| import OAI.Geometry.IsometricImmersion.Energy.DirectedBounds |
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| namespace OAI |
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| noncomputable section |
| open scoped ContDiff |
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| namespace SmoothLocal.Weighted |
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| open SmoothLocal.Geometry |
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| def directedPrincipal (A : Coord → ℝ) (ell : ℕ) (epsilon : ℝ) (p : Coord) : ℝ := |
| -((ell : ℝ) + 1 / 2) * coordPartial 1 A p + epsilon * p 0 * coordPartial 0 A p / 2 |
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| def directedSResidual (B r : Coord → ℝ) (_ : ℕ) (epsilon : ℝ) (p : Coord) : ℝ := |
| (coordPartial 1 (coordinatePrimitive B) p + epsilon + epsilon * p 0 * B p) / 2 - r p |
|
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| theorem primitive_directedS_expansion |
| {B : Coord → ℝ} {R a b : ℝ} {p : Coord} |
| (hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) (hR : 0 < R) |
| (hp : p ∈ coordinateRectangle R a b) (A C r : Coord → ℝ) |
| (hA : DifferentiableAt ℝ A p) (edge lambda epsilon : ℝ) (ell : ℕ) |
| (hedge : p 1 < edge) |
| (hC : C p = ((ell : ℝ) + 1) * coordPartial 1 A p + A p * r p) : |
| multiplierS A C (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p = |
| directedWeight edge lambda (coordinatePrimitive B) p * |
| (directedPrincipal A ell epsilon p - weightH0 edge lambda p * A p / 2 + |
| A p * directedSResidual B r ell epsilon p) := by |
| have hI : DifferentiableAt ℝ (coordinatePrimitive B) p := |
| ((coordinatePrimitive_contDiffOn hB).contDiffAt |
| ((coordinateRectangle_isOpen R a b).mem_nhds hp)).differentiableAt (by simp) |
| rw [directedS_transverse_expansion A B C edge lambda epsilon ell (r p) hA hI |
| (sub_pos.mpr hedge).ne' hC, coordinatePrimitive_partial_t hB hR hp] |
| rfl |
|
|
| theorem directedSResidual_bound {B r : Coord → ℝ} {ell : ℕ} {epsilon R MI MB Mr : ℝ} |
| {p : Coord} (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) (hR : 0 ≤ R) |
| (ht : |p 0| ≤ R) (hI : |coordPartial 1 (coordinatePrimitive B) p| ≤ MI) |
| (hB : |B p| ≤ MB) (hr : |r p| ≤ Mr) : |
| |directedSResidual B r ell epsilon p| ≤ (MI + 1 + R * MB) / 2 + Mr := by |
| have hMB : 0 ≤ MB := (abs_nonneg _).trans hB |
| have hprod : |epsilon * p 0 * B p| ≤ R * MB := by |
| rw [abs_mul, abs_mul, abs_of_nonneg heps] |
| have het : epsilon * |p 0| ≤ R := |
| (mul_le_mul_of_nonneg_left ht heps).trans (by nlinarith) |
| exact (mul_le_mul het hB (abs_nonneg _) hR) |
| unfold directedSResidual |
| calc |
| _ ≤ |(coordPartial 1 (coordinatePrimitive B) p + epsilon + epsilon * p 0 * B p) / 2| + |
| |r p| := abs_sub _ _ |
| _ = |coordPartial 1 (coordinatePrimitive B) p + epsilon + epsilon * p 0 * B p| / 2 + |
| |r p| := by rw [abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| _ ≤ ((|coordPartial 1 (coordinatePrimitive B) p| + |epsilon|) + |
| |epsilon * p 0 * B p|) / 2 + |r p| := by |
| have hsum : |coordPartial 1 (coordinatePrimitive B) p + epsilon + epsilon * p 0 * B p| ≤ |
| (|coordPartial 1 (coordinatePrimitive B) p| + |epsilon|) + |epsilon * p 0 * B p| := |
| (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl) |
| exact add_le_add (div_le_div_of_nonneg_right hsum (by norm_num : (0 : ℝ) ≤ 2)) le_rfl |
| _ ≤ (MI + 1 + R * MB) / 2 + Mr := by rw [abs_of_nonneg heps]; linarith |
|
|
| theorem directedPrincipal_product {G K : Coord → ℝ} {p : Coord} |
| (hG : DifferentiableAt ℝ G p) (hK : DifferentiableAt ℝ K p) |
| (hGne : G p ≠ 0) (ell : ℕ) (epsilon : ℝ) : |
| directedPrincipal (fun q => G q * K q) ell epsilon p = |
| G p * (-((ell : ℝ) + 1 / 2) * coordPartial 1 K p + |
| epsilon * p 0 * coordPartial 0 K p / 2) + G p * K p * |
| ((-((ell : ℝ) + 1 / 2) * coordPartial 1 G p + |
| epsilon * p 0 * coordPartial 0 G p / 2) / G p) := by |
| unfold directedPrincipal |
| rw [HessianCalculus.coordPartial_mul_at hG hK 1, |
| HessianCalculus.coordPartial_mul_at hG hK 0] |
| field_simp [hGne] |
| ring |
|
|
| theorem directed_geometric_expression |
| {G K : Coord → ℝ} {p : Coord} {rho Ky kappa q epsilon : ℝ} |
| (ell : ℕ) (hrho : rho ≠ 0) |
| (hKs : coordPartial 1 K p = Ky / rho) |
| (hKt : coordPartial 0 K p = 2 * kappa * p 0 - q * Ky) : |
| G p * (-((ell : ℝ) + 1 / 2) * coordPartial 1 K p + |
| epsilon * p 0 * coordPartial 0 K p / 2) = |
| G p * (-(((ell : ℝ) + 1 / 2) / rho + epsilon * p 0 * q / 2) * Ky + |
| epsilon * kappa * (p 0) ^ 2) := by |
| rw [hKs, hKt] |
| field_simp [hrho] |
| ring |
|
|
| theorem normalized_loss_budget {H A MA L mu : ℝ} |
| (hH : 0 < H) (hL : 0 ≤ L) (hA : |A| ≤ MA) |
| (hsmall : L * MA ≤ 1 / 8) (hloss : L / H ≤ mu / 4) : |
| L * (H⁻¹ + H * A ^ 2) ≤ mu / 4 + H * |A| / 8 := by |
| have hLA : L * |A| ≤ 1 / 8 := |
| (mul_le_mul_of_nonneg_left hA hL).trans hsmall |
| have hLA2 : L * A ^ 2 ≤ |A| / 8 := by |
| have hm := mul_le_mul_of_nonneg_right hLA (abs_nonneg A) |
| nlinarith [sq_abs A] |
| have hmul := mul_le_mul_of_nonneg_left hLA2 hH.le |
| calc |
| _ = L / H + H * (L * A ^ 2) := by ring |
| _ ≤ mu / 4 + H * (|A| / 8) := add_le_add hloss hmul |
| _ = _ := by ring |
|
|
| theorem residual_abs_error_bound {A e E H : ℝ} |
| (he : |e| ≤ E) (hEH : E ≤ H / 8) : |
| -(H * |A| / 8) ≤ A * e := by |
| have habs : |A * e| ≤ H * |A| / 8 := by |
| rw [abs_mul] |
| have h := mul_le_mul_of_nonneg_left (he.trans hEH) (abs_nonneg A) |
| nlinarith only [h] |
| exact (abs_le.mp habs).1 |
|
|
| theorem normalized_negative_near_coercivity {H A MA L mu Q e E : ℝ} |
| (hH : 0 < H) (hL : 0 ≤ L) (hmu : 0 < mu) (hA : A ≤ 0) (hAM : |A| ≤ MA) |
| (hsmall : L * MA ≤ 1 / 8) (hloss : L / H ≤ mu / 4) |
| (he : |e| ≤ E) (hEH : E ≤ H / 8) (hQ : mu ≤ Q) : |
| mu / 2 + H * |A| / 4 ≤ Q - H * A / 2 + A * e - L * (H⁻¹ + H * A ^ 2) := by |
| have hcross := normalized_loss_budget hH hL hAM hsmall hloss |
| have herr := residual_abs_error_bound (A := A) he hEH |
| rw [abs_of_nonpos hA] at * |
| nlinarith |
|
|
| theorem normalized_positive_strip_coercivity {H A MA L mu Q e E : ℝ} |
| (hH : 0 < H) (hL : 0 ≤ L) (_ : 0 < mu) (hA : 0 ≤ A) (hAM : |A| ≤ MA) |
| (hsmall : L * MA ≤ 1 / 8) (hloss : L / H ≤ mu / 4) |
| (he : |e| ≤ E) (hEH : E ≤ H / 8) (hQ : mu ≤ Q) (hstrip : H * A ≤ mu / 3) : |
| mu / 2 ≤ Q - H * A / 2 + A * e - L * (H⁻¹ + H * A ^ 2) := by |
| have hcross := normalized_loss_budget hH hL hAM hsmall hloss |
| have herr := residual_abs_error_bound (A := A) he hEH |
| rw [abs_of_nonneg hA] at * |
| nlinarith |
|
|
| theorem normalized_negative_far_coercivity {H A MA L mu Q e E alpha Pmax : ℝ} |
| (hH : 0 < H) (hL : 0 ≤ L) (hmu : 0 < mu) (hA : A ≤ 0) (hAM : |A| ≤ MA) |
| (hsmall : L * MA ≤ 1 / 8) (hloss : L / H ≤ mu / 4) |
| (he : |e| ≤ E) (hEH : E ≤ H / 8) (hQ : -Pmax ≤ Q) |
| (halpha : alpha ≤ |A|) (hbudget : Pmax + mu ≤ H * alpha / 4) : |
| mu / 2 ≤ Q - H * A / 2 + A * e - L * (H⁻¹ + H * A ^ 2) := by |
| have hcross := normalized_loss_budget hH hL hAM hsmall hloss |
| have herr := residual_abs_error_bound (A := A) he hEH |
| have hmargin : Pmax + mu ≤ H * |A| / 4 := by |
| have hm := mul_le_mul_of_nonneg_left halpha hH.le |
| nlinarith |
| rw [abs_of_nonpos hA] at * |
| nlinarith |
|
|
| theorem actual_quadratic_coercivity {T S J W H lambda mu Q e A L x y : ℝ} |
| (hW : 0 < W) (hH : 0 < H) (_ : 0 < lambda) (_ : 0 < mu) |
| (hlH : lambda ≤ H) (hT : W * H / 4 ≤ T) |
| (hS : S = W * (Q - H * A / 2 + A * e)) |
| (hcross : J ^ 2 / (2 * T * W) ≤ L * (H⁻¹ + H * A ^ 2)) |
| (hmargin : mu / 2 ≤ Q - H * A / 2 + A * e - L * (H⁻¹ + H * A ^ 2)) : |
| min (lambda / 8) (mu / 2) * W * (x ^ 2 + y ^ 2) ≤ T * x ^ 2 + S * y ^ 2 + J * x * y := by |
| have hTpos : 0 < T := (show 0 < W * H / 4 by positivity).trans_le hT |
| have hcross' : J ^ 2 / (2 * T) ≤ W * (L * (H⁻¹ + H * A ^ 2)) := by |
| have hm := mul_le_mul_of_nonneg_right hcross hW.le |
| have he : J ^ 2 / (2 * T * W) * W = J ^ 2 / (2 * T) := by |
| field_simp [hTpos.ne', hW.ne'] |
| rw [he] at hm |
| nlinarith only [hm] |
| have hSlo : mu / 2 * W ≤ S - J ^ 2 / (2 * T) := by |
| have hm := mul_le_mul_of_nonneg_left hmargin hW.le |
| rw [hS] |
| nlinarith only [hm, hcross'] |
| have hTlo : lambda / 8 * W ≤ T / 2 := by |
| have hm := mul_le_mul_of_nonneg_left hlH hW.le |
| nlinarith only [hm, hT] |
| have hxlo : min (lambda / 8) (mu / 2) * W ≤ T / 2 := |
| (mul_le_mul_of_nonneg_right (min_le_left _ _) hW.le).trans hTlo |
| have hylo : min (lambda / 8) (mu / 2) * W ≤ S - J ^ 2 / (2 * T) := |
| (mul_le_mul_of_nonneg_right (min_le_right _ _) hW.le).trans hSlo |
| have hx := mul_le_mul_of_nonneg_right hxlo (sq_nonneg x) |
| have hy := mul_le_mul_of_nonneg_right hylo (sq_nonneg y) |
| have hq := quadratic_cross_absorption (S := S) (J := J) (u := x) (v := y) hTpos |
| nlinarith only [hx, hy, hq] |
|
|
| theorem actual_normalized_cross_loss {T J W H A Cj epsilon : ℝ} |
| (hW : 0 < W) (hH : 0 < H) (hCj : 0 ≤ Cj) (heps : 0 ≤ epsilon) |
| (hT : W * H / 4 ≤ T) |
| (hJ : |J| ≤ W * Cj * epsilon * (1 + H * |A|)) : |
| J ^ 2 / (2 * T * W) ≤ 4 * Cj ^ 2 * epsilon ^ 2 * (H⁻¹ + H * A ^ 2) := by |
| have hh := cross_loss_bound hW hH hCj heps hT hJ |
| have hTpos : 0 < T := (show 0 < W * H / 4 by positivity).trans_le hT |
| have he : J ^ 2 / (2 * T * W) = (J ^ 2 / (T * W)) / 2 := by |
| field_simp [hTpos.ne', hW.ne'] |
| rw [he] |
| nlinarith only [hh] |
|
|
| theorem primitive_directed_near_coercivity |
| {B : Coord → ℝ} {R a b : ℝ} {p : Coord} |
| (hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) (hR : 0 < R) |
| (hp : p ∈ coordinateRectangle R a b) (A C r : Coord → ℝ) |
| (hA : DifferentiableAt ℝ A p) (edge lambda epsilon : ℝ) (ell : ℕ) |
| (hedge : p 1 < edge) (hlambda : 0 < lambda) |
| (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) |
| (hC : C p = ((ell : ℝ) + 1) * coordPartial 1 A p + A p * r p) |
| (MI MB Mr MA MAs mu : ℝ) (hmu : 0 < mu) |
| (hIs : |coordPartial 1 (coordinatePrimitive B) p| ≤ MI) |
| (hBp : |B p| ≤ MB) (hr : |r p| ≤ Mr) |
| (hAp : |A p| ≤ MA) (hAs : |coordPartial 1 A p| ≤ MAs) |
| (hTbudget : 2 * (MI + epsilon + epsilon * R * MB) ≤ lambda) |
| (hEbudget : (MI + 1 + R * MB) / 2 + Mr ≤ lambda / 8) |
| (hsmall : 4 * (R * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1)) ^ 2 * |
| epsilon ^ 2 * MA ≤ 1 / 8) |
| (hloss : (4 * (R * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1)) ^ 2 * |
| epsilon ^ 2) / lambda ≤ mu / 4) |
| (hprincipal : mu ≤ directedPrincipal A ell epsilon p) |
| (hstrip : 0 < A p → weightH0 edge lambda p * A p ≤ mu / 3) |
| (x y : ℝ) : |
| min (lambda / 8) (mu / 2) * directedWeight edge lambda (coordinatePrimitive B) p * |
| (x ^ 2 + y ^ 2) ≤ |
| multiplierT B (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p * x ^ 2 + |
| multiplierS A C (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p * y ^ 2 + |
| multiplierJ A B C (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p * x * y := by |
| let W := directedWeight edge lambda (coordinatePrimitive B) p |
| let H := weightH0 edge lambda p |
| let Cj := R * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1) |
| let L := 4 * Cj ^ 2 * epsilon ^ 2 |
| let E := (MI + 1 + R * MB) / 2 + Mr |
| have hMI : 0 ≤ MI := (abs_nonneg _).trans hIs |
| have hMB : 0 ≤ MB := (abs_nonneg _).trans hBp |
| have hMr : 0 ≤ Mr := (abs_nonneg _).trans hr |
| have hMA : 0 ≤ MA := (abs_nonneg _).trans hAp |
| have hMAs : 0 ≤ MAs := (abs_nonneg _).trans hAs |
| have hCj : 0 ≤ Cj := by dsimp [Cj]; positivity |
| have hL : 0 ≤ L := by dsimp [L]; positivity |
| have hW : 0 < W := directedWeight_pos _ _ _ _ hedge |
| have hlH : lambda ≤ H := by |
| dsimp [H, weightH0] |
| exact le_add_of_nonneg_right (div_nonneg (by norm_num) (sub_pos.mpr hedge).le) |
| have hH : 0 < H := hlambda.trans_le hlH |
| have hCA : |C p - coordPartial 1 A p| ≤ (ell : ℝ) * MAs + MA * Mr := by |
| have he : C p - coordPartial 1 A p = |
| (ell : ℝ) * coordPartial 1 A p + A p * r p := by rw [hC]; ring |
| rw [he] |
| calc |
| _ ≤ |(ell : ℝ) * coordPartial 1 A p| + |A p * r p| := abs_add_le _ _ |
| _ = (ell : ℝ) * |coordPartial 1 A p| + |A p| * |r p| := by |
| rw [abs_mul, abs_mul, abs_of_nonneg (Nat.cast_nonneg ell)] |
| _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_left hAs (Nat.cast_nonneg ell)) |
| (mul_le_mul hAp hr (abs_nonneg _) hMA) |
| have hT := primitive_directedT_lower_bound hB hR hp edge lambda epsilon MI MB |
| heps hedge hIs hBp hTbudget |
| have hJ0 := primitive_directedJ_bound hB hR hp A C hA edge lambda epsilon |
| ((ell : ℝ) * MAs + MA * Mr) MA MI heps hedge hH.le hCA hAp hIs |
| have hJ : |multiplierJ A B C (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p| ≤ |
| W * Cj * epsilon * (1 + H * |A p|) := by |
| refine hJ0.trans ?_ |
| have hD : 0 ≤ (ell : ℝ) * MAs + MA * Mr + MA * MI := by positivity |
| have hh : 0 ≤ H * |A p| := mul_nonneg hH.le (abs_nonneg _) |
| have hbase : (ell : ℝ) * MAs + MA * Mr + MA * MI + H * |A p| ≤ |
| ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1) * (1 + H * |A p|) := by |
| nlinarith [mul_nonneg hD hh] |
| have hm := mul_le_mul_of_nonneg_left hbase (mul_nonneg (mul_nonneg hW.le heps) hR.le) |
| convert hm using 1 |
| dsimp only [W, H, Cj] |
| ring |
| have hcross := actual_normalized_cross_loss hW hH hCj heps hT hJ |
| have he : |directedSResidual B r ell epsilon p| ≤ E := |
| directedSResidual_bound heps heps1 hR.le (abs_lt.mpr hp.1).le hIs hBp hr |
| have hEH : E ≤ H / 8 := hEbudget.trans (by linarith) |
| have hLH : L / H ≤ mu / 4 := by |
| apply (div_le_iff₀ hH).2 |
| have hh : L ≤ (mu / 4) * lambda := (div_le_iff₀ hlambda).1 hloss |
| exact hh.trans (mul_le_mul_of_nonneg_left hlH (by positivity)) |
| have hmargin : mu / 2 ≤ directedPrincipal A ell epsilon p - H * A p / 2 + |
| A p * directedSResidual B r ell epsilon p - L * (H⁻¹ + H * (A p) ^ 2) := by |
| by_cases hn : A p ≤ 0 |
| · have hh := normalized_negative_near_coercivity hH hL hmu hn hAp hsmall hLH he hEH hprincipal |
| have hbonus : 0 ≤ H * |A p| / 4 := by positivity |
| linarith |
| · exact normalized_positive_strip_coercivity hH hL hmu (lt_of_not_ge hn).le hAp |
| hsmall hLH he hEH hprincipal (hstrip (lt_of_not_ge hn)) |
| exact actual_quadratic_coercivity hW hH hlambda hmu hlH hT |
| (primitive_directedS_expansion hB hR hp A C r hA edge lambda epsilon ell hedge hC) |
| hcross hmargin |
|
|
| end SmoothLocal.Weighted |
|
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| end |
|
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| end OAI |
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