import OAI.Geometry.LatticeCovering.Marginals namespace OAI section namespace SingleLatticeCovering.GaussianProjection open MeasureTheory ProbabilityTheory Set Filter Isotropization GaussianDensity Prekopa open scoped ENNReal Topology RealInnerProductSpace BigOperators lemma integrable_normalizedProjection {n k : ℕ} (μ : Measure (Isotropization.E n)) [IsFiniteMeasure μ] (s : ℝ) {r : ℝ} (hr : r ≠ 0) (g : MatrixSpace n k) : Integrable (normalizedProjection μ s r g) volume := by change Integrable (fun y => normalizedProjection μ s r g y) volume simp_rw [normalizedProjection_kernel] exact convolution_integrable μ ((s • colLinear g).continuous_of_finiteDimensional.measurable) (by have := sq_pos_of_ne_zero hr; positivity) lemma integral_normalizedProjection {n k : ℕ} (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ] (s : ℝ) {r : ℝ} (hr : r ≠ 0) (g : MatrixSpace n k) : (∫ y, normalizedProjection μ s r g y)=1 := by simp_rw [normalizedProjection_kernel] exact convolution_integral μ ((s • colLinear g).continuous_of_finiteDimensional.measurable) (by have := sq_pos_of_ne_zero hr; positivity) lemma continuous_normalizedProjection {n k : ℕ} (μ : Measure (Isotropization.E n)) [IsFiniteMeasure μ] (s r : ℝ) (g : MatrixSpace n k) : Continuous (normalizedProjection μ s r g) := (continuous_smoothedProjection_y μ s r g).const_mul _ lemma normalizedProjection_pos {n k : ℕ} (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ] (s : ℝ) {r : ℝ} (hr : r ≠ 0) (g : MatrixSpace n k) (y : Isotropization.E k) : 0 < normalizedProjection μ s r g y := by rw [normalizedProjection] apply mul_pos (inv_pos.mpr (normalizer_pos k (by have := sq_pos_of_ne_zero hr; positivity))) have hi : Integrable (fun x : Isotropization.E n => Real.exp (-r^2*‖y-s•colApply g x‖^2/2)) μ := by apply (integrable_const (1:ℝ)).mono' (by have hc : Continuous (fun x : Isotropization.E n => colApply g x) := continuous_colApply.comp (continuous_const.prodMk continuous_id) fun_prop) exact Filter.Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using gaussianKernel_bounded r (y-s•colApply g x)) exact integral_pos_iff_support_of_nonneg (fun x => (Real.exp_pos _).le) hi |>.mpr (by simp [Function.support,ne_of_gt (Real.exp_pos _)]) end SingleLatticeCovering.GaussianProjection end section namespace SingleLatticeCovering.GaussianProjection open MeasureTheory ProbabilityTheory Set Filter Isotropization GaussianDensity Prekopa Radial open scoped ENNReal Topology RealInnerProductSpace BigOperators noncomputable def scaleVariance {n : ℕ} (s r : ℝ) (x : Isotropization.E n) : ℝ := (s*‖x‖)^2+1/r^2 lemma scaleVariance_pos {n : ℕ} (s : ℝ) {r : ℝ} (hr : r ≠ 0) (x : Isotropization.E n) : 0 < scaleVariance s r x := by unfold scaleVariance have := sq_pos_of_ne_zero hr positivity lemma scaleVariance_integrable {n : ℕ} {μ : Measure (Isotropization.E n)} [IsFiniteMeasure μ] (hμ : MemLp id 2 μ) (s r : ℝ) : Integrable (scaleVariance s r) μ := by change Integrable (fun x => (s*‖x‖)^2+1/r^2) μ simp only [mul_pow] exact (hμ.norm.integrable_sq.const_mul _).add (integrable_const _) lemma scaleVariance_mean {n : ℕ} {μ : Measure (Isotropization.E n)} [IsProbabilityMeasure μ] (hμ : MemLp id 2 μ) (hiso : IsIsotropic μ) (s r : ℝ) : (∫ x, scaleVariance s r x ∂μ)=s^2*(n:ℝ)+1/r^2 := by unfold scaleVariance simp only [mul_pow] have hi : Integrable (fun x : Isotropization.E n => s^2*‖x‖^2) μ := hμ.norm.integrable_sq.const_mul _ rw [integral_add hi (integrable_const _),integral_const_mul, isotropic_norm_sq hμ hiso] simp noncomputable def normalizedModel {n k : ℕ} (μ : Measure (Isotropization.E n)) (s r : ℝ) (y : Isotropization.E k) : ℝ := (normalizer k (r^2/2))⁻¹*radialSmoothedDensity μ s r y lemma normalizedModel_mixture {n k : ℕ} (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ] (s : ℝ) {r : ℝ} (hr : r ≠ 0) (y : Isotropization.E k) : normalizedModel μ s r y=∫ x, density (1/(2*scaleVariance s r x)) y ∂μ := normalized_radialSmoothedDensity_mixture μ s hr y noncomputable def projectionError (n k : ℕ) (s r : ℝ) : ℝ := (normalizer k (r^2/2))⁻¹*|r*s| *Real.sqrt (Real.sqrt (n:ℝ))*Real.sqrt (k:ℝ) lemma projectionError_nonneg (n k : ℕ) (s r : ℝ) : 0 ≤ projectionError n k s r := by dsimp [projectionError,normalizer] positivity lemma exists_normalized_projection {n k : ℕ} {μ : Measure (Isotropization.E n)} [IsProbabilityMeasure μ] (hμ : MemLp id 2 μ) (hiso : IsIsotropic μ) (s : ℝ) {r : ℝ} (hr : r ≠ 0) : ∃ g : MatrixSpace n k, ∀ y : Isotropization.E k, |normalizedProjection μ s r g y-normalizedModel μ s r y| ≤ projectionError n k s r := by obtain ⟨g,hg⟩ := exists_uniform_smoothed_projection (k := k) hμ hiso s r refine ⟨g,fun y => ?_⟩ have hN : 0 < normalizer k (r^2/2) := normalizer_pos k (by have := sq_pos_of_ne_zero hr; positivity) have H := hg y rw [←radialSmoothedDensity_cast,←Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs] at H unfold normalizedProjection normalizedModel rw [←mul_sub,abs_mul,abs_of_nonneg (inv_nonneg.mpr hN.le)] have HH := mul_le_mul_of_nonneg_left H (inv_nonneg.mpr hN.le) exact HH.trans_eq (by dsimp [projectionError]; ring) theorem actual_model_shell {n k : ℕ} (hk : 2 ≤ k) {K : Set (Isotropization.E n)} (hK : IsCompact K) (hc : Convex ℝ K) (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ] (hdef : μ=(volume K)⁻¹ • volume.restrict K) (hμ : MemLp id 2 μ) (hiso : IsIsotropic μ) (s : ℝ) {r T : ℝ} (hr : r ≠ 0) (hT : 0 < T) : ∃ r₀ : ℝ, 0 < r₀ ∧ ∀ ε : ℝ, 0 < ε → ε < 1 → (∫ y : Isotropization.E k in {y | ‖y‖ ≤ (1-ε)*r₀}, normalizedModel μ s r y)+ (∫ y : Isotropization.E k in {y | (1+ε)*r₀ < ‖y‖}, normalizedModel μ s r y) ≤ (4*((k-1:ℕ)+1:ℝ)/((k-1:ℕ)*ε)*Real.exp (-((k-1:ℕ):ℝ)*ε^2/9))+ 2*(projectionError n k s r*volume.real (Metric.closedBall (0:Isotropization.E k) T)+ (k:ℝ)*(s^2*(n:ℝ)+1/r^2)/T^2) := by obtain ⟨g,hg⟩ := exists_normalized_projection (k := k) hμ hiso s hr have hmod : (normalizedModel μ s r : Isotropization.E k → ℝ)= fun y => ∫ x, density (1/(2*scaleVariance s r x)) y ∂μ := by funext y; exact normalizedModel_mixture μ s hr y have hm : Measurable (scaleVariance (n := n) s r) := by unfold scaleVariance; fun_prop have hMi : Integrable (normalizedModel μ s r : Isotropization.E k → ℝ) volume := by rw [hmod]; exact mixture_integrable μ hm (scaleVariance_pos s hr) have hMn (y : Isotropization.E k) : 0 ≤ normalizedModel μ s r y := by rw [hmod] exact integral_nonneg (fun x => (density_pos (by have := scaleVariance_pos s hr x; positivity) y).le) have hMr (x y : Isotropization.E k) (hxy : ‖x‖=‖y‖) : normalizedModel μ s r x=normalizedModel μ s r y := by unfold normalizedModel rw [radialSmoothedDensity_norm_eq μ s r hxy] obtain ⟨r₀,hr₀,H⟩ := radial_model_shell hk (logConcave_normalizedProjection hK hc μ hdef s r g) (normalizedProjection_pos μ s hr g) (continuous_normalizedProjection μ s r g) (integrable_normalizedProjection μ s hr g) (integral_normalizedProjection μ s hr g) hMi hMn hMr (projectionError_nonneg n k s r) hg T refine ⟨r₀,hr₀,fun ε hε hε1 => ?_⟩ have ht : (∫ y : Isotropization.E k in (Metric.closedBall (0:Isotropization.E k) T)ᶜ, normalizedModel μ s r y) ≤ (k:ℝ)*(s^2*(n:ℝ)+1/r^2)/T^2 := by simp_rw [normalizedModel_mixture μ s hr] have HH := mixture_tail_markov (k := k) μ hm (scaleVariance_pos s hr) (scaleVariance_integrable hμ s r) hT rwa [scaleVariance_mean hμ hiso] at HH linarith [H ε hε hε1] end SingleLatticeCovering.GaussianProjection end section namespace SingleLatticeCovering.GaussianDensity open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal Topology RealInnerProductSpace lemma probability_interval_lower {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] {v : Ω → ℝ} (hm : Measurable v) (a b : ℝ) : 1-μ.real {x | v x ≤ a}-μ.real {x | b ≤ v x} ≤ μ.real {x | a ≤ v x ∧ v x ≤ b} := by let B := {x | a ≤ v x ∧ v x ≤ b} have hB : MeasurableSet B := (measurableSet_le measurable_const hm).inter (measurableSet_le hm measurable_const) have hs : Bᶜ ⊆ {x | v x ≤ a} ∪ {x | b ≤ v x} := by intro x hx by_cases hxa : a ≤ v x · exact Or.inr (le_of_lt (lt_of_not_ge (fun hxb => hx ⟨hxa,hxb⟩))) · exact Or.inl (le_of_lt (lt_of_not_ge hxa)) have H := (measureReal_mono (μ := μ) hs).trans (measureReal_union_le _ _) rw [probReal_compl_eq_one_sub hB] at H change 1-μ.real {x | v x ≤ a}-μ.real {x | b ≤ v x} ≤ μ.real B linarith theorem mixture_shell_scales {Ω : Type*} [MeasurableSpace Ω] {k : ℕ} (μ : Measure Ω) [IsProbabilityMeasure μ] (hk : 0 < k) {v : Ω → ℝ} (hm : Measurable v) (hv : ∀ x, 0 < v x) {r₀ ε S : ℝ} (hr₀ : 0 < r₀) (hε : 0 < ε) (hε1 : ε < 1) (hshell : (∫ y : E k in {y | ‖y‖ ≤ (1-ε)*r₀}, ∫ x, density (1/(2*v x)) y ∂μ)+ (∫ y : E k in {y | (1+ε)*r₀ < ‖y‖}, ∫ x, density (1/(2*v x)) y ∂μ) ≤ S) : 1-S/(1-Real.exp (-(k:ℝ)*ε^2/16)) ≤ μ.real {x | ((1-ε)*r₀)^2/((1+ε)*(k:ℝ)) ≤ v x ∧ v x ≤ ((1+ε)*r₀)^2/((1-ε)*(k:ℝ))} := by have hkR : (0:ℝ) < k := by exact_mod_cast hk have hc : 0 < 1-Real.exp (-(k:ℝ)*ε^2/16) := by have hh : -(k:ℝ)*ε^2/16 < 0 := by nlinarith [sq_pos_of_pos hε] linarith [Real.exp_lt_one_iff.mpr hh] let a := ((1-ε)*r₀)^2/((1+ε)*(k:ℝ)) let b := ((1+ε)*r₀)^2/((1-ε)*(k:ℝ)) have heA : {x | v x*(1+ε)*(k:ℝ) ≤ ((1-ε)*r₀)^2}={x | v x ≤ a} := by ext x simp only [mem_ofPred_eq,a,le_div_iff₀ (mul_pos (show 0 < 1+ε by linarith) hkR)] ring_nf have heB : {x | ((1+ε)*r₀)^2 ≤ v x*(1-ε)*(k:ℝ)}={x | b ≤ v x} := by ext x simp only [mem_ofPred_eq,b,div_le_iff₀ (mul_pos (show 0 < 1-ε by linarith) hkR)] ring_nf have HA := mixture_scale_small (k := k) μ hm hv (show 0 ≤ (1-ε)*r₀ by positivity) hε hε1 have HB := mixture_scale_large (k := k) μ hm hv (show 0 ≤ (1+ε)*r₀ by positivity) hε hε1 rw [heA] at HA rw [heB] at HB have Hsum : μ.real {x | v x ≤ a}+μ.real {x | b ≤ v x} ≤ S/(1-Real.exp (-(k:ℝ)*ε^2/16)) := by apply (le_div_iff₀ hc).mpr nlinarith exact le_trans (by linarith) (probability_interval_lower μ hm a b) end SingleLatticeCovering.GaussianDensity end section namespace SingleLatticeCovering.GaussianDensity open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal lemma good_scale_anchors {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] {v : Ω → ℝ} (hm : Measurable v) (hi : Integrable v μ) (hn : ∀ x, 0 ≤ v x) (hmean : (∫ x, v x ∂μ) ≤ 2) {p u w : ℝ} (hbig : p ≤ μ.real {x | 1/2 ≤ v x}) (hgood : 1-p < μ.real {x | u ≤ v x ∧ v x ≤ w}) (hhalf : 1/2 ≤ μ.real {x | u ≤ v x ∧ v x ≤ w}) (huw : w ≤ 2*u) : 1/4 ≤ u ∧ u ≤ 4 := by let B := {x | u ≤ v x ∧ v x ≤ w} let A := {x | (1:ℝ)/2 ≤ v x} have hB : MeasurableSet B := (measurableSet_le measurable_const hm).inter (measurableSet_le hm measurable_const) have hsum := measureReal_union_add_inter (μ := μ) (s := A) hB have hunion : μ.real (A ∪ B) ≤ 1 := by simpa only [probReal_univ] using measureReal_mono (μ := μ) (subset_univ (A ∪ B)) have hp : 0 < μ.real (A ∩ B) := by change p ≤ μ.real A at hbig; change 1-p < μ.real B at hgood; linarith have hne : (A ∩ B).Nonempty := by by_contra h have he : A ∩ B=∅ := Set.not_nonempty_iff_eq_empty.mp h rw [he,measureReal_empty] at hp exact (lt_irrefl (0:ℝ)) hp obtain ⟨x,hxA,hxB⟩ := hne have hul : 1/4 ≤ u := by change 1/2 ≤ v x at hxA change u ≤ v x ∧ v x ≤ w at hxB linarith have H := mul_meas_ge_le_integral_of_nonneg (μ := μ) (Filter.Eventually.of_forall hn) hi u have hmono : μ.real B ≤ μ.real {x | u ≤ v x} := measureReal_mono (fun x hx => hx.1) have hh : 1/2 ≤ μ.real {x | u ≤ v x} := hhalf.trans hmono constructor · exact hul · nlinarith lemma scale_interval_ratio {k : ℕ} (hk : 0 < k) {r ε : ℝ} (hr : 0 < r) (hε : 0 ≤ ε) (hε1 : ε < 1) : let u := ((1-ε)*r)^2/((1+ε)*(k:ℝ)) let w := ((1+ε)*r)^2/((1-ε)*(k:ℝ)) 0 < u ∧ u ≤ w ∧ w/u=((1+ε)/(1-ε))^3 := by have hkR : (0:ℝ) < k := by exact_mod_cast hk dsimp only have hp : 0 < 1-ε := by linarith have hq : 0 < 1+ε := by linarith have hu : 0 < ((1-ε)*r)^2/((1+ε)*(k:ℝ)) := by positivity refine ⟨hu,?_,?_⟩ · apply (div_le_div_iff₀ (mul_pos hq hkR) (mul_pos hp hkR)).mpr have H : (1-ε)^3 ≤ (1+ε)^3 := pow_le_pow_left₀ hp.le (by linarith) 3 have HH := mul_le_mul_of_nonneg_right H (mul_nonneg (sq_nonneg r) hkR.le) nlinarith · field_simp lemma scale_ratio_le_two {ε : ℝ} (hε : 0 ≤ ε) (hε1 : ε ≤ 1/10) : ((1+ε)/(1-ε))^3 ≤ 2 := by have hp : 0 < 1-ε := by linarith have hh : (1+ε)/(1-ε) ≤ 11/9 := (div_le_iff₀ hp).mpr (by linarith) have H := pow_le_pow_left₀ (by positivity : 0 ≤ (1+ε)/(1-ε)) hh 3 norm_num at H ⊢ linarith end SingleLatticeCovering.GaussianDensity end section namespace SingleLatticeCovering.GaussianProjection open MeasureTheory ProbabilityTheory Set Filter Isotropization GaussianDensity Prekopa Radial open scoped ENNReal Topology RealInnerProductSpace BigOperators noncomputable def shellError (n k : ℕ) (s r T ε : ℝ) : ℝ := (4*((k-1:ℕ)+1:ℝ)/((k-1:ℕ)*ε)*Real.exp (-((k-1:ℕ):ℝ)*ε^2/9))+ 2*(projectionError n k s r*volume.real (Metric.closedBall (0:Isotropization.E k) T)+ (k:ℝ)*(s^2*(n:ℝ)+1/r^2)/T^2) noncomputable def shellLoss (n k : ℕ) (s r T ε : ℝ) : ℝ := shellError n k s r T ε/(1-Real.exp (-(k:ℝ)*ε^2/16)) lemma scaleVariance_measurable {n : ℕ} (s r : ℝ) : Measurable (scaleVariance (n := n) s r) := by unfold scaleVariance fun_prop theorem actual_good_scales {n k : ℕ} (hn : 0 < n) (hk : 2 ≤ k) {K : Set (Isotropization.E n)} (hK : IsCompact K) (hc : Convex ℝ K) (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ] (hdef : μ=(volume K)⁻¹ • volume.restrict K) (hμ : MemLp id 2 μ) (hiso : IsIsotropic μ) {s r T ε : ℝ} (hs : s^2*(n:ℝ)=1) (hr : 1 ≤ r) (hT : 0 < T) (hε : 0 < ε) (hε1 : ε ≤ 1/10) (hloss : shellLoss n k s r T ε < 1/(4*fourthMomentConstant)) (hloss8 : shellLoss n k s r T ε ≤ 1/8) : ∃ u w : ℝ, 1/4 ≤ u ∧ u ≤ 4 ∧ u ≤ w ∧ w/u=((1+ε)/(1-ε))^3 ∧ 1-shellLoss n k s r T ε ≤ μ.real {x | u ≤ scaleVariance s r x ∧ scaleVariance s r x ≤ w} := by have hr0 : r ≠ 0 := by linarith have hk0 : 0 < k := by omega obtain ⟨r₀,hr₀,hsh⟩ := actual_model_shell hk hK hc μ hdef hμ hiso s hr0 hT have hshell := hsh ε hε (by linarith) simp_rw [normalizedModel_mixture μ s hr0] at hshell have hv := scaleVariance_pos (n := n) s hr0 have hvm := scaleVariance_measurable (n := n) s r have hgood := mixture_shell_scales μ hk0 hvm hv hr₀ hε (by linarith) hshell let u := ((1-ε)*r₀)^2/((1+ε)*(k:ℝ)) let w := ((1+ε)*r₀)^2/((1-ε)*(k:ℝ)) change 1-shellLoss n k s r T ε ≤ μ.real {x | u ≤ scaleVariance s r x ∧ scaleVariance s r x ≤ w} at hgood obtain ⟨hu,huw,hratio⟩ := scale_interval_ratio hk0 hr₀ hε.le (show ε < 1 by linarith) change 0 < u at hu change u ≤ w at huw change w/u=((1+ε)/(1-ε))^3 at hratio have ratio2 : w ≤ 2*u := (div_le_iff₀ hu).mp (hratio ▸ scale_ratio_le_two hε.le hε1) have hbig : 1/(4*fourthMomentConstant) ≤ μ.real {x | 1/2 ≤ scaleVariance s r x} := by apply le_trans (uniform_body_radius_lower hn hK hc μ hdef hμ hiso) refine measureReal_mono (fun x hx => ?_) (measure_ne_top _ _) change (n:ℝ)/2 ≤ ‖x‖^2 at hx change 1/2 ≤ (s*‖x‖)^2+1/r^2 have H := mul_le_mul_of_nonneg_left hx (sq_nonneg s) have hq : 0 ≤ 1/r^2 := by positivity nlinarith have hmean : (∫ x, scaleVariance s r x ∂μ) ≤ 2 := by rw [scaleVariance_mean hμ hiso,hs] have hh : 1 ≤ r^2 := by nlinarith have hq : 1/r^2 ≤ 1 := (div_le_one (by positivity : 0 < r^2)).mpr hh linarith obtain ⟨hulo,huhi⟩ := good_scale_anchors μ hvm (scaleVariance_integrable hμ s r) (fun x => (hv x).le) hmean hbig (lt_of_lt_of_le (by linarith : 1-1/(4*fourthMomentConstant) < 1-shellLoss n k s r T ε) hgood) (le_trans (by linarith : (1:ℝ)/2 ≤ 1-shellLoss n k s r T ε) hgood) ratio2 exact ⟨u,w,hulo,huhi,huw,hratio,hgood⟩ end SingleLatticeCovering.GaussianProjection end section namespace SingleLatticeCovering.GaussianDensity open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal lemma normalizer_ratio {k : ℕ} {u w : ℝ} (hu : 0 < u) (hw : 0 < w) : normalizer k (1/(2*u))/normalizer k (1/(2*w))=(u/w)^((k:ℝ)/2) := by unfold normalizer have hp : (0:ℝ) < Real.pi := Real.pi_pos simp only [one_div,div_inv_eq_mul] rw [←Real.div_rpow (by positivity) (by positivity)] congr 1 field_simp lemma inverse_scale_ratio_lower {u w ε : ℝ} (hu : 0 < u) (hw : 0 < w) (hε : 0 ≤ ε) (hε1 : ε ≤ 1/10) (hratio : w/u=((1+ε)/(1-ε))^3) : 1-6*ε ≤ u/w := by have _ := hu have _ := hw have hp : 0 < 1-ε := by linarith have hq : 0 < 1+ε := by linarith have he : u/w=((1-ε)/(1+ε))^3 := by have H := congrArg (fun x : ℝ => x⁻¹) hratio rw [inv_div,←inv_pow,inv_div] at H exact H rw [he] have hh : 1-2*ε ≤ (1-ε)/(1+ε) := (le_div_iff₀ hq).mpr (by nlinarith) have H := pow_le_pow_left₀ (by linarith : 0 ≤ 1-2*ε) hh 3 have HB := one_add_mul_le_pow (a := -2*ε) (by linarith : -2 ≤ -2*ε) 3 norm_num only [Nat.cast_ofNat] at HB nlinarith lemma normalizer_ratio_lower {k : ℕ} {u w ε : ℝ} (hu : 0 < u) (huw : u ≤ w) (hε : 0 ≤ ε) (hε1 : ε ≤ 1/10) (hratio : w/u=((1+ε)/(1-ε))^3) (hsmall : (k:ℝ)*ε ≤ 1/48) : 7/8 ≤ normalizer k (1/(2*u))/normalizer k (1/(2*w)) := by have hw : 0 < w := hu.trans_le huw rw [normalizer_ratio hu hw] have hr0 : 0 < u/w := div_pos hu hw have hr1 : u/w ≤ 1 := (div_le_one hw).mpr huw have HB := one_add_mul_le_pow (a := -6*ε) (by linarith : -2 ≤ -6*ε) k have HP := pow_le_pow_left₀ (by linarith : 0 ≤ 1-6*ε) (inverse_scale_ratio_lower hu hw hε hε1 hratio) k have HR := Real.rpow_le_rpow_of_exponent_ge hr0 hr1 (show (k:ℝ)/2 ≤ k by have := Nat.cast_nonneg (α := ℝ) k; linarith) rw [Real.rpow_natCast] at HR have HBB : 1-6*(k:ℝ)*ε ≤ (1-6*ε)^k := by calc 1-6*(k:ℝ)*ε=1+(k:ℝ)*(-6*ε) := by ring _ ≤ (1+(-6*ε))^k := HB _ = (1-6*ε)^k := by congr 1; ring exact le_trans (by nlinarith) (HBB.trans (HP.trans HR)) end SingleLatticeCovering.GaussianDensity end section namespace SingleLatticeCovering.GaussianDensity open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal Topology RealInnerProductSpace lemma normalizer_variance_mono {k : ℕ} {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) : normalizer k (1/(2*u)) ≤ normalizer k (1/(2*v)) := by unfold normalizer have he (a : ℝ) : Real.pi/(1/(2*a))=Real.pi*(2*a) := by simp only [one_div,div_inv_eq_mul] simp only [he] apply Real.rpow_le_rpow (by positivity) _ (by positivity) exact mul_le_mul_of_nonneg_left (by linarith) Real.pi_pos.le lemma density_variance_le_peak {k : ℕ} {a v : ℝ} (ha : 0 < a) (hav : a ≤ v) (y : E k) : density (1/(2*v)) y ≤ (normalizer k (1/(2*a)))⁻¹ := by have hv : 0 < v := ha.trans_le hav apply le_trans _ (inv_anti₀ (normalizer_pos k (by positivity)) (normalizer_variance_mono ha hav)) unfold density apply mul_le_of_le_one_right (inv_nonneg.mpr (normalizer_pos k (by positivity)).le) apply Real.exp_le_one_iff.mpr exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (by positivity)) (sq_nonneg _) lemma integrable_mixture_at {Ω : Type*} [MeasurableSpace Ω] {k : ℕ} (μ : Measure Ω) [IsFiniteMeasure μ] {v : Ω → ℝ} (hm : Measurable v) {a : ℝ} (ha : 0 < a) (hav : ∀ x, a ≤ v x) (y : E k) : Integrable (fun x => density (1/(2*v x)) y) μ := by apply (integrable_const ((normalizer k (1/(2*a)))⁻¹)).mono' (by have H := (measurable_mixture_kernel (k := k) hm).comp (measurable_id.prodMk (measurable_const (a := y))) exact H.aestronglyMeasurable) exact Filter.Eventually.of_forall (fun x => by have hvx : 0 < 1/(2*v x) := by have := ha.trans_le (hav x); positivity simpa only [Real.norm_eq_abs,abs_of_nonneg (density_pos hvx y).le] using density_variance_le_peak ha (hav x) y) lemma density_scales_lower {k : ℕ} {u v w : ℝ} (hu : 0 < u) (huv : u ≤ v) (hvw : v ≤ w) (y : E k) : (normalizer k (1/(2*u))/normalizer k (1/(2*w)))*density (1/(2*u)) y ≤ density (1/(2*v)) y := by have hv : 0 < v := hu.trans_le huv have hw : 0 < w := hv.trans_le hvw have hNu : 0 < normalizer k (1/(2*u)) := normalizer_pos k (by positivity) have hNv : 0 < normalizer k (1/(2*v)) := normalizer_pos k (by positivity) have hNw : 0 < normalizer k (1/(2*w)) := normalizer_pos k (by positivity) have he : (normalizer k (1/(2*u))/normalizer k (1/(2*w)))*(normalizer k (1/(2*u)))⁻¹= (normalizer k (1/(2*w)))⁻¹ := by field_simp unfold density rw [←mul_assoc,he] apply mul_le_mul (inv_anti₀ hNv (normalizer_variance_mono hv hvw)) _ (Real.exp_pos _).le (inv_nonneg.mpr hNv.le) apply Real.exp_le_exp.mpr have hh : 1/(2*v) ≤ 1/(2*u) := one_div_le_one_div_of_le (by positivity) (by linarith) nlinarith [mul_le_mul_of_nonneg_right hh (sq_nonneg ‖y‖)] theorem mixture_lower {Ω : Type*} [MeasurableSpace Ω] {k : ℕ} (μ : Measure Ω) [IsFiniteMeasure μ] {v : Ω → ℝ} (hm : Measurable v) {a u w : ℝ} (ha : 0 < a) (hav : ∀ x, a ≤ v x) (hu : 0 < u) (huw : u ≤ w) (y : E k) : μ.real {x | u ≤ v x ∧ v x ≤ w}* (normalizer k (1/(2*u))/normalizer k (1/(2*w)))*density (1/(2*u)) y ≤ ∫ x, density (1/(2*v x)) y ∂μ := by have _ := huw classical let B := {x | u ≤ v x ∧ v x ≤ w} have hB : MeasurableSet B := (measurableSet_le measurable_const hm).inter (measurableSet_le hm measurable_const) let c := (normalizer k (1/(2*u))/normalizer k (1/(2*w)))*density (1/(2*u)) y have hh (x : Ω) : B.indicator (fun _ => c) x ≤ density (1/(2*v x)) y := by by_cases hx : x ∈ B · rw [indicator_of_mem hx] exact density_scales_lower hu hx.1 hx.2 y · rw [indicator_of_notMem hx] exact (density_pos (by have := ha.trans_le (hav x); positivity) y).le have H := integral_mono ((integrable_const c).indicator hB) (integrable_mixture_at μ hm ha hav y) hh rw [integral_indicator hB,integral_const,smul_eq_mul] at H simp only [measureReal_def,Measure.restrict_apply_univ] at H change μ.real B*c ≤ _ at H simpa only [B,c,mul_assoc] using H end SingleLatticeCovering.GaussianDensity end section namespace SingleLatticeCovering.GaussianProjection open MeasureTheory ProbabilityTheory Set Filter Isotropization GaussianDensity Prekopa Radial open scoped ENNReal Topology RealInnerProductSpace BigOperators theorem actual_smoothed_lower {n k D : ℕ} (hn : 0 < n) (hk : 2 ≤ k) {K : Set (Isotropization.E n)} (hK : IsCompact K) (hc : Convex ℝ K) (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ] (hdef : μ=(volume K)⁻¹ • volume.restrict K) (hμ : MemLp id 2 μ) (hiso : IsIsotropic μ) {s r T ε : ℝ} (hs : s^2*(n:ℝ)=1) (hr : 1 ≤ r) (hT : 0 < T) (hε : 0 < ε) (hε1 : ε ≤ 1/10) (hDε : (D:ℝ)*ε ≤ 1/48) (hloss : shellLoss n k s r T ε < 1/(4*fourthMomentConstant)) (hloss8 : shellLoss n k s r T ε ≤ 1/8) : ∃ u : ℝ, 1/4 ≤ u ∧ u ≤ 4 ∧ ∃ g : MatrixSpace n D, ∀ y : Isotropization.E D, (3/4)*density (1/(2*u)) y-projectionError n D s r ≤ normalizedProjection μ s r g y := by have hr0 : r ≠ 0 := by linarith obtain ⟨u,w,hulo,huhi,huw,hratio,hgood⟩ := actual_good_scales hn hk hK hc μ hdef hμ hiso hs hr hT hε hε1 hloss hloss8 have hu : 0 < u := by linarith have hratio' := normalizer_ratio_lower hu huw hε.le hε1 hratio hDε have hmass : 7/8 ≤ μ.real {x | u ≤ scaleVariance s r x ∧ scaleVariance s r x ≤ w} := by linarith have hm : Measurable (scaleVariance (n := n) s r) := scaleVariance_measurable s r have ha : 0 < 1/r^2 := by positivity have hav (x : Isotropization.E n) : 1/r^2 ≤ scaleVariance s r x := by unfold scaleVariance; nlinarith [sq_nonneg (s*‖x‖)] have hmodel (y : Isotropization.E D) : (3/4)*density (1/(2*u)) y ≤ normalizedModel μ s r y := by rw [normalizedModel_mixture μ s hr0] apply le_trans _ (mixture_lower μ hm ha hav hu huw y) have hmul := mul_le_mul hmass hratio' (by norm_num : (0:ℝ) ≤ 7/8) (measureReal_nonneg) have hp : (3/4:ℝ) ≤ μ.real {x | u ≤ scaleVariance s r x ∧ scaleVariance s r x ≤ w}* (normalizer D (1/(2*u))/normalizer D (1/(2*w))) := by nlinarith exact mul_le_mul_of_nonneg_right hp (density_pos (by positivity) y).le obtain ⟨g,hg⟩ := exists_normalized_projection (k := D) hμ hiso s hr0 refine ⟨u,hulo,huhi,g,fun y => ?_⟩ have H := (abs_le.mp (hg y)).1 linarith [hmodel y] end SingleLatticeCovering.GaussianProjection end section namespace SingleLatticeCovering.GaussianDensity open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal RealInnerProductSpace noncomputable def densityFloor (k : ℕ) (R : ℝ) : ℝ := (normalizer k (1/8))⁻¹*Real.exp (-8*(R+1)^2) lemma densityFloor_pos (k : ℕ) (R : ℝ) : 0 < densityFloor k R := by unfold densityFloor exact mul_pos (inv_pos.mpr (normalizer_pos k (by norm_num))) (Real.exp_pos _) lemma variance_density_floor {k : ℕ} {u R : ℝ} (hu : 1/4 ≤ u) (hu4 : u ≤ 4) (hR : 0 ≤ R) {y : E k} (hy : ‖y‖ ≤ 2*(R+1)) : densityFloor k R ≤ density (1/(2*u)) y := by have _ := hR have hu0 : 0 < u := by linarith have hb : 0 < 1/(2*u) := by positivity have hbu : 1/(2*u) ≤ 2 := (div_le_iff₀ (by positivity)).mpr (by linarith) have hN : normalizer k (1/(2*u)) ≤ normalizer k (1/8) := by convert normalizer_variance_mono hu0 hu4 using 1 ; norm_num unfold densityFloor density apply mul_le_mul (inv_anti₀ (normalizer_pos k hb) hN) _ (Real.exp_pos _).le (inv_nonneg.mpr (normalizer_pos k hb).le) apply Real.exp_le_exp.mpr have hy2 := pow_le_pow_left₀ (norm_nonneg y) hy 2 nlinarith [mul_le_mul_of_nonneg_right hbu (sq_nonneg ‖y‖)] lemma variance_density_nearby {k : ℕ} {u R δ : ℝ} (hu : 1/4 ≤ u) (hR : 0 ≤ R) (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/(100*(R+1))) {y w : E k} (hy : ‖y‖ ≤ 2*R) (hw : ‖w-y‖ ≤ δ) : (3/4)*density (1/(2*u)) y ≤ density (1/(2*u)) w := by have hu0 : 0 < u := by linarith have hb : 0 < 1/(2*u) := by positivity have hbu : 1/(2*u) ≤ 2 := (div_le_iff₀ (by positivity)).mpr (by linarith) have hδ1 : δ ≤ 1/100 := by apply hδsmall.trans apply one_div_le_one_div_of_le (by norm_num) nlinarith have hδR : δ*(R+1) ≤ 1/100 := by have H := (le_div_iff₀ (by positivity : 0 < 100*(R+1))).mp hδsmall nlinarith have hwN : ‖w‖ ≤ ‖y‖+δ := by have H := norm_add_le (w-y) y rw [sub_add_cancel] at H linarith have hw2 := pow_le_pow_left₀ (norm_nonneg w) hwN 2 have hdq : δ^2 ≤ δ/100 := by nlinarith have hxy : ‖w‖^2-‖y‖^2 ≤ 1/10 := by have H := mul_le_mul_of_nonneg_right hy hδ nlinarith have he : -(1/(2*u))*‖y‖^2-1/4 ≤ -(1/(2*u))*‖w‖^2 := by have H := mul_le_mul_of_nonneg_left hxy hb.le nlinarith have hexp : (3/4)*Real.exp (-(1/(2*u))*‖y‖^2) ≤ Real.exp (-(1/(2*u))*‖w‖^2) := by have hl : (3/4:ℝ) ≤ Real.exp (-(1/4:ℝ)) := by linarith [Real.add_one_le_exp (-(1/4:ℝ))] calc _ ≤ Real.exp (-(1/4:ℝ))*Real.exp (-(1/(2*u))*‖y‖^2) := mul_le_mul_of_nonneg_right hl (Real.exp_pos _).le _ = Real.exp (-(1/(2*u))*‖y‖^2-1/4) := by rw [←Real.exp_add]; congr 1; ring _ ≤ _ := Real.exp_le_exp.mpr he unfold density nlinarith [mul_le_mul_of_nonneg_left hexp (inv_nonneg.mpr (normalizer_pos k hb).le)] end SingleLatticeCovering.GaussianDensity end section namespace SingleLatticeCovering.GaussianProjection open MeasureTheory ProbabilityTheory Set Filter Isotropization GaussianDensity open scoped ENNReal Topology RealInnerProductSpace BigOperators lemma surjective_of_smoothed_lower {n k : ℕ} (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ] (A : Isotropization.E n →ₗ[ℝ] Isotropization.E k) {b : ℝ} (hb : 0 < b) (hlower : ∀ y : Isotropization.E k, ‖y‖=1 → (normalizer k b)⁻¹*Real.exp (-b) < ∫ x, density b (y-A x) ∂μ) : Function.Surjective A := by by_contra h have hr : A.range ≠ ⊤ := fun htop => h (LinearMap.range_eq_top.mp htop) have ho : A.rangeᗮ ≠ ⊥ := fun hbot => hr (Submodule.orthogonal_eq_bot_iff.mp hbot) obtain ⟨z,hz,hz0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot ho let u := ‖z‖⁻¹ • z have hu : ‖u‖=1 := norm_smul_inv_norm hz0 have huo : u ∈ A.rangeᗮ := A.rangeᗮ.smul_mem _ hz have hfar (x : Isotropization.E n) : 1 ≤ ‖u-A x‖^2 := by have hh : ⟪u,A x⟫=0 := by rw [real_inner_comm] exact (Submodule.mem_orthogonal _ _).mp huo (A x) (LinearMap.mem_range_self _ _) rw [norm_sub_sq_real,hu,hh] nlinarith [sq_nonneg ‖A x‖] have hi : Integrable (fun x => density b (u-A x)) μ := by apply (integrable_const ((normalizer k b)⁻¹)).mono' (by exact ((continuous_density k b).comp (continuous_const.sub A.continuous_of_finiteDimensional)).aestronglyMeasurable) exact Filter.Eventually.of_forall (fun x => by rw [Real.norm_eq_abs,abs_of_nonneg (density_pos hb _).le] unfold density apply mul_le_of_le_one_right (inv_nonneg.mpr (normalizer_pos k hb).le) exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hb.le) (sq_nonneg _))) have hp (x : Isotropization.E n) : density b (u-A x) ≤ (normalizer k b)⁻¹*Real.exp (-b) := by unfold density apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr (normalizer_pos k hb).le) apply Real.exp_le_exp.mpr nlinarith [hfar x] have H := integral_mono hi (integrable_const ((normalizer k b)⁻¹*Real.exp (-b))) hp simp only [integral_const,probReal_univ,smul_eq_mul,one_mul] at H exact (not_lt_of_ge H) (hlower u hu) end SingleLatticeCovering.GaussianProjection end section noncomputable section end end end OAI