Open problem MAIS-O39 · posed in MAIS-A3 as Problem 4.9 · Status: open.
Tags: interpretability · sparse autoencoders · mechanistic interpretability · superposition · optimization · statistics. Difficulty: ★★★.
Authored by: Claude Fable 5 directed by Lionel Levine · Audited by: GPT 5.6 Sol.
A sparse autoencoder never solves the sparse-coding program it is named for. Instead of computing the optimal code for each data point, it applies one learned affine-plus-ReLU map — cheap, but not optimal. O'Neill, Gumran, and Klindt [OGK25] proved this shortcut can produce inaccurate codes; whether it moves the minimizing dictionary, the object interpretability actually reads off, is open.
The setting is a superposition model: data $y=\Phi x+\xi$, where the dictionary $\Phi$ belongs to $U_{n,m}$, the set of $n\times m$ matrices with unit columns (here $v_1,\dots,v_m$), the codes $x$ are sparse and nonnegative, and $\xi$ is Gaussian noise; the exact class of models — support law, coefficient law, noise level — is specified in MAIS-A3. Two objectives over dictionaries $\Psi\in U_{n,M}$, with unit columns $u_1,\dots,u_M$, compete. Exact coding: $F_\lambda(\Psi)=\mathbb{E}\ \min_{z\ge0}\bigl[\tfrac12\lVert y-\Psi z\rVert^2+\lambda\lVert z\rVert_1\bigr]$. Amortized coding: $G_\lambda(\Psi,W,b)=\mathbb{E}\bigl[\tfrac12\lVert y-\Psi\ c(y)\rVert^2+\lambda\lVert c(y)\rVert_1\bigr]$ with $c(y)=\mathrm{ReLU}(Wy+b)$, the encoder $W\in\mathbb{R}^{M\times n}$ and $b\in\mathbb{R}^M$ optimized jointly with $\Psi$. Say $\Psi$ $\varepsilon$-recovers $\Phi$ if some injection $\tau$ has $\langle u_{\tau(i)},v_i\rangle\ge1-\varepsilon$ for all $i$. Call an atom $u_j$ live if, with positive probability in $y$, some exact code — a minimizer of the inner program defining $F_\lambda$ — puts mass on it; liveness is judged by the exact code for every dictionary, including one produced by amortized training, so the learned encoder plays no role in the definition. A live atom is an $(\varepsilon,\delta)$-merge of two features if it lies within $\varepsilon$ of a combination $\alpha v_i+\beta v_{i'}$ with $\alpha,\beta\ge\delta$ while its inner product with each feature stays at most $1-\delta$.
Problem (MAIS-A3, Problem 4.9). Fix a superposition model $\mathcal S$ and $\lambda>0$. Let $\mathcal M_F=\mathrm{argmin}_{U_{n,M}}F_\lambda$, and let $\mathcal A_G$ be the set of dictionaries $\Psi\in U_{n,M}$ for which some minimizing sequence $(\Psi_r,W_r,b_r)$ for $G_\lambda$ has $\Psi_r\to\Psi$. Compare $\mathcal M_F$ with $\mathcal A_G$.
- Under the hypotheses of the agenda's Conjecture 4.4 (MAIS-O36: independent Bernoulli($p$) supports, coefficients $\mathrm{Unif}[1,2]$, no noise, coherence and density bounds), is there an absolute constant $C'$ such that every $\Psi\in\mathcal A_G$ $(C'\lambda)$-recovers $\Phi$?
- Exhibit a superposition model, or prove none exists, for which — for some $\varepsilon,\delta>0$ — some $\Psi\in\mathcal A_G$ contains an $(\varepsilon,\delta)$-merge while no $\Psi\in\mathcal M_F$ does (with the same $M$, $\lambda$).
The compactness of $U_{n,M}$ makes $\mathcal A_G$ nonempty even when the unconstrained encoder parameters diverge; settling attainment of $G_\lambda$ remains part of the problem.
Part 2 asks whether amortization can create a failure mode absent from exact coding: a merged atom the ideal estimator would never produce. Sun, Wang, and Hu [SWH26] document such dictionary distortions empirically across amortized architectures; the exact comparison of the population sets $\mathcal M_F$ and $\mathcal A_G$ is untouched. For the estimator definitions and the amortization literature, see MAIS-A3.
References
- [OGK25] C. O'Neill, A. Gumran, and D. Klindt, Compute optimal inference and provable amortisation gap in sparse autoencoders, International Conference on Machine Learning (ICML) 2025. arXiv:2411.13117
- [SWH26] Y. Sun, Z. Wang, and W. Hu, The price of amortized inference in sparse autoencoders, International Conference on Learning Representations (ICLR) 2026.
Related: MAIS-O36 (the exact-coding recovery conjecture that part 1 extends) · MAIS-O3 (the headline identifiability problem) · MAIS-O43 (numerics run on the amortized objective $G_\lambda$).