Topographic Training Concentrates Causal Circuits Without Improving Neuron Monosemanticity

Gautam Ranka, Shubham Pandere*, Aiden Ross D'souza*

IvLabs, VNIT Nagpur, India  ·  *Equal contribution

Mechanistic Interpretability Workshop @ ICML 2026, Seoul

📄 Paper 💻 Code ▶ Video

TL;DR: A sub-0.1k-parameter training-time penalty (TopoLoss) concentrates a vision transformer's causal circuits into tight spatial clusters — 2.79× more causally sufficient than random neuron sets — yet leaves individual neurons no more monosemantic. Interpretability gains here are a property of circuits, not neurons.

2.79× causal sufficiency of topographic clusters vs. random
d = 0.95 effect size (p < 0.0001)
<0.1k extra parameters added by TopoLoss

Abstract

Mechanistic interpretability of vision transformers seeks to decompose model computation into human-readable units, but learned representations entangle many concepts in each neuron. Feature superposition is widely treated as the central obstacle to this decomposition, yet most mitigations (sparse autoencoders, dictionary learning) are post-hoc and leave the underlying network unchanged. We ask whether a spatial-locality training loss (TopoLoss) can act as a lightweight, training-time prior that improves interpretability of standard mech-interp tools. Training ViT on ImageNet-100 across multiple TopoLoss weights α, we measure causal sufficiency of topographic clusters via activation patching and feature geometry via sparse autoencoders fit to the same residual stream. At α = 1.0, topographic clusters are 2.79× more causally sufficient than random unit sets of the same size, with the effect increasing monotonically in α. SAE L0 sparsity decreases by 11% and dead-feature fraction rises 19-fold, yet standard neuron-level monosemanticity scores are unchanged, indicating that topographic pressure acts at circuit level, concentrating causal mass into spatially local structures without disentangling individual neurons. This dissociation suggests current neuron-level monosemanticity metrics are insensitive to a class of real interpretability gains, and positions cheap architectural priors as a viable training-time complement to post-hoc tooling.

Method: TopoLoss

$$\mathcal{L} = \mathcal{L}_{\text{CE}} + \mathcal{L}_{\text{topo}}, \qquad \mathcal{L}_{\text{topo}} = \alpha \sum_{(i,j)\in\mathcal{E}} \lVert w_i - w_j \rVert^2$$

TopoLoss adds a weight-smoothness penalty between units that are adjacent on a fixed 2D topographic grid, pushing functionally similar units to co-locate — echoing cortical maps in biological vision. Applied to the attention-projection weights (attn.proj) of all 12 ViT-S/16 blocks; units are arranged on a $\sqrt{d} \times \sqrt{d}$ grid matching the head dimension. α controls pressure strength: it adds fewer than 0.1k parameters and costs about 1.6 pp accuracy at the strongest setting (α = 1.0).

Setup: ViT-S/16 trained on ImageNet-100, α ∈ {0.0, 0.1, 1.0}, three seeds {42, 123, 456}. A 4-layer TinyViT (d = 128) serves as a preliminary scale probe.

Three-panel overview figure. Left: TopoLoss schematic showing a random baseline weight grid transforming into a topographically organised grid after adding the L_topo penalty. Centre: SAE sparsity plot — mean active features (L0) decreases 11% at alpha=1.0 while dead-feature fraction rises sharply (19-fold). Right: causal purity plot — cluster/random patching ratios for Baseline, TopoWeak, and TopoStrong; mean rises from 1.68 to 2.79x at alpha=1.0 (p less than 0.0001).
Figure 1. (a) TopoLoss penalises weight differences between adjacent units, pushing functionally similar units to co-localise. (b) SAE L0 decreases monotonically with α, while dead-feature fraction rises 19-fold at α = 1.0. (c) Per-class causal patching ratios. Topographic clusters at α = 1.0 are 2.79× more causally sufficient than random unit sets (p < 0.0001, d = 0.95).

Three Interpretability Axes

We audit TopoLoss on three independent questions about what changes when you impose topographic structure at training time.

H1

Monosemanticity

Do individual neurons become more class-selective?

Metric: entropy-based selectivity Mᵤ

NULL
H2

Superposition

Does SAE-measured feature complexity decrease?

Metric: SAE L0 norm + dead-feature fraction

SUPPORTED *
H3

Causal Purity

Are topographic clusters more causally sufficient than random sets?

Metric: activation-patching logit-delta ratio

SUPPORTED

H3 — Causal circuits concentrate

SUPPORTED

Topographic clusters carry disproportionate causal weight. At α = 1.0, topographic clusters reach a patching ratio of 2.79 ± 1.24 vs 1.68 ± 0.42 at baseline — a 66% increase. The effect is monotone in α and robust across all three seeds independently (per-seed ratios: 2.53, 2.89, 2.97). Multi-class expansion (n = 60): p < 0.0001, d = 0.95.

1.68 ± 0.42 Baseline — near the random line
Scatter plot of per-class causal patch ratios for Baseline (alpha=0), TopoWeak (alpha=0.1), and TopoStrong (alpha=1.0). Mean ratio rises monotonically: 1.68, 2.27, 2.79. Both TopoWeak and TopoStrong are significantly above baseline (p less than 0.0001). The dashed line marks ratio=1, the random baseline.
Figure 2. H3: Causal patching ratios across 20 classes × 3 seeds (n = 60). Each point is the mean cluster/random logit-delta ratio for one class × seed combination. Both TopoWeak (α = 0.1) and TopoStrong (α = 1.0) are significantly above baseline (p < 0.0001, Cohen’s d ≥ 0.61). The dashed line marks the random-cluster baseline (ratio = 1).

Spatial coherence: The top-k = 16 class units at α = 1.0 are 6.1% more spatially concentrated than baseline (mean grid distance 9.948 vs 10.599; p = 0.0005, d = 0.66), confirming the selected units genuinely co-localise on the topographic sheet.

H2 — Superposition drops

SUPPORTED *

At α = 1.0, SAE L0 decreases 11% (1133.8 vs 1273.9; p = 0.007, d = 7.06), and dead-feature fraction rises 19-fold (3.8% vs 0.2%). The L0 reduction is distributed across depth (layers 3–10, peak at layer 10). The α = 0.1 model shows neither effect reliably, confirming a threshold.

Caveat: We treat the L0 reduction as suggestive, not confound-free. The α = 1.0 model costs ~1.6 pp accuracy (61.2% → 59.6%), and OLS regression of L0 on validation accuracy across all 9 ViT-S checkpoints yields R² = 0.72 (p = 0.004) — accuracy explains substantial L0 variance. Two signs the effect isn’t purely accuracy-driven are reported in the paper. An accuracy-matched ablation is left to future work.
Two line charts showing SAE L0 and dead-feature fraction per ViT layer 0 through 11 for Baseline (alpha=0, blue), TopoWeak (alpha=0.1, orange dashed), and TopoStrong (alpha=1.0, green). L0 for alpha=1.0 diverges below baseline at layer 3 and remains lower through layer 10. Dead-feature fraction for alpha=1.0 is sharply elevated, especially in early layers.
Figure 3. Layer-wise SAE L0 (seed 42). The α = 1.0 curve diverges from baseline at layer 3 and remains below through layer 10; α = 0.1 tracks baseline. Layer 6 (used for main-text H2/H3 analyses) lies in the middle of the effect band.

H1 — Neurons don’t change

NULL

Entropy-based selectivity Mᵤ is flat across α: 0.234 ± 0.002 (α = 0) vs 0.236 ± 0.002 (α = 1.0); Δ = 0.0019, not significant. The SAE-feature-level Mᵤ is also null, including at a larger 25k-image evaluation.

Interpretation: Monosemanticity measures how one neuron spreads across classes — a marginal statistic insensitive to the joint arrangement of many neurons. TopoLoss reorganises neurons into functionally coherent spatial clusters (H3) without changing per-neuron selectivity (H1). Consistent with the superposition hypothesis: spatial co-location, not per-neuron selectivity, is the relevant unit of circuit purity.

Takeaway

A penalty of fewer than 0.1k parameters, added to the training loss, systematically shifts circuit geometry in a way existing tools can measure (d = 0.95 for H3) — at a modest cost (~1.6 pp accuracy at α = 1.0) and without disentangling individual neurons. This positions cheap architectural priors as a viable training-time complement to post-hoc tooling like SAEs, and suggests current neuron-level monosemanticity metrics are blind to a class of real interpretability gains.

Scale caveat: In a 4-layer TinyViT the H2 trend replicates (L0 −2.1% at α = 1.0) but H3 saturates at just 1.42×, far below the ViT-S result of 2.79× — consistent with a depth/scale requirement for circuit formation. Limitations: single dataset (ImageNet-100), partial accuracy confound on H2, single seed for TinyViT.

Video Walkthrough

Citation

@inproceedings{ranka2026topographic,
  title     = {Topographic Training Concentrates Causal Circuits
               Without Improving Neuron Monosemanticity},
  author    = {Ranka, Gautam and Pandere, Shubham and D'souza, Aiden Ross},
  booktitle = {Mechanistic Interpretability Workshop at the 43rd
               International Conference on Machine Learning (ICML)},
  year      = {2026}
}