A transformer can be built from operators that are legible by construction -- bounded, named units that read as fuzzy set operations rather than dense activations -- but legibility must be pressed for during training, and the pressure has a failure mode. A crispness penalty meant to sharpen a bounded operator into a decisive detector instead collapses it into a dead constant. An identity, E[v(1-v)] = mu(1-mu) - var, shows why -- the penalty is a variance-minimizer blind to the difference between a live detector and a constant -- and names the fix: a per-channel variance floor, the target legibility metric written as a loss, which recovers both legibility and quality. A learned per-unit fraction then retires the hand-set reserved-GELU partition of prior work: given the choice the model keeps no unit as pure GELU and routes 87% of its load-bearing computation through crisp operators. The result is the most legible transformer we have built -- 78% of its feed-forward operands and 50% of its attention value channels are crisp-and-contextual detectors, and per-head legibility rises from 18% in shallow layers to 78% in deep ones. Read in the correct rotated per-layer frame, these units separate a clean detection (what a unit responds to) from a harder naming (what its output decodes to); and because the objective makes each unit crisp and sparse, edits to them are far more local -- 50-184x in the deep layers where the edit sites concentrate -- and can target explicit conjunctions a single neuron cannot express. Finally, a between-unit decorrelation pressure exposes a legibility dial: it trades a circuit's reuse for independence at no quality cost, turning concepts into single, surgically editable units and a prediction into a short explanation read off a handful of named operations. Quality holds at parity with a conventional baseline throughout.
Sparse autoencoders (SAEs) decompose internal activations of neural networks into sparse linear combinations of learned features by fitting an overcomplete dictionary $\mathbf{W}\in\mathbb{R}^{m\times n}$ with $m<n$, and inferring a sparse code $\mathbf{x}\in\mathbb{R}^n$ from $\mathbf{h}\approx\mathbf{W}\mathbf{x}$. This inference problem closely resembles the canonical setup of compressed sensing, but dense decoders requires $O(mn)$ learned values, which becomes costly at large feature counts. We introduce Expander SAEs: TopK SAEs whose decoder and tied encoder are supported on a left-$d$-regular expander mask with $d\ll m$, learning only $dn$ decoder values while keeping the sparse-coding problem $(m,n,k)$ fixed. The same structure reduces storage and turns the matching-pursuit correlation step $\mathbf{W}^\top \mathbf{r}$ in OMP into an $O(dn)$ gather-and-reduce operation. Our experiments show that across Pythia-70M/160M, Qwen2.5-3B, and Llama-3.2-1B residual-stream activations, varying $d$ traces a consistent storage--fidelity frontier, and that at the most compressed modern-LM setting, Qwen2.5-3B with $d=7$ uses $293\times$ fewer learned decoder values than the full dense decoder while retaining $84$% of dense CE-loss recovered. Control experiments show that the improved storage--fidelity tradeoff is driven by sparse, diverse decoder support structure rather than by fewer learned decoder values, and that when sparse and dense decoders are compared at matched parameter count, part of the remaining gap comes from encoder amortisation. On the theoretical side, we show that expansion and column flatness are sufficient for identifiability of noiseless $k$-sparse codes, and we derive complementary sufficient conditions under which OMP recovers the support exactly.
Named Entity Recognition (NER) in low-resource languages suffers from limited supervision and a lack of high-quality pretrained embeddings. Biological olfaction, which relies on sparse combinatorial coding through receptor and glomerular organization, offers a compelling paradigm for learning robust representations under uncertainty. In this paper, we introduce a receptor-glomerular bottleneck - a novel, biologically-inspired olfactory architecture - between standard token embeddings and a BiLSTM-CRF sequence model. We evaluate our architecture across six multilingual datasets trained entirely from scratch (without pre-trained embeddings) under varied data-scale conditions, including a strict 1k-sentence low-resource control. Our results demonstrate that introducing a representation bottleneck yields F1 score improvements under severe data scarcity, primarily by acting as a powerful regularizer. Under the 1k capped training condition, at least one olfactory-inspired configuration achieves the highest mean F1 score across all six datasets. While these improvements represent near-ties with generic bottleneck controls for most languages, the olfactory architecture provides a significant advantage in languages like Bangla (+6.23% F1 over standard baseline and +8.47% F1 over the best control baseline) where generic bottlenecks degrade performance. We also observe improvements in the ultra-low-resource Telugu setting (+4.43% F1) at full-scale, and find that sparse specialization naturally emerges within the receptor layer. Our findings suggest that structured sparse coding inspired by olfactory networks serves as an effective inductive bias and regularizer when representations must be learned from limited or noisy supervision.
Sparse Autoencoders (SAEs) have found success parsing neural representations into interpretable concepts, providing a basis for understanding and control. However, what exactly SAEs extract, and, correspondingly, the scientific conclusions we can draw from them, are not obvious. Empirically, the proof is in the pudding: SAEs learn interpretable features. Theoretically, we lack a clear account of what properties a 'concept' must satisfy for an SAE to extract it. There has been extensive identifiability work studying the conditions under which sparse coding recovers ground-truth features; however, these approaches tends to focus on simple data-generating models (e.g. sparse independent features) which poorly approximate the internet-swallowing language-model representations on which SAEs are trained. Here, avoiding data-generating models, we ask simply what properties any dictionary learning optimum must satisfy. Concretely, we extend local optimality analyses (Gribonval & Schnass, 2010) to the nonnegative joint-optimisation problem that vanilla SAEs approximate, and derive constraints relating optimal SAE features to their distributions. We use these constraints to explain a range of observed SAE behaviours - hierarchical splitting & absorption, the structure of residuals, and dense antipodal features - each reflecting how L1+nonnegativity interact with data to structure optimal dictionaries. Finally, we construct a novel large-dictionary convex problem and explore the wide atom-per-datapoint limit. In sum, we hope to tease model assumptions from unexpected observations, letting us learn more from SAEs' successes and provide principles for designing their successors.