SAE Phase Diagram: 0 Dictionary Recoveries in 3,500 Fits

Sparse autoencoders are the workhorse of mechanistic interpretability: train one on a model's activations, read off the "features." The field's quiet assumption is that a well-trained SAE recovers the true underlying features. A new preprint by Alexis D. Plascencia put that assumption into a controlled experiment — the MAIS-O43 open problem — and the result is a clean, uncomfortable null: across 3,500 independently initialized fits, zero full-dictionary recoveries and zero feature merges. Not fewer than expected. Zero.

The problem: when do SAEs merge features?

When two true features systematically co-occur in the data, an SAE can absorb or merge them into one learned atom, which silently corrupts any interpretation you build on top. The MAIS-O43 open problem specifies a synthetic setup where the ground-truth dictionary is known: generate data from a known dictionary with a controlled nesting fraction γ, vary the sparsity penalty λ and dictionary size M, and map out where recovery degrades into merging. Because the truth is synthetic and known, you can measure recovery exactly — median best cosine similarity between learned atoms and true features, against a 0.95 recovery criterion.

The results: a diffuse phase nobody predicted

Plascencia ran 200 independently initialized fits across ten grid cells, then stress-tested with 3,300 more across the full 165-cell grid using standard minibatch Adam. The findings:

• Reconstruction is nearly perfect — the SAEs are doing their job on the loss surface.
• Feature recovery fails everywhere — median best cosine of 0.5–0.7 against the 0.95 criterion. Learned atoms are simply not the true features.
• Learned codes are ~10× denser than the ground truth — the "sparse" in sparse autoencoder is relative, not absolute.
• Zero merges, too. The failure mode isn't the predicted merging; it's something else entirely — a reproducible diffuse phase.

mindmap
  root((MAIS-O43 outcome))
    Predicted
      Recovery phase
        cosine ≥ 0.95
      Merge phase
        nested features absorbed
    Observed across 3,500 fits
      Neither
        0 recoveries
        0 merges
      Diffuse phase
        recon ≈ perfect
        best cosine 0.5–0.7
        codes ~10x denser
            

Here's the claim you can't get from the abstract: the phase diagram of trained SAEs may be fundamentally different from the phase diagram of the objective's minimizers. It's already known that the global optimum of the exact sparse-coding objective merges nested features in the two-feature case. If trained SAEs were reaching those minima, we'd see merging. We don't — so gradient descent with minibatch Adam isn't finding the optima whose pathology everyone was worried about. The merging debate was, in a sense, forensics on the wrong suspect: optimization dynamics, not the objective, decide what an SAE learns. That's good news (real SAEs don't merge) wrapped around bad news (they don't recover either — they sit in a stable, diffuse compromise that reconstructs well while mislabeling the features you're interpreting).

Limitations worth flagging

• Synthetic only. Real activations aren't drawn from a fixed synthetic dictionary; real SAEs on real models might behave better or worse. The paper says nothing directly about production interpretability runs.
• Grid coverage is asymmetric — 200 deep fits on 10 of 165 cells, then 3,300 shallower fits across the full grid. The full-grid sweep could in principle miss cell-specific behavior that only appears with more initializations.
• One optimizer family. All main results use standard minibatch Adam. The claim is specifically that this training regime lands in the diffuse phase; better-optimized or second-order methods weren't tested.
• The 0.95 criterion is strict. A cosine of 0.5–0.7 is bad, but the paper doesn't report downstream-task performance of the "failed" features — interpretation quality and cosine-to-truth are related but not identical.

Why this matters if you build things

If you use SAE features for monitoring, steering, or auditing models — anything like the latent-channel detection stack we wrote about in the latent collusion work — this result puts a price tag on your confidence: a feature label from a trained SAE is a property of the optimization trajectory, not a recovered ground-truth concept. Reconstruction loss won't warn you; the paper's models reconstruct nearly perfectly while missing the true dictionary by a wide cosine margin. Treat SAE feature labels as consistent-but-arbitrary coordinates, validate them behaviorally (causal patching, ablation effect sizes), and don't read phase transitions off objective analysis alone. For the interpretability field at large, the actionable question just changed from "when does merging happen?" to "what does the optimizer actually converge to, and is it stable enough to trust?"