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Discovering Cross-Language Reasoning Invariance in LLMs with Geometry-Invariant Sparse Autoencoders

Igor Bogdanov, Changcheng Huang

arXiv:2608.23809v1cs.LGcs.AIcs.CL

TL;DR

The paper asks whether multilingual mathematical reasoning relies on shared internal features or language-specific computations. It compares baseline and contrastive SAEs across five models and tests shared features by causal patching. GI-SAE broadly increases geometric similarity, but functional interchangeability remains model- and architecture-dependent.

  • Problem

    It is unclear whether multilingual mathematical reasoning uses shared internal representations or language-specific computations, and whether geometric similarity predicts functional interchangeability.

  • Method

    The study selects aligned layers with CKA, trains baseline and GI-SAE models on six-language reasoning traces, and causally patches shared feature values while measuring KL divergence.

  • Results

    GI-SAE improves CKA at nearly every layer, but KL/feature improves only in the convergent profile, with an 83% win rate, and not in the saturated profile, with 6%.

  • Takeaways & Limitations

    Geometric similarity should not be used as a proxy for functional interchangeability; cross-language feature sharing requires per-model and per-architecture analysis.

  • Takeaways & Limitations

    The study uses MGSM only, has roughly 60 valid problems for Llama and Phi versus 151–202 for Qwen, and evaluates models up to 4B parameters.

Abstract

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Multilingual language models can solve the same mathematical problem in different languages, but it remains unclear whether they rely on shared features or on language-specific computations that only produce similar outputs. We study this question in five models from four families using the Multilingual Grade School Math (MGSM) dataset, with problems solved in English, German, French, Spanish, Russian, and Chinese, retaining problems with valid reasoning traces in all six languages and replaying those traces through the model to record representations at multiple layers. For each model, we first use Centered Kernel Alignment (CKA) to identify layers with cross-language alignment. At each selected layer, we train two sparse autoencoders (SAE): a baseline reconstruction-only model and a contrastive variant introduced in this work, the Geometry-Invariant SAE (GI-SAE). GI-SAE supplements the reconstruction loss with an Information Noise-Contrastive Estimation (InfoNCE) loss that trains the encoder to produce similar activations for traces of the same problem, regardless of language or token position. We then test whether the resulting shared features are functionally interchangeable by swapping their values between languages during the model's forward pass and measuring the resulting change in output, quantified by Kullback-Leibler (KL) divergence per feature. Although GI-SAE yields higher CKA and Jaccard similarity at nearly every layer, higher geometric similarity does not consistently imply greater functional interchangeability. We find that cross-language feature sharing is model- and architecture-dependent in this sample and appears at different depths in different models. GI-SAE primarily amplifies cross-language structure already present: the pattern is model-specific, with strengthening in Qwen, no functional benefit in Gemma, and mixed layer-dependent effects in Llama and Phi.

1. Introduction

The paper asks whether multilingual mathematical reasoning uses shared internal features or language-specific computations, and introduces a causal framework to test functional interchangeability across languages. Across five models, GI-SAE increases geometric similarity broadly, but functional gains depend on baseline sharing.

  • Cross-language shared features could enable one interpretability analysis across languages, whereas language-specific computations would require separate analyses.
  • The study addresses whether shared features are functionally interchangeable, whether sharing differs across model families, and whether geometric metrics predict interchangeability.
  • GI-SAE adds an InfoNCE loss to reconstruction training, encouraging same-problem activations to align across languages, and tests swapping through causal patching with KL divergence.
  • GI-SAE outcomes stratify by baseline shared fraction: 83% win rate for convergent profiles, no systematic benefit for low-sharing profiles, and 6% for saturated profiles.

2. Related Work

Prior work links multilingual models to shared latent spaces and identifies language-specific or shared SAE features, but behavioral consistency does not establish shared internal representations. This paper therefore evaluates cross-language SAE features through causal interchangeability rather than geometric similarity alone.

  • Research reports language-specific output-steering neurons alongside evidence that multilingual transformers route non-English inputs through shared latent spaces.
  • Existing multilingual SAE studies identify language-specific features and shared grammatical concepts, while this work asks whether shared features are functionally interchangeable.
  • Causal interventions test functional necessity because CKA can assign high similarity to representations that function differently.

3. Methodology and Measurement Framework

The framework selects cross-language-aligned layers, trains baseline and contrastive sparse autoencoders on residual-stream activations, and causally patches shared features. It separates representation geometry, feature overlap, and behavioral interchangeability using complementary metrics.

  • Residual Stream: Residual-stream vectors are cumulative states read and modified by downstream layers, so all SAE training and interventions operate on this representation.
  • Layer Selection via CKA Survey: Linear CKA selects contiguous layer ranges with highest mean pairwise alignment using pre-answer residual matrices matched by problem ID.
  • Sparse Autoencoders: Baseline SAEs reconstruct activations, while GI-SAE adds InfoNCE so same-problem activations across languages or token positions become positive pairs and different problems become negatives.
  • Patching Convention: Backward offsets from the last reasoning token address cross-language differences in absolute token positions and target answer-proximal reasoning structure.
  • Causal Feature Patching: Causal patching encodes target and donor states, swaps shared active feature values, decodes the patched state, and measures output disruption during a forward pass.
  • Evaluation Metrics: KL/shared feature normalizes cross-language disruption by intervened feature count and same-language disruption; lower values indicate greater interchangeability.
  • Decision Rule: GI-SAE wins at a layer when KL/featGI is lower than KL/featbaseline.
  • Evaluation Metrics: CKA measures rotation-invariant matrix geometry, Jaccard and shared fraction measure active-index overlap, and causal patching tests whether swapping shared values preserves behavior.

4. Experimental Setup

The experiments compare five models from four architecture families on a common six-language MGSM reasoning protocol. Analyses are conditional on problems producing valid reasoning traces and correct answers in every language.

  • Models and Families: The study evaluates Qwen3-1.7B, Qwen3-4B, Llama-3.2-3B, Gemma-3-4B, and Phi-3-mini-4k, spanning 1.7B to 4B parameters.
  • Layer Selection: Layer sweep ranges differ because models reach peak cross-language CKA at different depths; Phi-3 peaks near 31% depth, unlike others at 88–100%.
  • Multilingual Protocol: MGSM contains 250 grade-school math problems solved independently in English, German, French, Spanish, Russian, and Chinese.
  • Validity Criterion: A problem is retained only when the model solves it correctly and produces an extractable reasoning trace in all six languages.
  • Training Protocol: Valid problems are split into train, validation, and test sets at 70/15/15%, while both SAE variants use shared optimization settings including learning rate 10^-4 and batch size 256.

5. Results

GI-SAE increases cross-language geometric similarity, but functional interchangeability depends on baseline sharing and varies across models and layers. Qwen benefits consistently, Gemma does not, and Llama and Phi show mixed depth-dependent patterns.

  • Cross-language feature sharing: GI-SAE increases sharing for Qwen and Phi, leaves Gemma unchanged, and only marginally improves Llama.Gemma remains at 87–96% baseline sharing; Qwen rises to 16–69% and Phi to 9–39% with GI-SAE.
  • Geometric similarity: GI-SAE improves CKA at nearly every observation across all five models.The figure compares CKA gains with KL/feature changes to distinguish geometric from functional benefit.
  • Model-specific functional effects: Qwen shows joint geometric and functional gains, with GI-SAE wins on 11/11 and 14/14 test layers.CKA changes range from +0.14–0.23 for Qwen3-1.7B and +0.04–0.32 for Qwen3-4B.
  • Model-specific functional effects: Gemma gains CKA by +0.25–0.41 but GI-SAE wins on 0/16 test layers.Gemma occupies the lower-right quadrant, indicating geometric improvement without functional benefit.
  • Model-specific functional effects: Llama and Phi show depth-dependent functional effects despite geometric gains.Llama wins on 7/17 layers, mainly at L23–L27; Phi wins at L1–L6 but loses from L7 onward.

6. Discussion

GI-SAE’s geometric gains do not uniformly translate into functional interchangeability: outcomes depend on baseline sharing and model profile, within a study whose taxonomy and evaluation scope remain limited.

  • Profile-dependent effects: GI-SAE wins most often in the convergent profile, but shows little or no benefit in low-sharing and saturated profiles.The phase diagram compares KL/feature benefit against baseline shared fraction; points above the dashed line indicate GI-SAE wins.
  • Interpretive scope: In low-sharing layers, GI-SAE can substantially raise CKA without improving whether shared features can be swapped without disrupting output.The reported CKA increase reaches Δ up to +0.67 in Llama early layers.
  • Limitations and future directions: The taxonomy is descriptive because each non-Qwen family contributes one checkpoint, while Qwen provides only a within-generation scale check.This constrains how confidently the observed profiles can be treated as family-wide patterns.
  • Limitations and future directions: The study is limited to MGSM, has fewer valid problems for Llama and Phi, and evaluates models up to 4B parameters.The passage also notes fixed hyperparameters and several scope constraints for generalization.
  • Profile-dependent effects: The three representative profiles differ in baseline sharing and layerwise CKA and KL/feature behavior across Gemma, Qwen3-4B, and Phi-3-mini.Lower KL/feature indicates greater functional interchangeability, with shading marking GI-SAE advantage or disadvantage.

7. Conclusion

The paper introduces GI-SAE and evaluates cross-language feature sharing across five models and four families. It finds that geometric similarity is not a reliable proxy for functional interchangeability, which varies by model and profile.

  • Conclusion: GI-SAE and a baseline SAE are evaluated under a common six-language mathematical reasoning protocol across five models from four families.The study compares progressive Qwen, saturated Gemma, and mixed Llama and Phi sharing patterns.

Impact Statement

This foundational mechanistic-interpretability study examines existing open-source language models without introducing new capabilities, models, or datasets. Its potential impact is improved understanding of multilingual model behavior for future interpretability and auditing research.

  • The study is foundational mechanistic-interpretability research on internal representations in existing open-source language models.
  • It does not introduce new model capabilities, generate content, or release new pretrained models or datasets with direct deployment implications.
  • The empirical study covers five models from four families using 78 layer observations and two SAE variants.
  • Qwen3-1.7B reports GI-SAE wins on 11/11 layers, while Qwen3-4B reports wins on 14/14 test layers.
  • Across the control evaluation, the overall pass rate is 89% (25/28), with GI-SAE passing the different-problem control on 7/7 layers versus 5/7 for baseline.

D.1. Per-Model Experimental Pipeline

The pipeline replays valid multilingual reasoning traces, trains baseline and GI-SAE models, evaluates geometric similarity, and causally patches shared feature values. Results are restricted to problems solved correctly with extractable traces in all six languages.

  • D.1. Per-Model Experimental Pipeline: The pipeline extracts activations, trains two SAE variants, evaluates CKA and Jaccard, performs causal patching, and measures downstream effects.
  • D.1. Per-Model Experimental Pipeline: Reasoning traces are replayed exactly, and residual-stream vectors are recorded at target layers and backward-aligned token positions.
  • D.1. Per-Model Experimental Pipeline: Shared features are active in both target and donor encodings, and patching swaps donor values only at those shared indices.
  • D.1. Per-Model Experimental Pipeline: The patched residual replaces the target state, after which the forward pass continues through downstream layers and KL divergence is measured on final reasoning-token distributions.
  • D.1. Per-Model Experimental Pipeline: A problem is retained only when the model solves it correctly and produces an extractable reasoning trace in all six languages.
  • D.1. Per-Model Experimental Pipeline: Valid problem counts vary substantially across models: 202 for Qwen-4B, 151 for Qwen-1.7B, 137 for Gemma, 63 for Phi, and 60 for Llama.

F. SAE Training Diagnostics

GI-SAE training remains broadly healthy but trades off reconstruction quality against contrastive invariance. Its functional benefits depend on the baseline sharing profile, with strong gains in convergent layers and no systematic benefit in low-sharing or saturated profiles.

  • F. SAE Training Diagnostics: Baseline SAEs maintain more than 99% alive features, while GI-SAE remains above 90% in most layers but drops to 84–88% in selected Llama and late Qwen3-1.7B layers.
  • F. SAE Training Diagnostics: GI-SAE validation loss is typically 1.5–2.0× baseline, with spikes above 3× at selected Qwen3-1.7B layers.
  • F. SAE Training Diagnostics: With contrastive weight w=1.0 on Qwen3-1.7B L20, CKA improved from 0.503 to 0.673 and KL/feature improved from 0.0070 to 0.0003 relative to baseline.
  • F. SAE Training Diagnostics: In the convergent profile, raw KL drops from 0.43 to 0.25 while sharing increases from 42% to 60%; in the saturated profile, raw KL rises from 1.58 to 3.52 with sharing near 89%.
  • F. SAE Training Diagnostics: The low-sharing confidence interval [33%, 64%] straddles 50%, while Qwen intervals lie above 50% and Gemma lies below 50%.

I. Qualitative Feature Audit

The qualitative audit identifies shared SAE features tied to answer boundaries, answer-transition language, and reasoning-step or format markers. A normalized-space sensitivity check preserves the aggregate Qwen conclusion but makes some low- and mixed-profile layer verdicts convention-sensitive.

  • I. Qualitative Feature Audit: The audit identifies consistently shared features active across at least four of six languages at the pre-answer token.
  • I. Qualitative Feature Audit: Answer-boundary markers fire on punctuation at the reasoning-to-answer transition, which is language-invariant because final answers are numeric.
  • I. Qualitative Feature Audit: Answer-transition phrase feature F3882 captures the syntactic role introducing an answer across language-specific forms.
  • I. Qualitative Feature Audit: Step and format markers remain active even at L3, suggesting early detection of discourse structure.
  • I. Qualitative Feature Audit: The aggregate win rate across valid layers remains 4/6 under both native-scale and normalized-space patching conventions.
  • I. Qualitative Feature Audit: Normalized-space patching changes absolute shared-feature counts and flips several individual low- and mixed-profile layers.
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