| import OAI.Probability.ProjectionMoments.SphereDensity |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory |
| open scoped ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| namespace ProjectionMoments |
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| section |
| attribute [fun_prop] measurable_ballAverage |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma replicaConstant_real_le [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {ν : Measure E} (hν : ν Set.univ ≤ 1) {a : ℝ} {m : ℕ} |
| (hd : 1 ≤ Module.finrank ℝ E) (ha : a ≤ Module.finrank ℝ E) |
| (hm : m ≤ Module.finrank ℝ E) : |
| NoiselessRegression.replicaConstant ν a m ≤ (128 : ℝ≥0∞)^Module.finrank ℝ E := by |
| apply le_trans _ (NoiselessRegression.replicaConstant_le hν hd le_rfl hm) |
| have hr : (4 : ℝ)^a ≤ (4 : ℝ)^(Module.finrank ℝ E : ℝ) := |
| Real.rpow_le_rpow_of_exponent_le (by norm_num) ha |
| have hc : ENNReal.ofReal ((3 : ℝ)^Module.finrank ℝ E * 4^a) ≤ |
| ENNReal.ofReal ((3 : ℝ)^Module.finrank ℝ E * 4^(Module.finrank ℝ E : ℝ)) := |
| ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left hr (by positivity)) |
| exact add_le_add le_rfl (mul_le_mul' hc le_rfl) |
|
|
| theorem frame_moment_unit {ν : Measure E} [IsFiniteMeasure ν] {a : ℝ} |
| (hν : NoiselessRegression.HasLocalMassBound ν a) |
| (hs : ν.support ⊆ Metric.closedBall 0 1) (hmass : ν Set.univ ≤ 1) |
| {m n : ℕ} (hd : 1 ≤ Module.finrank ℝ E) (ha : a ≤ Module.finrank ℝ E) |
| (hm : m ≤ Module.finrank ℝ E) (hmargin : (n : ℝ)+m ≤ a) |
| (u : Fin m → E) (hu : Orthonormal ℝ u) |
| (hac : ∀ᵐ A ∂frameLaw u, ν.map (coordinateMap A) ≪ (volume : Measure (Fin m → ℝ))) : |
| ∫⁻ A, labelMoment ν m n A ∂frameLaw u ≤ |
| (2 : ℝ≥0∞)^m * ENNReal.ofReal (Real.sqrt (2 * Module.finrank ℝ E))^(m*n) * |
| (128 : ℝ≥0∞)^(Module.finrank ℝ E*n) := by |
| rw [← orbitMoment_eq_frame] |
| apply (frame_gaussian_comparison ν hm hd u hu hac).trans |
| apply mul_le_mul' le_rfl |
| apply (gaussian_labelMoment_bound hν hs m n hmargin).trans |
| simpa only [one_mul, pow_mul] using |
| mul_le_mul' hmass (pow_le_pow_left' (replicaConstant_real_le hmass hd ha hm) n) |
|
|
| lemma lintegral_inv_smul_label {m : ℕ} (f : Label m → ℝ≥0∞) |
| (hf : Measurable f) {R : ℝ} (hR : 0 < R) : |
| ∫⁻ y, f (R⁻¹ • y) = ENNReal.ofReal (R^m) * ∫⁻ y, f y := by |
| rw [← lintegral_map hf (by fun_prop), Measure.map_addHaar_smul volume (inv_ne_zero hR.ne'), |
| finrank_euclideanSpace_fin, inv_pow, inv_inv, abs_of_nonneg (pow_nonneg hR.le m), |
| lintegral_smul_measure, smul_eq_mul] |
|
|
| lemma ballAverage_label_scaled {ν : Measure E} [IsFiniteMeasure ν] |
| {K a : ℝ} (hK : 0 < K) (z : E) {R η : ℝ} (hR : 0 < R) (hη : 0 < η) |
| (m n : ℕ) (A : Fin m → E) : |
| (∫⁻ y : Label m, ballAverage ν m η (A,y)^(n+1)) = |
| (ENNReal.ofReal (K*R^(a-m)))^(n+1) * ENNReal.ofReal (R^m) * |
| ∫⁻ y : Label m, ballAverage (normalizedRescale ν K a z R) m (η/R) (A,y)^(n+1) := by |
| let μ := normalizedRescale ν K a z R |
| have : IsFiniteMeasure μ := normalizedRescale_finite ν K a z R (by positivity) |
| rw [← lintegral_add_left_eq_self (fun y : Label m => ballAverage ν m η (A,y)^(n+1)) (rowMap A z)] |
| simp_rw [ballAverage_normalizedRescale K a hK z hR hη, mul_pow] |
| rw [lintegral_const_mul _ (by fun_prop), lintegral_inv_smul_label (fun y : Label m => ballAverage (normalizedRescale ν K a z R) m (η/R) (A,y)^(n+1)) (by fun_prop) hR, |
| mul_assoc] |
|
|
| theorem ballAverage_integrated_scaled {ν : Measure E} [IsFiniteMeasure ν] |
| {K a : ℝ} (hν : AllBallMass ν K a) (hK : 0 < K) |
| (z : E) {R η : ℝ} (hR : 0 < R) (hη : 0 < η) |
| (hs : ν.support ⊆ Metric.closedBall z R) |
| (m n : ℕ) (hmargin : (n : ℝ)+m ≤ a) |
| (hd : 1 ≤ Module.finrank ℝ E) (ha : a ≤ Module.finrank ℝ E) |
| (hm : m ≤ Module.finrank ℝ E) : |
| ∫⁻ x : (Fin m → E) × Label m, ballAverage ν m η x^(n+1) |
| ∂(Measure.pi (fun _ => stdGaussian E)).prod volume ≤ |
| ENNReal.ofReal (K*R^(a-m))^(n+1) * ENNReal.ofReal (R^m) * |
| (128 : ℝ≥0∞)^(Module.finrank ℝ E*n) := by |
| let μ := normalizedRescale ν K a z R |
| have : IsFiniteMeasure μ := normalizedRescale_finite ν K a z R (by positivity) |
| rw [lintegral_prod _ (by fun_prop)] |
| simp_rw [ballAverage_label_scaled (ν := ν) (a := a) hK z hR hη m n] |
| rw [lintegral_const_mul _ (by fun_prop)] |
| apply mul_le_mul' le_rfl |
| rw [← lintegral_prod _ (show AEMeasurable (fun x : (Fin m → E) × Label m => |
| ballAverage μ m (η/R) x^(n+1)) _ by fun_prop)] |
| apply (ballAverage_integrated_moment (normalizedRescale_local hν hK z hR) |
| (normalizedRescale_support K a z hR hs) m n hmargin (div_pos hη hR)).trans |
| have hmass := normalizedRescale_mass hν hK z hR hs |
| simpa only [one_mul, pow_mul] using mul_le_mul' hmass |
| (pow_le_pow_left' (replicaConstant_real_le hmass hd ha hm) n) |
|
|
| theorem lowerDensity_integrated_scaled {ν : Measure E} [IsFiniteMeasure ν] |
| {K a : ℝ} (hν : AllBallMass ν K a) (hK : 0 < K) |
| (z : E) {R : ℝ} (hR : 0 < R) (hs : ν.support ⊆ Metric.closedBall z R) |
| (m n : ℕ) (hmargin : (n : ℝ)+m ≤ a) |
| (hd : 1 ≤ Module.finrank ℝ E) (ha : a ≤ Module.finrank ℝ E) |
| (hm : m ≤ Module.finrank ℝ E) : |
| ∫⁻ x : (Fin m → E) × Label m, lowerDensity ν m x^(n+1) |
| ∂(Measure.pi (fun _ => stdGaussian E)).prod volume ≤ |
| ENNReal.ofReal (K*R^(a-m))^(n+1) * ENNReal.ofReal (R^m) * |
| (128 : ℝ≥0∞)^(Module.finrank ℝ E*n) := by |
| have hp (v : ℕ → ℝ≥0∞) : (liminf v atTop)^(n+1) = |
| liminf (fun j => v j^(n+1)) atTop := |
| (show Monotone (fun x : ℝ≥0∞ => x^(n+1)) from |
| (fun _ _ h => pow_le_pow_left' h (n+1))).map_liminf_of_continuousAt |
| v (ENNReal.continuous_pow (n+1)).continuousAt |
| simp only [lowerDensity, hp] |
| apply (lintegral_liminf_le (fun j => (measurable_ballAverage ν m _).pow_const _)).trans |
| apply liminf_le_of_frequently_le' |
| exact Filter.Frequently.of_forall (fun j => ballAverage_integrated_scaled hν hK z hR |
| (inv_pos.mpr (by positivity)) hs m n hmargin hd ha hm) |
|
|
| theorem gaussian_labelMoment_scaled {ν : Measure E} [IsFiniteMeasure ν] |
| {K a : ℝ} (hν : AllBallMass ν K a) (hK : 0 < K) |
| (z : E) {R : ℝ} (hR : 0 < R) (hs : ν.support ⊆ Metric.closedBall z R) |
| (m n : ℕ) (hmargin : (n : ℝ)+m ≤ a) |
| (hd : 1 ≤ Module.finrank ℝ E) (ha : a ≤ Module.finrank ℝ E) |
| (hm : m ≤ Module.finrank ℝ E) : |
| ∫⁻ A, labelMoment ν m n A ∂Measure.pi (fun _ => stdGaussian E) ≤ |
| ENNReal.ofReal (K*R^(a-m))^(n+1) * ENNReal.ofReal (R^m) * |
| (128 : ℝ≥0∞)^(Module.finrank ℝ E*n) := by |
| have he (A : Fin m → E) : labelMoment ν m n A = |
| ∫⁻ y : Label m, lowerDensity ν m (A,y)^(n+1) := |
| (PiLp.volume_preserving_toLp (Fin m)).lintegral_comp |
| (show Measurable (fun y : Label m => lowerDensity ν m (A,y)^(n+1)) by fun_prop) |
| simp_rw [he] |
| rw [← lintegral_prod _ (show AEMeasurable (fun x : (Fin m → E) × Label m => |
| lowerDensity ν m x^(n+1)) _ by fun_prop)] |
| exact lowerDensity_integrated_scaled hν hK z hR hs m n hmargin hd ha hm |
|
|
| theorem frame_moment_scaled {ν : Measure E} [IsFiniteMeasure ν] |
| {K a : ℝ} (hν : AllBallMass ν K a) (hK : 0 < K) |
| (z : E) {R : ℝ} (hR : 0 < R) (hs : ν.support ⊆ Metric.closedBall z R) |
| (m n : ℕ) (hmargin : (n : ℝ)+m ≤ a) |
| (hd : 1 ≤ Module.finrank ℝ E) (ha : a ≤ Module.finrank ℝ E) |
| (hm : m ≤ Module.finrank ℝ E) (u : Fin m → E) (hu : Orthonormal ℝ u) |
| (hac : ∀ᵐ A ∂frameLaw u, ν.map (coordinateMap A) ≪ (volume : Measure (Fin m → ℝ))) : |
| ∫⁻ A, labelMoment ν m n A ∂frameLaw u ≤ |
| (2 : ℝ≥0∞)^m * ENNReal.ofReal (Real.sqrt (2*Module.finrank ℝ E))^(m*n) * |
| (ENNReal.ofReal (K*R^(a-m))^(n+1) * ENNReal.ofReal (R^m) * |
| (128 : ℝ≥0∞)^(Module.finrank ℝ E*n)) := by |
| rw [← orbitMoment_eq_frame] |
| exact (frame_gaussian_comparison ν hm hd u hu hac).trans |
| (mul_le_mul' le_rfl (gaussian_labelMoment_scaled hν hK z hR hs m n hmargin hd ha hm)) |
|
|
| end |
|
|
| lemma frame_cost_power {d m n : ℕ} (hd : 1 ≤ d) (hm : m ≤ d) |
| (a K R : ℝ) (hK : 0 ≤ K) (hR : 0 < R) : |
| (2 : ℝ)^m * (Real.sqrt (2*d))^(m*n) * |
| ((K*R^(a-m))^(n+1) * R^m * (128 : ℝ)^(d*n)) ≤ |
| ((256 : ℝ)^d * (256*Real.sqrt d)^((m:ℝ)*(1-1/((n:ℝ)+1))) * K * |
| R^(a-(m:ℝ)*(1-1/((n:ℝ)+1))))^(n+1) := by |
| have hq : (n : ℝ)+1 ≠ 0 := by positivity |
| have hexp : ((m:ℝ)*(1-1/((n:ℝ)+1)))*((n:ℝ)+1) = (m*n:ℕ) := by |
| push_cast |
| field_simp |
| ring |
| have hRexp : (a-(m:ℝ)*(1-1/((n:ℝ)+1)))*((n:ℝ)+1) = |
| (a-m)*((n:ℝ)+1)+m := by |
| field_simp |
| ring |
| have hscale : (K*R^(a-m))^(n+1)*R^m = |
| K^(n+1)*R^((a-(m:ℝ)*(1-1/((n:ℝ)+1)))*((n:ℝ)+1)) := by |
| rw [mul_pow, mul_assoc, ← Real.rpow_mul_natCast hR.le, |
| ← Real.rpow_natCast R m, ← Real.rpow_add hR] |
| congr 2 |
| rw [Nat.cast_add, Nat.cast_one, hRexp] |
| have h2 : (2 : ℝ)^m ≤ 256^d := by |
| apply (pow_le_pow_left₀ (by norm_num) (by norm_num : (2 : ℝ) ≤ 256) m).trans |
| exact pow_le_pow_right₀ (by norm_num) hm |
| have h128 : (128 : ℝ)^(d*n) ≤ 256^(d*n) := |
| pow_le_pow_left₀ (by norm_num) (by norm_num) (d*n) |
| have hcoef : (2 : ℝ)^m * 128^(d*n) ≤ 256^(d*(n+1)) := by |
| apply (mul_le_mul h2 h128 (by positivity) (by positivity)).trans_eq |
| rw [← pow_add] |
| congr 1 |
| ring |
| have hsqrt : Real.sqrt (2*d) ≤ 256*Real.sqrt d := by |
| rw [Real.sqrt_mul (by norm_num)] |
| apply mul_le_mul_of_nonneg_right _ (Real.sqrt_nonneg _) |
| exact (Real.sqrt_le_iff.mpr ⟨by norm_num, by norm_num⟩) |
| have hS : ((256 : ℝ)^d * (256*Real.sqrt d)^((m:ℝ)*(1-1/((n:ℝ)+1))) * K * |
| R^(a-(m:ℝ)*(1-1/((n:ℝ)+1))))^(n+1) = |
| 256^(d*(n+1)) * (256*Real.sqrt d)^(m*n) * |
| (K^(n+1)*R^((a-(m:ℝ)*(1-1/((n:ℝ)+1)))*((n:ℝ)+1))) := by |
| simp only [mul_pow] |
| rw [← pow_mul, ← Real.rpow_mul_natCast (by positivity), |
| ← Real.rpow_mul_natCast hR.le] |
| simp only [Nat.cast_add, Nat.cast_one] |
| rw [hexp, Real.rpow_natCast] |
| ring |
| rw [hS] |
| calc |
| _ = (2^m*128^(d*n))*(Real.sqrt (2*d))^(m*n) * |
| (K^(n+1)*R^((a-(m:ℝ)*(1-1/((n:ℝ)+1)))*((n:ℝ)+1))) := by |
| rw [← hscale] |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (mul_le_mul hcoef (pow_le_pow_left₀ (Real.sqrt_nonneg _) hsqrt (m*n)) |
| (by positivity) (by positivity)) (by positivity) |
|
|
| lemma ennreal_root_bound {q : ℕ} (hq : q ≠ 0) {v : ℝ≥0∞} {B : ℝ} |
| (hB : 0 ≤ B) (h : v ≤ ENNReal.ofReal B ^ q) : |
| v ^ (1/(q:ℝ)) ≤ ENNReal.ofReal B := by |
| have hq' : (q : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hq |
| have hh := ENNReal.rpow_le_rpow h (by positivity : 0 ≤ 1/(q:ℝ)) |
| rwa [← ENNReal.ofReal_pow hB, |
| ENNReal.ofReal_rpow_of_nonneg (pow_nonneg hB q) (by positivity), |
| ← Real.rpow_natCast_mul hB, mul_one_div_cancel hq', Real.rpow_one] at hh |
|
|
| lemma frame_cost_root {d m n : ℕ} (hd : 1 ≤ d) (hm : m ≤ d) |
| (a K R : ℝ) (hK : 0 ≤ K) (hR : 0 < R) {v : ℝ≥0∞} |
| (h : v ≤ (2 : ℝ≥0∞)^m * ENNReal.ofReal (Real.sqrt (2*d))^(m*n) * |
| (ENNReal.ofReal (K*R^(a-m))^(n+1) * ENNReal.ofReal (R^m) * |
| (128 : ℝ≥0∞)^(d*n))) : |
| v ^ (1/((n:ℝ)+1)) ≤ |
| ENNReal.ofReal ((256 : ℝ)^d * (256*Real.sqrt d)^((m:ℝ)*(1-1/((n:ℝ)+1))) * K * |
| R^(a-(m:ℝ)*(1-1/((n:ℝ)+1)))) := by |
| have hb := ENNReal.ofReal_le_ofReal (frame_cost_power (n := n) hd hm a K R hK hR) |
| simp (discharger := positivity) only [ENNReal.ofReal_mul, ENNReal.ofReal_pow, |
| ENNReal.ofReal_ofNat] at hb |
| simp (discharger := positivity) only [ENNReal.ofReal_mul, ENNReal.ofReal_pow] at h |
| have hB : 0 ≤ (256 : ℝ)^d * (256*Real.sqrt d)^((m:ℝ)*(1-1/((n:ℝ)+1))) * K * |
| R^(a-(m:ℝ)*(1-1/((n:ℝ)+1))) := by positivity |
| have hb' : v ≤ (ENNReal.ofReal ((256 : ℝ)^d * |
| (256*Real.sqrt d)^((m:ℝ)*(1-1/((n:ℝ)+1))) * K * |
| R^(a-(m:ℝ)*(1-1/((n:ℝ)+1)))))^(n+1) := by |
| simp (discharger := positivity) only [ENNReal.ofReal_mul, ENNReal.ofReal_pow, |
| ENNReal.ofReal_ofNat] |
| exact h.trans hb |
| have hh := ennreal_root_bound (show n+1 ≠ 0 by omega) hB hb' |
| simpa only [Nat.cast_add, Nat.cast_one] using hh |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| omit [BorelSpace E] in |
| lemma measure_eq_zero_of_allBall_zero [BorelSpace E] {ν : Measure E} {a : ℝ} |
| (hν : AllBallMass ν 0 a) (z : E) {R : ℝ} (hR : 0 < R) |
| (hs : ν.support ⊆ Metric.closedBall z R) : ν = 0 := by |
| apply Measure.measure_univ_eq_zero.mp |
| have he : ν Set.univ = ν (Metric.closedBall z R) := by |
| apply measure_congr |
| filter_upwards [ν.support_mem_ae] with x hx |
| exact propext (iff_of_true (Set.mem_univ x) (hs hx)) |
| rw [he] |
| apply le_antisymm _ bot_le |
| simpa using hν z R hR |
|
|
| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| @[simp] lemma lowerDensity_zero [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (m : ℕ) (x : (Fin m → E) × Label m) : |
| lowerDensity (0 : Measure E) m x = 0 := by |
| simp [lowerDensity, ballAverage, vectorSmoothing] |
|
|
| @[simp] lemma coordinateDensity_zero (m : ℕ) (x : (Fin m → E) × (Fin m → ℝ)) : |
| coordinateDensity (0 : Measure E) m x = 0 := lowerDensity_zero m _ |
|
|
| lemma coordinateDensity_frame_support (ν : Measure E) [IsFiniteMeasure ν] {m : ℕ} |
| (z : E) {R : ℝ} (hR : 0 ≤ R) (hs : ν.support ⊆ Metric.closedBall z R) |
| (A : Fin m → E) (hA : Orthonormal ℝ A) (y : Fin m → ℝ) |
| (hy : R < ‖WithLp.toLp 2 (y-coordinateMap A z)‖) : |
| coordinateDensity ν m (A,y) = 0 := by |
| apply lowerDensity_eq_zero_of_far ν m A z R (WithLp.toLp 2 y) hs |
| have he : WithLp.toLp 2 (y-coordinateMap A z) = WithLp.toLp 2 y - rowMap A z := rfl |
| rw [he] at hy |
| exact (show ‖rowMap A‖ * R ≤ R from by nlinarith [norm_rowMap_le hA]).trans_lt hy |
|
|
| theorem integrated_frame_moment : |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ (d k q : ℕ) (β K R : ℝ) |
| (ν : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ν] |
| (z : EuclideanSpace ℝ (Fin d)) (u : Fin k → EuclideanSpace ℝ (Fin d)), |
| 1 ≤ k → 2 ≤ q → k+q ≤ d → 0 < β → β ≤ d → |
| 1 ≤ β-k-((q:ℝ)-2) → 0 ≤ K → 0 < R → |
| ν.support ⊆ Metric.closedBall z R → AllBallMass ν K β → |
| Orthonormal ℝ u → |
| (∀ᵐ A ∂frameLaw u, ν.map (coordinateMap A) ≪ (volume : Measure (Fin k → ℝ))) → |
| Measurable (coordinateDensity ν k) ∧ |
| (∀ᵐ A ∂frameLaw u, |
| (volume : Measure (Fin k → ℝ)).withDensity (fun y => coordinateDensity ν k (A,y)) = |
| ν.map (coordinateMap A)) ∧ |
| (∀ (A : Fin k → EuclideanSpace ℝ (Fin d)), Orthonormal ℝ A → |
| ∀ y : Fin k → ℝ, |
| R < ‖WithLp.toLp 2 (y-coordinateMap A z)‖ → coordinateDensity ν k (A,y) = 0) ∧ |
| (∫⁻ A, ∫⁻ y : Fin k → ℝ, coordinateDensity ν k (A,y)^q ∂volume ∂frameLaw u) ^ (1/(q:ℝ)) ≤ |
| ENNReal.ofReal (C^d * (C*Real.sqrt d)^((k:ℝ)*(1-1/(q:ℝ))) * |
| K * R^(β-(k:ℝ)*(1-1/(q:ℝ)))) := by |
| refine ⟨256, by norm_num, ?_⟩ |
| intro d k q β K R ν _ z u hk hq hdim hβ hβd hmargin hK hR hs hν hu hac |
| refine ⟨measurable_coordinateDensity ν k, ?_, ?_, ?_⟩ |
| · filter_upwards [hac] with A hA |
| exact coordinateDensity_is_density ν k A hA |
| · intro A hA y hy |
| exact coordinateDensity_frame_support ν z hR.le hs A hA y hy |
| · rcases eq_or_lt_of_le hK with hK0 | hK0 |
| · have hzero : ν = 0 := measure_eq_zero_of_allBall_zero (hK0 ▸ hν) z hR hs |
| subst ν |
| simp only [coordinateDensity_zero, zero_pow (by omega : q ≠ 0), lintegral_zero] |
| rw [ENNReal.zero_rpow_of_pos (by positivity : 0 < 1/(q:ℝ))] |
| exact bot_le |
| · have hn : q-1+1 = q := by omega |
| have hnR : ((q-1:ℕ):ℝ)+1 = q := by exact_mod_cast hn |
| have hmm : ((q-1:ℕ):ℝ)+k ≤ β := by linarith |
| have h := frame_moment_scaled hν hK0 z hR hs k (q-1) hmm |
| (by simpa using (show 1 ≤ d by omega)) (by simpa using hβd) |
| (by simpa using (show k ≤ d by omega)) u hu hac |
| have ht := frame_cost_root (d := d) (m := k) (n := q-1) (by omega) (by omega) |
| β K R hK hR (by simpa only [finrank_euclideanSpace_fin] using h) |
| simpa only [labelMoment, hn, hnR] using ht |
|
|
| end ProjectionMoments |
|
|
| end |
|
|
| end OAI |
|
|
