith |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| apply mul_le_mul _ (hV _) (norm_nonneg _) (Real.exp_pos _).le |
| apply Real.exp_le_exp.mpr |
| exact ((le_abs_self _).trans (vectorEnergy_bound hW hV y s)).trans |
| (add_le_add le_rfl (mul_le_mul_of_nonneg_left hy' hC)) |
| · exact integrable_const _ |
| · exact ae_of_all _ fun s => by fun_prop |
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| lemma vectorLogPartition_gaussian_variance (hWm : Measurable W) (hVm : Measurable V) |
| {K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) : |
| variance (vectorLogPartition ρ W V) (stdGaussian E) ≤ (Real.pi^2/8)*C^2 := |
| stdGaussian_variance_le_of_fderiv_bound |
| (vectorLogPartition_hasFDerivAt ρ hWm hVm hC hW hV) |
| (vectorLogDerivative_continuous ρ hWm hVm hC hW hV) hC |
| (vectorLogDerivative_bound ρ hWm hVm hW hV) |
|
|
| end SphericalPerceptronFreeEnergy |
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| end |
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| end OAI |
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