f = ReLU(W_enc (x - b_dec) + b_enc)

x_hat = W_dec f + b_dec.

L = ||x - x_hat||² + λ ||f||₁.

Advanced Technical Expansion I | SAE Optimisation, Sparsity Mechanics and Reconstruction Fidelity

An SAE is easy to describe and difficult to train well. The model is asked to satisfy three competing demands: reconstruct a dense activation faithfully, use only a small number of dictionary atoms for each example, and allocate a very large dictionary without leaving most features dead or redundant. This expansion treats those demands as an optimisation system rather than as one generic “interpretability loss”.

A. The reconstruction–sparsity frontier is the primary engineering object

For any fixed dictionary width and activation site, increasing sparsity usually worsens reconstruction. Reducing sparsity usually improves reconstruction and makes the code harder to interpret or store. Plot reconstruction quality against active feature count rather than reporting one checkpoint.

The resulting Pareto frontier allows fairer comparison among architectures. One SAE is better than another when it achieves lower reconstruction error at comparable sparsity, lower sparsity at comparable fidelity, or stronger semantic/causal quality at similar operating points.

Comparing an extremely sparse model with a much denser one using only MSE confounds two objectives.

B. L1 regularisation creates coefficient shrinkage

For positive feature coefficient f, an L1 penalty adds gradient pressure of roughly λ whenever f is active. The feature must overcome this pressure even after the encoder has correctly detected it. Coefficients can therefore be systematically smaller than the reconstruction-optimal magnitude.

This is analogous to the soft-thresholding behaviour of LASSO in simple linear problems. It encourages exact or near-zero coefficients but biases active magnitudes.

If activation magnitude is interpreted as feature strength, shrinkage matters. Thresholded schemes such as JumpReLU were motivated partly by separating feature selection from magnitude estimation.

C. TopK implements a hard per-example sparsity budget

TopK computes all candidate pre-activations and retains exactly k. The gradient flows through selected features while unselected features receive no direct reconstruction gradient for that example.

This creates competition. A feature can die because another atom consistently outranks it on the examples where it would otherwise be useful. Initialisation and early training dynamics therefore matter.

TopK’s main interpretive convenience is clear L0 control. Its main assumption is equally clear: every observation deserves the same active feature count.

D. Variable-complexity inputs challenge fixed k

A punctuation token in a simple context and a token participating in a complex reasoning state need not require the same number of explanatory features. Fixed-k selection can allocate unnecessary atoms to simple states and insufficient atoms to complex ones.

BatchTopK or threshold-based approaches allow active count to vary. The price is that sparsity becomes less predictable per token and can depend on batch statistics or threshold calibration.

Evaluate the distribution of active count, not just its mean. Long tails reveal which contexts consume interpretability capacity.

E. JumpReLU creates an explicit detection threshold

For pre-activation a and threshold θ, a schematic JumpReLU feature is zero below θ and approximately a above θ. Unlike ordinary shifted ReLU max(0,a-θ), the output can retain the full magnitude after crossing the threshold.

The discontinuity makes ordinary gradient treatment of θ nontrivial, so practical training uses surrogate or specialised gradient methods. The conceptual payoff is cleaner separation between whether a feature is active and how much it contributes.

Google DeepMind reported JumpReLU as a major architecture in Gemma Scope, where balancing feature detection with magnitude estimation reduced error relative to earlier approaches.

F. Threshold distributions reveal feature-specific operating points

Different learned features can require different thresholds because their pre-activation noise and signal distributions differ. A common threshold assumes one detection problem for every atom.

Inspect threshold against feature frequency, decoder norm, reconstruction contribution and semantic coherence. Extremely high thresholds can create rare features that look pure simply because only extreme examples survive.

Monosemanticity and selectivity should therefore be evaluated across threshold regimes rather than treating one operating point as intrinsic.

G. Dictionary scale degeneracy is not merely mathematical housekeeping

Without decoder norm control, the pair (f_j,d_j) can be rescaled while preserving reconstruction. Sparsity penalties on f then create incentives unrelated to semantic quality.

Unit decoder norms make feature activation magnitude more comparable across atoms, though encoder scaling and bias still influence statistics. If a feature dashboard compares activations, it should state these conventions.

Representation coordinates only become interpretable after their gauge freedoms are fixed.

H. Tied and untied encoder–decoder weights express different assumptions

A tied SAE uses encoder directions related directly to decoder directions. An untied SAE lets detection and reconstruction use different vectors. Untying increases flexibility because the direction that recognises a feature need not be identical to the direction that adds it back to activation space.

This can improve reconstruction and complicates interpretation: which vector is “the feature”? Usually the decoder direction describes the contribution to native activation, while the encoder row describes the detector.

Feature semantics can therefore have two geometries: detection boundary and reconstructed effect.

I. Encoder bias defines a feature’s activation boundary

For a ReLU-style encoder, a feature activates when a linear projection plus bias crosses zero or another threshold. Bias therefore controls prevalence separately from decoder meaning.

Two features with similar decoder directions can have very different activation sets because their encoder biases differ. Matching features by decoder cosine alone can miss this.

A feature evidence record should include both decoder geometry and empirical activation behaviour.

J. Decoder bias is a learned baseline state

If the decoder bias approximates the mean activation, sparse features explain deviations from that baseline. This can improve efficiency because the common component does not need to be reconstructed repeatedly by active atoms.

But “mean activation” can vary by domain, token position or model mode. A single global bias can leave systematic residuals that dictionary atoms absorb.

Evaluate bias residuals across data slices. Domain-dependent baselines can produce features that are really dataset-offset corrections.

K. Activation preprocessing changes the dictionary problem

Centering, normalising, whitening or clipping native activations alters which directions carry large reconstruction loss. Whitening makes low-variance native directions more important relative to raw MSE, potentially exposing different feature dictionaries.

There is no neutral preprocessing. Raw activation geometry reflects the model’s own scale; transformed geometry can be easier to optimise or compare.

Report preprocessing as part of the interpretability method, not as a hidden data-cleaning step.

L. MSE weights high-variance directions most heavily

Euclidean reconstruction loss sums squared error across native activation coordinates. Directions carrying large variance contribute more expected loss. Rare low-variance directions can be ignored with little penalty even when they are behaviourally important.

This explains why behavioural metrics such as loss recovered are necessary. It also motivates weighted reconstruction objectives when specific subspaces are known to matter.

Weighting should be justified carefully: prioritising one downstream behaviour can distort the SAE away from a general decomposition.

M. Cosine reconstruction can preserve direction and ignore magnitude

Some analyses care mainly about activation direction rather than norm. Cosine reconstruction or angular metrics can be informative diagnostics. Training solely on cosine error would allow large norm mistakes.

Use multiple reconstruction metrics if the native activation norm has behavioural meaning. A dictionary that points correctly but scales badly can produce large downstream errors in residual addition.

N. Loss recovered measures function through the base model

One evaluation replaces a native activation with a baseline or zeroed state and measures the increase in language-model loss. Then replace it with the SAE reconstruction. If reconstruction restores most of the lost performance, the SAE preserved a large fraction of behaviourally relevant information at that site.

The metric depends on the baseline, dataset and hook location. It is possible to recover next-token loss while losing information important for a rare downstream capability not represented in the evaluation corpus.

Loss recovered is a functional fidelity metric, not a completeness certificate.

O. Activation MSE and model loss can disagree

Construct two reconstruction errors with the same norm. One lies in a downstream-null direction and barely changes logits. The other aligns with a sensitive readout direction and changes predictions substantially.

This demonstrates why activation-space distance should be paired with Jacobian- or behaviour-aware metrics. The base model defines a non-isotropic importance geometry over activation errors.

At high stakes, evaluate the specific behaviours for which the SAE will be used.

P. Explained variance can be stratified by feature class

Global explained variance weights common examples heavily. Partition the evaluation data by language, domain, token type, capability or safety-relevant class. Compute reconstruction and loss recovery per slice.

A dictionary trained on general text may explain ordinary prose extremely well and code poorly. The average can hide this mismatch.

The interpretation scope should follow the weakest relevant slice, not the global mean.

Q. Rare-event fidelity deserves a separate benchmark

If the SAE will support monitoring of hallucination, jailbreak or deception-related behaviours, construct targeted evaluation sets containing those phenomena. Measure reconstruction and relevant feature coverage there.

Targeted sets should not be used to estimate natural prevalence. Their job is sensitivity analysis: can the dictionary represent the rare state when it occurs?

Interpretability infrastructure intended for safety must be tested on the tail, not only average language.

R. Dead features are an optimisation symptom with several causes

A feature can be dead because its encoder never wins TopK competition, its threshold is too high, its decoder direction is redundant, the corpus lacks its triggering state or early optimisation pushed it into an unhelpful region.

Count dead features under a large evaluation corpus and define the activation threshold used to call a feature alive. “Never fired in one million tokens” and “mathematically incapable of firing” are different claims.

Dead-feature diagnostics should be stratified by training stage because some features die early and never recover.

S. Resampling dead features injects new hypotheses into the dictionary

One repair reinitialises dead decoder directions toward current reconstruction residuals. The idea is intuitive: if existing features fail to explain a residual pattern, give an unused atom a chance to model it.

Resampling changes the optimisation trajectory and can destabilise feature identity. A feature index before and after resampling should not be assumed to mean the same thing.

Track feature lineage or reset provenance when dictionary slots are reused.

T. Feature absorption is a competition problem

Suppose broad feature A activates on all legal text and narrow feature B should represent case citations. If A already reconstructs citation examples reasonably, B may receive little unique residual error and never specialise.

This can be addressed by greater dictionary capacity, more targeted data, alternative sparsity competition or losses that reward explaining structured residuals. Every repair changes the inferred ontology.

Absorption is especially relevant for rare features nested inside common categories.

U. Feature splitting can reduce reconstruction error without increasing useful explanatory power

A broad feature may split by punctuation, position or superficial form because those distinctions reduce MSE. Human interpretability can become worse even though reconstruction improves.

Measure whether splits correspond to distinct causal roles or downstream behaviour. Fine dictionaries should earn their granularity through use, not only reconstruction.

This is why expansion factor must be evaluated against semantic and mechanistic metrics as well as MSE.

V. Feature redundancy can make dictionary width look more productive than it is

Two atoms with nearly identical decoder directions and strongly correlated activation patterns may be redundant. Redundancy can improve optimisation robustness and inflate feature count.

Cluster decoder atoms, calculate activation correlation and test substitution: suppress one atom and ask whether another carries the same reconstruction contribution.

Report effective feature count at several redundancy thresholds rather than only raw dictionary width.

W. Decoder coherence can predict ambiguity

High cosine similarity among decoder atoms means several features reconstruct along similar native directions. Their semantic distinction must then come largely from encoder detection or contextual activation patterns.

High coherence is not automatically bad; related features can share native effects. It weakens the claim that decoder direction alone identifies a unique feature.

Match geometric and activation-space diagnostics before declaring duplicates.

X. Feature activation distributions are usually highly non-Gaussian

Sparse features have a mass at zero and a long-tailed positive distribution when active. Means and standard deviations can therefore be misleading summaries.

Report activation frequency, conditional quantiles given active, extreme values and domain-specific distributions. Thresholds for monitoring should be set on the empirical tail rather than assuming Gaussian z-scores.

Outliers can be genuine rare semantics or reconstruction artefacts; inspect examples.

Y. Extreme activations can dominate feature descriptions

Top-k example galleries select the most extreme tail. This can make a feature appear more coherent than its moderate activation region. A feature might strongly detect Python code and moderately activate on unrelated punctuation contexts.

Sample across activation quantiles. Ask whether the semantic description predicts the entire dose-response or only the maxima.

A robust feature label should have calibrated scope across activation strength.

Z. TopK creates rank competition whose threshold changes by example

In TopK, the effective threshold is the kth-largest pre-activation for that example. The same raw pre-activation can be selected in a sparse example and suppressed in a crowded one.

This means feature presence is contextually relative to competition. A dashboard treating non-selection as evidence of semantic absence can overinterpret the code.

Threshold-based SAEs and TopK SAEs therefore have different presence semantics even if their average L0 is identical.

AA. BatchTopK introduces cross-example competition

When a batch shares one activation budget, feature selection for one token can depend indirectly on activations from another token in the same batch. This can allocate sparsity efficiently and makes the encoding sensitive to batch composition during training.

Deployment usually needs deterministic per-example encoding, so implementation details determine how batch-level training maps to inference. Document the procedure.

Batch composition is not merely an optimisation detail when it determines which features receive gradients.

AB. Matryoshka training treats dictionary width as a hierarchy rather than separate models

A Matryoshka representation can optimise nested prefixes or subsets so a small dictionary captures high-value coarse structure while larger sets add detail. This can make feature resolution explicit and reduce the need to train many unrelated widths.

The ordering itself becomes meaningful: early features receive stronger pressure because they must work in smaller prefixes. Inspect whether early atoms are broader, more frequent or more important.

Gemma Scope 2 uses state-of-the-art Matryoshka training as part of its newer interpretability suite.

AC. Low-rank parameterisation reduces compute and can restrict dictionary freedom

A massive d×m encoder/decoder is expensive when m is millions. Low-rank factorisation or structured parameter sharing can reduce memory and matrix-multiply cost.

The restriction can act as regularisation and can prevent the dictionary from representing arbitrary independent directions. Evaluate whether semantic feature quality changes, especially for rare or fine-grained atoms.

Efficiency architecture is part of the feature hypothesis.

AD. Activation caching can dominate the experiment

Training SAEs on large language models often requires collecting enormous activation datasets. Caching permits repeated SAE experiments without rerunning the base model and consumes substantial storage and I/O.

Sampling strategy determines which tokens enter the cache. Sequence positions, document boundaries, padding, system prompts and instruction data can all change activation distribution.

Cache provenance should include base-model checkpoint, tokenizer, context length, activation hook, dataset mix and preprocessing.

AE. Training on cached activations creates a frozen-observer assumption

The base model is fixed while the SAE learns. This is useful because the interpretability tool cannot alter the representation it studies. It also means the SAE never sees how activations would change under feature interventions during training.

Dictionary learning is observational unless causal evaluation is added later. Reconstruction alone does not train atoms to correspond to interventions.

AF. Optimisation schedules can change feature allocation

Learning rate, sparsity warm-up, threshold schedule and dead-feature resampling determine early competition. Features that secure useful regions of activation space early can crowd out later candidates.

Run multiple seeds and schedule variants. If semantic features appear only under one narrow schedule, the dictionary is not strongly determined by the data.

Optimisation stability is evidence about identifiability in practice.

AG. Sparsity warm-up can prevent early collapse and can bias the eventual ontology

Applying strong sparsity from the first step can kill features before they discover useful directions. Starting with weaker sparsity allows the dictionary to learn reconstruction structure, then encourages competition.

The path matters because the optimisation landscape is non-convex. A dictionary reached through warm-up can differ from one trained with fixed sparsity even at the same final operating point.

Report schedule, not only final λ or k.

AH. Feature-frequency regularisation can redistribute capacity

Some training variants discourage highly unequal feature usage or explicitly revive rare atoms. This can improve dictionary coverage and can force the model to use unnatural atoms merely to satisfy utilisation targets.

Uniform feature frequency is not a natural law. Real semantic features are long-tailed. The objective should avoid dead capacity without pretending all concepts are equally common.

AI. Cross-seed reconstruction similarity is weaker than feature similarity

Two SAEs can reconstruct nearly identical activations while using different sparse codes. Reconstruction agreement therefore does not establish dictionary agreement.

Compare decoder subspaces, matched atom directions, activation sets and semantic descriptions separately. The fourth owner in this batch, Representation Similarity Analysis, provides tools for comparing whole learned spaces.

A stable microscope should be evaluated both by the image it reconstructs and by the features it claims to resolve.

AJ. A full SAE operating curve

For several sparsity settings and widths, plot: activation MSE, explained variance, loss recovered, mean L0, median L0, dead-feature fraction, feature-frequency entropy, decoder coherence and interpretation score.

No one point dominates every metric. The curve reveals whether semantic coherence improves only by sacrificing behaviour or whether a method genuinely shifts the frontier.

Choose operating points based on the receiver job: circuit tracing, monitoring, explanation, steering or broad exploratory mapping.

AK. Worked laboratory: scaling degeneracy and decoder normalisation

Take one learned atom with coefficient f and decoder vector d. Multiply d by ten and divide f by ten. Reconstruction is unchanged. Under an L1 penalty, the new coefficient costs one tenth as much.

Then impose unit decoder norm and repeat. The degeneracy disappears. This tiny calculation explains why dictionary-normalisation conventions are not optional under coefficient penalties.

AL. Worked laboratory: L1 shrinkage versus JumpReLU magnitude

Create a one-feature reconstruction problem with noisy pre-activation. Solve with soft-thresholding/L1 and compare to a hard or jump threshold. Measure detection accuracy and amplitude bias.

The L1 solution tends to shrink active amplitude. A jump threshold can preserve amplitude after selection while creating a harder detection boundary. This illustrates the conceptual trade-off without a huge model.

AM. Worked laboratory: TopK under variable input complexity

Generate examples with between one and ten true sparse factors. Encode all with fixed k=5. Simple examples are forced to select extra atoms; complex examples drop some true factors. Compare with a thresholded or variable-budget encoder.

The experiment demonstrates the assumption hidden in fixed-k sparsity and motivates reporting active-count distributions for real data.

AN. Worked laboratory: reconstruction MSE versus functional sensitivity

Construct a downstream linear readout with one highly sensitive direction. Create two reconstruction errors of equal Euclidean norm: one aligned with the sensitive direction, one orthogonal.

The first changes output strongly; the second barely matters. This proves why functional fidelity cannot be inferred from MSE alone.

AO. Worked laboratory: dictionary width and feature splitting

Generate data from a hierarchy: broad factor A has two subtypes A1 and A2 with subtle residual differences. Fit dictionaries of increasing width and sparsity.

A narrow dictionary may learn A. A wider dictionary may split A1 and A2. Evaluate whether both representations reconstruct well and which supports the desired receiver. “More features” becomes a resolution choice rather than an automatic improvement.

AP. The SAE optimisation evidence card

• Native site: model, layer, sublayer and tensor.
• Data: activation corpus and sampling policy.
• Preprocessing: centering, normalisation and dtype.
• Dictionary: width, expansion factor and parameterisation.
• Sparsity: mechanism and operating target.
• Threshold: global, feature-specific, fixed or learned.
• Norm gauge: decoder/encoder constraints.
• Optimiser: schedule, warm-up and feature revival.
• Reconstruction: MSE, explained variance and angular diagnostics.
• Functional fidelity: loss recovered and task slices.
• Utilisation: dead, rare, common and redundant features.
• Operating frontier: metrics across sparsity/width settings.
• Cost: activation storage, compute and inference overhead.

An SAE should therefore be judged as a measurement instrument before it is used as an explanatory language. Its reconstruction, sparsity and optimisation behaviour determine what it is capable of seeing. The second advanced layer asks what the recovered atoms mean, whether they recur, whether they form circuits and whether interventions support the explanations we attach to them.

Advanced Technical Expansion II | Feature Validation, Monosemanticity, Circuit Evidence and Dictionary Reliability

Once an SAE reconstructs well and sparsely, the harder scientific question begins: what do its features mean, and how reproducible is that meaning? A dictionary can be numerically excellent and semantically unstable. A feature can have a beautiful activation gallery and little causal relevance. Two independent training runs can reconstruct the same activation with different dictionaries. This expansion therefore treats interpretation as an evidence pipeline rather than a naming exercise.

AQ. Top activations generate hypotheses; held-out examples test them

Feature interpretation often begins by sorting examples by activation. Suppose the strongest examples contain Python list comprehensions. A natural hypothesis is “this feature detects list-comprehension syntax”. The correct next step is not to publish the label. It is to create a held-out test containing list comprehensions, other Python syntax, analogous constructs in other languages, brackets without comprehensions, English uses of similar punctuation and code comments.

If the feature activates selectively on the intended construct, the description survives. If it fires broadly on square brackets, the label must narrow. If it activates on iteration structure across languages, the label can broaden in a different direction.

This is the minimum standard for monosemanticity claims: a description should predict activation beyond the examples that inspired it.

AR. Activation quantiles reveal whether the label survives outside the extreme tail

The top twenty examples can look perfectly coherent even when the feature’s middle activation range mixes several meanings. Divide positive activations into quantile bands and inspect each. Also inspect examples just below threshold.

A feature with coherent top 0.1 percent and noisy remaining activations may still be useful as a high-threshold detector. It should not be described as globally monosemantic without qualification.

Monitoring systems should calibrate at the threshold actually used. Feature semantics can be operating-point dependent.

AS. Counterexamples are more informative than another dozen confirming examples

Search for inputs predicted by the semantic label that fail to activate, and inputs outside the label that activate strongly. False positives and false negatives reveal hidden conditions.

A purported “quotation” feature may fail on single quotes and activate on code strings. A “refusal” feature may actually track formal safety-policy vocabulary. Counterexamples turn broad labels into operational descriptions.

The best interpretability note often includes an explicit “does not mean” field.

AT. Automated descriptions must be scored on prediction, not eloquence

An LLM interpreter can read activation examples and produce a concise label. The label can sound convincing because language models are good at summarising patterns. Use a separate held-out set to test whether the description predicts activation.

One evaluation asks the interpreter to estimate which examples should activate given only the description. Correlation or ranking agreement with actual SAE activations provides a simulation score.

Simulation scores are useful and can reward surface descriptions that predict activation without capturing the feature’s causal role. They belong alongside, not above, intervention evidence.

AU. Human annotation adds judgement and requires reliability measurement

Humans can evaluate whether activation examples share a coherent concept, compare candidate labels and identify subtle context. Different annotators may disagree on granularity or ontology.

Use multiple annotators for a diagnostic subset, report agreement and preserve disagreements. High disagreement can mean the feature is genuinely mixed or simply that the concept is hard to verbalise.

Human-readable is a receiver property. A feature can be mechanistically useful and difficult to name.

AV. Embedding-based coherence metrics inherit another model’s geometry

One way to score monosemanticity embeds the feature’s activating examples using an external semantic encoder and measures how tightly they cluster. This scales and imports the external encoder’s own biases and anisotropy.

If the external encoder collapses technical distinctions, a technically mixed feature can look coherent. If it separates paraphrases oddly, a coherent feature can look diffuse.

Embedding-based metrics are evidence about coherence under that external representation, not a model-independent semantic truth.

AW. Tversky Monosemanticity Score offers a label-free alternative

July 2026 work, Measuring Monosemanticity in Sparse Autoencoders via Latent Activation Coherence, proposes the Tversky Monosemanticity Score. It operationalises monosemanticity through coherence among binarised latent activation sets rather than requiring external concept labels or pretrained semantic embeddings.

The authors report experiments across vision and vision-language representations, TopK and BatchTopK regimes, sparsity levels and expansion factors, and motivate the metric partly by sensitivity of embedding-based alternatives to encoder anisotropy.

This is progress on measurement, not a declaration that monosemanticity has one universally accepted scalar definition. Different metrics can prefer different feature granularities.

AX. Feature consistency is a separate axis from monosemanticity

A feature can be highly coherent in one SAE and fail to appear in another independently trained SAE. That weakens the idea that the feature is uniquely determined by the activation distribution.

ACL 2026 work, Mechanistic Interpretability Should Prioritize Feature Consistency in Sparse Autoencoders, argues that run-to-run consistency should be reported alongside reconstruction and sparsity. The paper proposes Pairwise Dictionary Mean Correlation Coefficient as an assignment-based consistency metric and reports high consistency under some TopK configurations.

Consistency does not prove semantic correctness. It tells us the optimisation repeatedly finds similar features, which is evidence for a more canonical decomposition.

AY. Feature matching is an assignment problem when dictionaries have the same width

Given two dictionaries, compute a similarity matrix between feature atoms—decoder cosine, activation correlation or another measure. A one-to-one assignment can maximise total similarity, for example using a Hungarian-style matching algorithm.

The resulting matched similarities show how much of one dictionary has a close counterpart in another. Low matches can indicate instability, split/merge differences or different granularities.

One-to-one assignment becomes artificial when widths differ or when one broad feature in one SAE corresponds to several finer features in another.

AZ. Many-to-many matching is necessary for feature splitting and merging

Suppose dictionary A has one “mathematics notation” atom while dictionary B has separate LaTeX, algebraic-symbol and theorem-reference atoms. No one-to-one match captures the relationship.

Compare subspaces spanned by candidate groups, aggregate activation patterns or semantic hierarchies. Feature lineage can be represented as a graph with split and merge edges.

This is especially important when comparing expansion factors or Matryoshka resolutions.

BA. Decoder cosine and activation correlation answer different matching questions

Decoder cosine asks whether two atoms write in similar native activation directions. Activation correlation asks whether they fire on similar examples. Two features can write similarly while using different detection boundaries; two can fire together while reconstructing different directions.

Semantic matching should consider both. High agreement across geometry, activation and description is stronger evidence than any one signal.

When signals disagree, preserve the disagreement rather than forcing one match.

BB. Feature consistency should be tested across data as well as random seed

Train SAEs on different samples from the same distribution and on deliberately different domains. A stable feature should recur when its supporting data remain present. Domain-specific features should change when their domain disappears.

If a supposedly universal syntax feature changes radically between corpora, either the dictionary is unstable or the feature depends on distributional context more than the label implies.

Consistency is always relative to the data-generating environment.

BC. Feature consistency across model versions is an interface problem

An SAE feature used in a monitoring system cannot be assumed to survive base-model retraining, instruction tuning, pruning or quantisation. Even if behaviour remains similar, activation geometry can move.

Relearn or re-match dictionaries after model changes. Recalibrate feature thresholds. Store model and dictionary version together.

A semantic label is not a permanent coordinate address.

BD. Model compression can preserve behaviour and alter feature geometry

Weight pruning, quantisation and distillation can leave aggregate benchmark performance largely intact while changing internal representation. Recent work uses SAE dictionaries as probes for whether features survive compression.

Geometric feature survival does not automatically imply identical causal importance. A matched feature can remain recognisable while the compressed model uses it differently.

This again separates representation similarity from functional equivalence.

BE. Feature descriptions should survive paraphrase and adversarial counterexamples

If a feature is labelled “uncertainty”, test several linguistic forms of uncertainty, direct certainty statements, negations, quoted uncertainty and meta-discussion of the word itself.

A robust feature should respond to the intended underlying pattern rather than one lexical trigger. Conversely, a lexical feature is perfectly legitimate if it is labelled lexical.

Adversarial interpretation tests protect against labels that simply repeat the most visible token in top examples.

BF. Concept erasure tests whether the feature carries usable concept information

One causal-style evaluation suppresses a candidate feature and measures whether a probe or downstream behaviour loses the associated concept while unrelated capabilities remain stable.

Successful selective deletion supports the feature interpretation. Failure can mean the feature label is wrong, the concept is redundantly represented, or the intervention does not isolate the feature cleanly.

Concept deletion is therefore evidence, not a binary oracle.

BG. Dose–response curves are stronger than one steering coefficient

Steer a feature over several magnitudes, including values inside the natural activation distribution. Measure target behaviour and collateral effects. A monotonic response in the predicted behaviour is stronger evidence than one dramatic output at an extreme coefficient.

Nonlinear responses can reveal thresholds or interactions with other features. At high magnitudes, outputs can become nonsensical because the intervention is off-manifold.

Report the natural activation quantiles alongside intervention strength.

BH. Random-direction controls reveal whether steering success is feature-specific

Adding any sufficiently large vector to a residual stream can change output. Compare candidate-feature steering with random directions matched for norm, principal directions and semantically unrelated SAE features.

If the target behaviour changes uniquely for the candidate direction, causal specificity is stronger. If many directions cause similar effects, the interpretation may reflect generic activation disruption.

Controls turn steering demonstrations into experiments.

BI. Reconstruction controls are essential for SAE interventions

To intervene on one SAE feature, researchers often encode a native activation, modify one coefficient, then decode back. The round trip itself introduces reconstruction error.

Compare the modified reconstruction with an unmodified SAE reconstruction, not only with the original native activation. This isolates the feature intervention from baseline SAE error.

For sensitive behaviours, also compare with matched reconstruction-noise controls.

BJ. Feature necessity can be hidden by redundancy

Ablating one SAE feature may not affect behaviour because another feature or native direction compensates. The label can still be correct as a redundant representation.

Test groups of correlated or semantically related features. Use mediation analysis to distinguish direct contribution from redundant representation.

“Ablation has no effect” is not automatically “feature is meaningless”.

BK. Feature sufficiency can be exaggerated by unnatural activation

A large positive intervention can force a behaviour because it overwhelms normal computation. This proves the downstream network is sensitive to the direction, not that ordinary feature activation is sufficient.

Use natural-range activation and matched contexts. Test whether the feature predicts and influences behaviour when it occurs naturally.

Causal claims should remain within the activation regime tested.

BL. Feature circuits require edges, not just nodes

A dictionary gives candidate nodes in a computation. A circuit requires knowing how activity in one feature affects another feature or output through attention, MLPs and residual updates.

Build candidate edges from model weights, transcoder contributions, attribution scores or causal patching. Validate important edges through intervention.

A list of interpretable features without their transformations is an ontology, not a mechanism.

BM. Transcoders turn feature dictionaries into sparse transition models

A transcoder predicts an output activation from an input activation through sparse latent features. This can associate a sparse feature with a particular transformation implemented by an MLP or layer transition.

Skip transcoders can model changes over a broader span. Cross-layer transcoders can connect earlier state to later state. Gemma Scope 2 combines these tools to move from local features toward multi-step behavioural analysis.

Predictive fidelity and sparse circuit interpretability should be measured separately, just as SAE reconstruction and monosemanticity are.

BN. Cross-layer lineage can split and merge like a biological genealogy

An early feature can fan out into several later features as the model resolves context. Several early lexical cues can merge into one later semantic state. Treating layer-to-layer matching as one-to-one misses this.

Represent feature lineage as a graph. Edge weights can combine activation predictability, decoder alignment and causal influence.

The graph becomes a hypothesis of information flow, not a final proof of algorithmic structure.

BO. Sparse circuits can still be too large for human comprehension

A token may activate dozens or hundreds of features across many layers. A behaviour can involve thousands of relevant feature transitions. Sparsity makes analysis easier and does not guarantee a small human-sized explanation.

Use receiver-specific pruning: keep features that materially affect this output, this safety property or this capability. Hierarchical summaries can compress low-level circuits into reusable modules.

Interpretability needs its own representation hierarchy.

BP. Behavioural completeness is the eventual standard

A feature system is more useful when it can account for a large fraction of behaviour variation. Reconstruction fidelity is only one proxy. Ask whether important model decisions can be predicted from sparse feature states and whether interventions on the sparse representation reproduce the native model’s responses.

A sparse explanation can be locally compelling and globally incomplete. Report residual unexplained behaviour.

Mechanistic interpretability should have a missing-mass concept: which behaviour remains outside the proposed feature/circuit model?

BQ. Nonlinear native features can be fragmented by a linear SAE

Most SAEs reconstruct activations as linear combinations of decoder atoms after a nonlinear sparse encoder. If the native semantic structure lies on curved or multiplicative manifolds, several linear atoms may approximate one nonlinear feature.

Feature splitting can therefore reflect model mismatch rather than semantic hierarchy. Nonlinear dictionary models could fit better and become harder to interpret.

The linear-feature hypothesis should be tested through residual structure and nonlinear probes.

BR. Non-negative sparse codes impose their own ontology

ReLU, TopK-positive and JumpReLU-style SAEs often produce non-negative coefficients. A signed native direction may then require separate positive and negative atoms or be represented through baseline shifts.

This can improve interpretability—feature “present” has a simple positive meaning—and can duplicate opposing states. A signed or symmetric sparse model makes a different representational choice.

Do not assume the sign structure of the SAE is the sign structure of the model’s native features.

BS. Rare-feature discovery needs active sampling

A feature firing once per ten million tokens will be poorly characterised by random galleries. Search corpora using preliminary labels, generate targeted prompts or condition sampling on activation.

Then return to an unbiased corpus to estimate prevalence and false positives. Discovery distribution and evaluation distribution must remain separate.

Rare-feature science is closer to anomaly detection than ordinary averaging.

BT. Monitoring thresholds need calibration and base rates

A feature can be used as a monitor only after thresholding. Even a detector with high sensitivity and specificity can produce many false alarms when the target state is extremely rare.

Evaluate precision-recall at realistic base rates. Calibrate thresholds on deployment-like traffic and revalidate after model or dictionary changes.

Feature interpretability does not bypass ordinary statistical decision theory.

BU. Feature monitoring should preserve an evidence route to raw context

If a feature triggers an alert, operators should see the source prompt, relevant tokens, neighbouring active features, activation magnitude and model response within privacy and access constraints.

The feature label alone is not sufficient evidence. World-return requires the ability to inspect the underlying behaviour that made the monitor fire.

This reduces the risk of taking an interpretability shorthand as ground truth.

BV. Interpretability can support safety without becoming a bypass manual

Research on refusal, jailbreak-related and faithfulness-associated features can help scientists understand model behaviour. Public educational treatment should explain evaluation principles and evidence boundaries without providing operational recipes for disabling safeguards.

The distinction is simple: describe how a feature-based safety analysis is validated; do not turn steering demonstrations into instructions for defeating controls.

Mechanistic transparency and responsible deployment are complementary goals.

BW. A feature reliability matrix is more informative than one interpretability score

For each important feature, score or annotate several axes: semantic coherence, held-out description accuracy, cross-seed consistency, cross-domain stability, reconstruction contribution, causal steering, causal necessity, redundancy, frequency and calibration.

Different receiver jobs care about different axes. A monitor needs high precision and stability. A circuit analysis needs causal relevance. A conceptual explainer prioritises semantic coherence.

No single scalar should erase this multidimensional reliability profile.

BX. Worked experiment: held-out semantic simulation

Select one hundred SAE features stratified by frequency. Generate descriptions using only a discovery split. On a separate set of text examples, ask an interpreter using the description to rank expected feature activation.

Compare predicted ranking with actual activation. Include simple lexical baselines and shuffled descriptions. Features whose labels fail held-out simulation should not be promoted to strong semantic claims.

Stratify by activation quantile to test whether descriptions cover only extreme examples.

BY. Worked experiment: dictionary consistency across seeds

Train five TopK SAEs with identical data and hyperparameters but different seeds. Compute pairwise atom similarity matrices using activation correlation or another defined feature relation. Solve one-to-one assignments and report the distribution of matched similarity.

Then inspect unmatched features for split/merge structure. Compare consistency with semantic-description similarity. This mirrors the 2026 feature-consistency agenda and reveals whether apparent concepts are reproducible.

BZ. Worked experiment: expansion factor and feature genealogy

Train SAEs at 4×, 8×, 16× and 32× expansion with matched sparsity frontiers. Match features across widths. Build a genealogy showing which broad features persist, which split and which disappear.

Measure whether child features have more coherent descriptions, distinct causal effects and improved reconstruction. This distinguishes useful semantic refinement from arbitrary overfragmentation.

CA. Worked experiment: intervention specificity with random controls

Choose a feature with a strong description and a measurable target behaviour. Steer it over natural-range magnitudes. Compare with random SAE features matched by frequency and decoder norm, plus random native directions.

Measure target response and a battery of collateral behaviours. A feature earns stronger causal status when it changes the intended behaviour more specifically than controls.

CB. Worked experiment: rare-feature enrichment and unbiased validation

Use activation search to find many examples of a rare feature. Interpret it from the enriched set. Then draw a large unbiased sample from ordinary traffic and measure the feature’s actual prevalence, false-positive contexts and activation distribution.

The two-stage design combines efficient discovery with honest base-rate estimation.

CC. The SAE interpretation evidence card

• Feature identity: model, layer, SAE version and index.
• Decoder geometry: direction and norm convention.
• Encoder rule: detector, bias, threshold or TopK semantics.
• Frequency: unbiased firing rate and active magnitude quantiles.
• Description: scoped human-readable hypothesis.
• Discovery evidence: top and representative examples.
• Counterexamples: false positives and false negatives.
• Held-out simulation: predictive validity of the description.
• Monosemanticity metric: method and its assumptions.
• Cross-seed consistency: matched-feature reliability.
• Cross-data stability: domains and languages tested.
• Cross-width lineage: splitting/merging relationships.
• Causal evidence: steering, ablation or mediation.
• Collateral effects: unintended behavioural changes.
• Redundancy: other features carrying similar information.
• Monitoring calibration: precision, recall and base rate if deployed.

CD. Final release gate for a longform SAE owner

1. The article separates sparse reconstruction from semantic interpretation.
2. It explains L1, TopK, BatchTopK and JumpReLU as different sparsity mechanisms.
3. It treats expansion factor as resolution rather than truth.
4. It reports reconstruction–sparsity trade-offs and functional fidelity.
5. It covers dead features, absorption, splitting, merging and redundancy.
6. It fixes scaling/gauge ambiguities before interpreting magnitudes.
7. It distinguishes decoder direction from encoder detection.
8. It includes activation-site and dataset provenance.
9. It treats monosemanticity as a measurement problem with current 2026 evidence.
10. It treats run-to-run feature consistency as a separate reproducibility axis.
11. It requires held-out semantic prediction and counterexamples.
12. It distinguishes association, sufficiency, necessity and mediation.
13. It controls for reconstruction artefacts during causal interventions.
14. It connects local features to transcoders/circuits without claiming dictionary completeness.
15. It preserves uncertainty when several dictionaries fit equally well.
16. It includes deployment calibration if features are used as monitors.
17. It avoids turning interpretability examples into operational safeguard-bypass instructions.

The mature interpretation of sparse autoencoders is therefore neither “SAEs solved model interpretability” nor “SAEs are merely arbitrary coordinate changes”. They are learned measurement instruments with a strong structural hypothesis: dense neural states can often be usefully reconstructed through a sparse, overcomplete vocabulary. That hypothesis can be tested. Reconstruction tells us whether the vocabulary carries the state; sparsity tells us whether it is economical; consistency tells us whether the vocabulary recurs; monosemanticity tells us whether humans can name its components; and intervention tells us whether those components matter to computation.

An SAE feature earns its name only after it survives more than reconstruction. It should recur, predict its own activation, withstand counterexamples and—when mechanistic claims are made—change the model in the way its interpretation predicts.