THEORY-1036: In a model that fits y = Σ v_i x_i² with ỹ = Σ_r (W_r·x)² + b, on independent inputs of zero mean, unit variance and common fourth moment k, the expected squared error at the optimal bias is (k − 1)Σ(n_i − v_i)² − 2Σn_i² + 2Σn_i²/C_i, in the squared embedding lengths n_i and capacities C_i = (w_i·w_i)²/Σ_j(w_i·w_j)², and its minimum with the capacities summing to the number of neurons D gives C_i = clip((k − 1)v_i/((k − 3)λ) − 2/(k − 3), 0, 1): for k ≤ 3 each feature is given a whole dimension or none, and for k > 3 a feature is shared between neurons exactly when 2λ/(k − 1) < v_i < λ

Source

Scherlis, Sachan, Jermyn, Benton and Shlegeris (2022), Polysemanticity and Capacity in Neural Networks, arXiv 2210.01892 v4, LIT-1625, read in full in NOTE-1205: §3 and Appendices A and C, with the capacity bound of Appendix F and the feasibility result of Appendix H.

What was actually shown

The derivation, re-done. Write A = WᵀW, so ỹ = xᵀAx + b, and M = A − diag(v), so ỹ − y = xᵀMx + b.

1. For independent x_i with mean 0, variance 1 and fourth moment k, E[xᵀMx] = tr M and Var[xᵀMx] = (k − 1)Σ_i M_ii² + 2Σ_{i≠j} M_ij². The optimal b is −tr M, leaving E[L] = (k − 1)Σ_i(n_i − v_i)² + 2Σ_{i≠j}(w_i·w_j)², with n_i = A_ii (the paper’s eq. 12).
2. By the definition of capacity, Σ_j(w_i·w_j)² = n_i²/C_i, so Σ_{j≠i}(w_i·w_j)² = n_i²/C_i − n_i², which gives the title’s form (eq. 13).
3. At fixed C, ∂L/∂n_i = 2(k − 1)(n_i − v_i) − 4n_i + 4n_i/C_i = 0, so n_i = (k − 1)v_i/((k − 3) + 2/C_i). Since n is optimal for every C, dL/dC_i = ∂L/∂C_i = −2n_i²/C_i² (eq. 14), never positive, so the capacities saturate their bound, Σ C_i = D.
4. An interior optimum has a common value of n_i/C_i; call it λ (the paper’s λ in eq. 15, which is √(λ₄/2) for the multiplier λ₄ of its eq. 4). Substituting n_i = λC_i in step 3 gives C_i = (k − 1)v_i/((k − 3)λ) − 2/(k − 3), clipped to [0, 1] at the corners, with λ fixed by Σ C_i = D.
5. For k > 3, C_i ≤ 0 when v_i ≤ 2λ/(k − 1) and C_i ≥ 1 when v_i ≥ λ. For k < 3 the expression decreases in v_i and the loss is concave in each C_i (−2n_i²/C_i² rises in magnitude with C_i), so the optimum is at corners: the D most important features get a dimension each.
6. The allocation is realizable: any C with 0 ≤ C_i ≤ 1 and Σ C_i = D is the capacity vector of a semiorthogonal W (Theorem 1), for which n_i ∝ C_i, and columns can be rescaled to set the constant.

The numerical check. Described in the history note: the loss formula against Monte Carlo, and the allocation against a numerical solution.

What could have come out otherwise. The interference term need not have reduced to a function of each feature’s own capacity, and then the allocation would not separate. The fourth moment could have entered the two terms alike, leaving sparsity no effect on sharing.

What this does not say

• Not about trained networks. One hidden layer, a scalar target that is a sum of squares, a quadratic activation, independent features.
• Not about ReLU or GeLU models, which the paper treats only numerically, in figures.
• Not that a shared feature is polysemantic in every basis by this result alone. That is the geometry of §4: this model’s solution is a single block, which is polysemantic in any basis.

Connections

• LIT-1583 (§6) and NOTE-1202. The appendix’s two-feature, one-neuron case: with a = b, v₁ = v₂ = 1, D = 1 and N = 2, step 4 gives C₁ = C₂ = 1/2 and λ = 2(k − 1)/(k + 1), and v = 1 < λ exactly when k > 3, so equal features of high kurtosis share the neuron, as the appendix states.
• LIT-323. Elhage et al.’s observation that sparser inputs give more superposition, derived here for this model through k.