| import Mathlib |
| import OAI.Probability.SKGap.Entropy.ScalarEntropyTarget |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set Filter |
| open scoped NNReal ENNReal Topology |
| noncomputable section |
| open Real Set |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped Topology NNReal ENNReal BoundedContinuousFunction |
| namespace SKGap |
|
|
| lemma entropyTestBound_q_pos {P : Measure ℝ} [IsProbabilityMeasure P] |
| {d D : ℝ} {s : ℝ≥0} (hs : s ≠ 0) (hD : EntropyTestBound P (gaussianReal d s) D) : |
| 0 < ∫ y, tanh y^2 ∂P := by |
| have hn : 0 ≤ ∫ y, tanh y^2 ∂P := integral_nonneg (fun y => sq_nonneg (tanh y)) |
| by_contra hh |
| have hzero : (∫ y, tanh y^2 ∂P) = 0 := le_antisymm (le_of_not_gt hh) hn |
| let F : ℕ → ℝ → ℝ := fun n y => exp (-(n:ℝ)*tanh y^2) |
| have hFn (n : ℕ) (y : ℝ) : ‖F n y‖ ≤ 1 := by |
| rw [Real.norm_eq_abs, abs_of_pos (exp_pos _)] |
| exact exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (Nat.cast_nonneg n)) (sq_nonneg _)) |
| have hFi (n : ℕ) : Integrable (F n) (gaussianReal d s) := |
| Integrable.of_bound (by dsimp [F]; fun_prop) 1 (ae_of_all _ (hFn n)) |
| have h_lower (n : ℕ) : exp (-D) ≤ ∫ y, F n y ∂gaussianReal d s := by |
| have h := entropyTestBound_continuous hD (f := fun y => -(n:ℝ)*tanh y^2) |
| (by fun_prop) (integrable_tanh_sq.const_mul _) (hFi n) |
| rw [integral_const_mul, hzero, mul_zero, zero_sub] at h |
| have hz := integral_exp_pos (hFi n) |
| calc |
| _ ≤ exp (log (∫ y, F n y ∂gaussianReal d s)) := exp_le_exp.mpr (by linarith) |
| _ = _ := exp_log hz |
| have := nullSingletonClass_gaussianReal (μ := d) hs |
| have hlim : Tendsto (fun n => ∫ y, F n y ∂gaussianReal d s) atTop (𝓝 0) := by |
| have hh : Tendsto (fun n => ∫ y, F n y ∂gaussianReal d s) atTop |
| (𝓝 (∫ _ : ℝ, (0:ℝ) ∂gaussianReal d s)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ => (1:ℝ)) |
| · intro n; exact (hFi n).aestronglyMeasurable |
| · exact integrable_const 1 |
| · intro n; exact ae_of_all _ (hFn n) |
| · filter_upwards [(gaussianReal d s).ae_ne 0] with y hy |
| have hm : tanh y ≠ 0 := fun hz => hy (tanh_injective (by simpa only [tanh_zero] using hz)) |
| have hc : 0 < tanh y^2 := sq_pos_of_ne_zero hm |
| have hnat : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop := tendsto_natCast_atTop_atTop |
| have hneg := tendsto_neg_atTop_atBot.comp (hnat.atTop_mul_const hc) |
| exact tendsto_exp_atBot.comp (by simpa only [Function.comp_def, neg_mul] using hneg) |
| simpa using hh |
| have hle := ge_of_tendsto hlim (Eventually.of_forall h_lower) |
| exact (not_le_of_gt (exp_pos (-D))) hle |
|
|
| @[fun_prop] lemma continuous_entropyTransportLog {u d s : ℝ} (hu : u < 1) : |
| Continuous (entropyTransportLog u d s) := by |
| have hc : Continuous (entropyTransportDeriv u) := by unfold entropyTransportDeriv; fun_prop |
| have hl := hc.log (fun y => (entropyTransportDeriv_pos hu y).ne') |
| unfold entropyTransportLog |
| exact ((continuous_const.mul ((continuous_id.sub continuous_const).mul continuous_tanh)).sub |
| (continuous_const.mul (continuous_tanh.pow 2))).add hl |
| |
| theorem test_transport_lower {P : Measure ℝ} [IsProbabilityMeasure P] |
| (hP : Integrable id P) {u d : ℝ} {s : ℝ≥0} (hu : u < 1) (hs : s ≠ 0) |
| {D : ℝ} (hD : EntropyTestBound P (gaussianReal d s) D) : |
| u / (s : ℝ) * (∫ y, (y - d) * tanh y ∂P) - |
| u ^ 2 / (2 * (s : ℝ)) * (∫ y, tanh y ^ 2 ∂P) + |
| (1 - ∫ y, tanh y ^ 2 ∂P) * log (1 - u) ≤ D := by |
| have hlog := integral_mono |
| (((integrable_const (1 : ℝ)).sub integrable_tanh_sq).mul_const (log (1 - u))) |
| (integrable_log_entropyTransportDeriv (P := P) hu) (log_entropyTransportDeriv_lower hu) |
| simp only [Pi.sub_apply] at hlog |
| rw [integral_mul_const, integral_sub (integrable_const 1) integrable_tanh_sq, |
| integral_const, probReal_univ, smul_eq_mul, one_mul] at hlog |
| have h := entropyTestBound_normalized hD (continuous_entropyTransportLog hu) (integrable_entropyTransportLog hP hu d s) |
| (integrable_exp_entropyTransportLog hu hs) (integral_exp_entropyTransportLog hu hs) |
| unfold entropyTransportLog at h |
| rw [integral_add (f := fun y => u / (s : ℝ) * ((y - d) * tanh y) - |
| u ^ 2 / (2 * (s : ℝ)) * tanh y ^ 2) |
| (((integrable_sub_mul_tanh hP d).const_mul _).sub (integrable_tanh_sq.const_mul _)) |
| (integrable_log_entropyTransportDeriv hu), |
| integral_sub ((integrable_sub_mul_tanh hP d).const_mul _) (integrable_tanh_sq.const_mul _), |
| integral_const_mul, integral_const_mul] at h |
| linarith |
| |
| lemma continuous_gaussianRatioLog {d : ℝ} {s r : ℝ≥0} (hs : s ≠ 0) (hr : r ≠ 0) : |
| Continuous (gaussianRatioLog d s r) := by |
| have he : gaussianRatioLog d s r = fun y => (log (s : ℝ) - log (r : ℝ)) / 2 + |
| (1 / (2 * (s : ℝ)) - 1 / (2 * (r : ℝ))) * (y - d)^2 := funext (gaussianRatioLog_formula hs hr) |
| rw [he]; fun_prop |
| |
| theorem test_variance_comparison {P : Measure ℝ} [IsProbabilityMeasure P] |
| {d : ℝ} {s r : ℝ≥0} (hs : s ≠ 0) (hr : r ≠ 0) |
| (hP : Integrable (fun y => (y - d) ^ 2) P) |
| {D : ℝ} (hD : EntropyTestBound P (gaussianReal d s) D) : |
| (log (s : ℝ) - log (r : ℝ)) / 2 + |
| (1 / (2 * (s : ℝ)) - 1 / (2 * (r : ℝ))) * (∫ y, (y - d) ^ 2 ∂P) ≤ |
| D := by |
| have h := entropyTestBound_normalized hD (continuous_gaussianRatioLog hs hr) (integrable_gaussianRatioLog hs hr hP) |
| (integrable_exp_gaussianRatioLog hs hr) (integral_exp_gaussianRatioLog hs hr) |
| simp_rw [gaussianRatioLog_formula hs hr] at h |
| rw [integral_add (integrable_const _) (hP.const_mul _), integral_const_mul] at h |
| simpa using h |
| |
| theorem test_variance_lower {P : Measure ℝ} [IsProbabilityMeasure P] |
| {d : ℝ} {s : ℝ≥0} (hs : s ≠ 0) |
| (hP : Integrable (fun y => (y - d) ^ 2) P) |
| {D : ℝ} (hD : EntropyTestBound P (gaussianReal d s) D) {v : ℝ} (hv : 0 < v) |
| (hvar : ∫ y, (y - d) ^ 2 ∂P = (s : ℝ) * v) : |
| (v - 1 - log v) / 2 ≤ D := by |
| let r : ℝ≥0 := s * ⟨v, hv.le⟩ |
| have hr : r ≠ 0 := mul_ne_zero hs (NNReal.coe_ne_zero.mp hv.ne') |
| have h := test_variance_comparison hs hr hP hD |
| change (log (s : ℝ) - log ((s : ℝ) * v)) / 2 + |
| (1 / (2 * (s : ℝ)) - 1 / (2 * ((s : ℝ) * v))) * |
| (∫ y, (y - d) ^ 2 ∂P) ≤ _ at h |
| have hsp : (s : ℝ) ≠ 0 := NNReal.coe_ne_zero.mpr hs |
| rw [log_mul hsp hv.ne', hvar] at h |
| convert h using 1 |
| field_simp |
| ring |
| |
| theorem test_scalar_inequality {P : Measure ℝ} [IsProbabilityMeasure P] |
| (hP : Integrable (fun y : ℝ => y ^ 2) P) |
| {j d : ℝ} {s : ℝ≥0} (hj0 : 0 < j) (hj1 : j < 1) (hs : s ≠ 0) |
| (hcons : j * (∫ y, tanh y ^ 2 ∂P) ≤ (s : ℝ)) |
| {D : ℝ} (hD : EntropyTestBound P (gaussianReal d s) D) : |
| let q := ∫ y, tanh y ^ 2 ∂P |
| let a := ∫ y, (y - d) * tanh y ∂P |
| let T := j * (a - (s : ℝ) * (1 - q)) ^ 2 / (2 * (s : ℝ) * ((s : ℝ) + j * q)) |
| T ≤ D ∧ |
| (a ≠ (s : ℝ) * (1 - q) → T < D) := by |
| dsimp only |
| let q := ∫ y, tanh y ^ 2 ∂P |
| let a := ∫ y, (y - d) * tanh y ∂P |
| let k := j * q / (s : ℝ) |
| let B := j * (1 - q) |
| let C := j * a / (s : ℝ) |
| have hsp : 0 < (s : ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr hs) |
| have hq0 : 0 < q := entropyTestBound_q_pos hs hD |
| have hq1 : q < 1 := entropy_q_lt_one |
| have hk0 : 0 < k := div_pos (mul_pos hj0 hq0) hsp |
| have hk1 : k ≤ 1 := (div_le_one hsp).mpr hcons |
| have hB0 : 0 < B := mul_pos hj0 (sub_pos.mpr hq1) |
| have hB1 : B < 1 := by dsimp [B]; nlinarith |
| have htransport (u : ℝ) (hu : u < 1) : |
| u * (C - B) - k * u ^ 2 / 2 - B * entropyEll u ≤ j * D := by |
| have h := mul_le_mul_of_nonneg_left (test_transport_lower (secondMoment_integrable hP) hu hs hD) hj0.le |
| change j * (u / (s : ℝ) * a - u ^ 2 / (2 * (s : ℝ)) * q + (1 - q) * log (1 - u)) ≤ j * D at h |
| convert h using 1 |
| dsimp [k, B, C, entropyEll] |
| field_simp |
| ring |
| have hvariance (hC : 1 < C) : C ^ 2 / k - 1 - log (C ^ 2 / k) ≤ 2 * (j * D) := by |
| let v := (∫ y, (y - d) ^ 2 ∂P) / (s : ℝ) |
| have hcauchy := entropy_moment_cauchy hP d hq0 |
| have hcv : C ^ 2 / (j * k) ≤ v := by |
| have heq : C ^ 2 / (j * k) * (s : ℝ) = a ^ 2 / q := by |
| dsimp [C, k] |
| field_simp |
| dsimp [v] |
| rw [le_div_iff₀ hsp, heq, div_le_iff₀ hq0] |
| simpa only [q, a, mul_comm] using hcauchy |
| have hv : 0 < v := (div_pos (pow_pos (by linarith : 0 < C) 2) (mul_pos hj0 hk0)).trans_le hcv |
| have hvar : (∫ y, (y - d) ^ 2 ∂P) = (s : ℝ) * v := by dsimp [v]; field_simp |
| have hbound := test_variance_lower hs (centered_secondMoment_integrable hP d) hD hv hvar |
| exact entropy_variance_scaled_lower hj0 hj1.le hk0 hk1 hC hcv hbound |
| have hopt := entropy_scalar_optimization hk0 hk1 hB0 hB1 |
| (mul_nonneg hj0.le (entropyTestBound_nonneg hD)) htransport hvariance |
| have heq : (C - B) ^ 2 / (2 * (1 + k)) = |
| j * (j * (a - (s : ℝ) * (1 - q)) ^ 2 / (2 * (s : ℝ) * ((s : ℝ) + j * q))) := by |
| dsimp [C, B, k] |
| field_simp |
| rw [heq] at hopt |
| refine ⟨le_of_mul_le_mul_left hopt.1 hj0, ?_⟩ |
| intro ha |
| have hCB : C ≠ B := by |
| intro h |
| apply ha |
| dsimp [C, B] at h |
| have he := (div_eq_iff hsp.ne').mp h |
| nlinarith |
| exact lt_of_mul_lt_mul_left (hopt.2 hCB) hj0.le |
|
|
| theorem scalar_strict_witness {P : Measure ℝ} [IsProbabilityMeasure P] |
| (hP : Integrable (fun y : ℝ => y^2) P) |
| {j d : ℝ} {s : ℝ≥0} (hj : 0 < j) (hj1 : j < 1) (hs : s ≠ 0) |
| (hcons : j * (∫ y, tanh y^2 ∂P) ≤ (s:ℝ)) (hne : P ≠ gaussianReal d s) : |
| ∃ f : ℝ →ᵇ ℝ, scalarEntropyTarget P j d s < |
| (∫ y, f y ∂P) - log (∫ y, exp (f y) ∂gaussianReal d s) := by |
| by_contra h |
| push Not at h |
| have hD : EntropyTestBound P (gaussianReal d s) (scalarEntropyTarget P j d s) := h |
| have hop := test_scalar_inequality hP hj hj1 hs hcons hD |
| have he : (∫ y, (y-d)*tanh y ∂P) = (s:ℝ)*(1-∫ y, tanh y^2 ∂P) := by |
| by_contra hh |
| exact lt_irrefl _ (hop.2 hh) |
| have hzero : scalarEntropyTarget P j d s = 0 := by simp [scalarEntropyTarget, he] |
| rw [hzero] at hD |
| exact hne (entropyTestBound_zero_eq hD) |
|
|
| end SKGap |
|
|
| end |
| end |
| end |
| end |
|
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| end OAI |
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