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Five Architectures, One Shared Space Cross-Model Representational Alignment Without Weight Merging
Oladimeji Anthonio, Dimeji Olawuyi, Oloruntoba Ajayi, Temilola Aderemi
TL;DR
Weight-based merging is not suited to independently trained models with different architectures, prompting the question of whether their activations can provide a shared representation while the models remain frozen. The paper tests that correspondence, constructs a common space, and finds strong final-layer retrieval but no routing advantage over always using the strongest model.
Problem
Weight averaging depends on models sharing a parameterization, leaving open whether independently trained models with different architectures can be related through activations while remaining frozen.
Source: §1 · p. 1Method
The authors test cross-model representational correspondence and use Generalized Procrustes Analysis to build a shared space from frozen-model activations.
Source: §1 · p. 2Results
At the final layer, shared-space retrieval reaches 0.91–0.93, but routing does not outperform always using the strongest model.
Source: §10 · p. 12Takeaways & Limitations
For these five models, activation alignment supports a shared representational space, but the space provides no better way to choose which model to use on the tested tasks.
Source: §10 · p. 12The study covers only 1–2B-parameter base checkpoints, with small probe and task sets; larger or instruction-tuned models were not tested.
Source: §9 · p. 12
Abstract
from arXiv · showhide
Combining language models is usually done by averaging weights, which works only when the models share a common starting point. We ask whether five frozen, independently trained models with different internal designs (four transformers from four organizations and one state-space model) can be related through their activations alone, with no weight merging. Using permutation tests, an independent replication probe set, depth sweeps, pooling ablations and confound checks, we find statistically robust correspondence between how the five models organize the same text. It is strongest at each model's final layer and weaker for the state-space model, whose weakest pairing is still 35-43 times chance. We use this to build a shared space with Generalized Procrustes Analysis, in which no model is a fixed reference. At the final layer, mean held-out retrieval accuracy over the twenty ordered model pairs is 0.91 to 0.93 on two probe sets (chance 0.03), beating pairwise linear maps (0.82 to 0.85), but the space is worse than those maps in the middle layers. It transfers to ordinary English of unfamiliar grammatical form and fails on mathematical and logical notation. We then test whether the space supports a router that picks among the five models on gradeable tasks. Across four task types we find no case where routing beats always using the strongest model, and we report this as a negative result. A correctness signal that looked strong turned out to be category membership in disguise. All experiments ran on one laptop without a GPU.
1 Introduction
Weight averaging depends on models sharing a parameterization, which does not hold for models differing in architecture, training data, and size. This paper asks whether frozen models can instead be related through their activations, then tests representational correspondence, builds a shared space, and evaluates routing.
1 Introduction
- Weight averaging and permutation-based merging rely on shared parameterization, limiting their applicability to models with different architectures, training data, and parameter counts.The paper therefore investigates activations rather than model weights.Source: §1 · p. 1
- The paper asks whether very different frozen models can be related through their representations of the same input, without merging weights.The motivation is that a small trainable component could combine their outputs while preserving each model’s original behavior.Source: §1 · p. 1
- The study tests cross-model correspondence among five small models from different groups, including a state-space model with a different sequence-mixing mechanism from the transformers.The experiments are designed to test whether early signals hold up and to identify where apparent signals are artifacts.Source: §1 · p. 2
- The authors build a shared space using Generalized Procrustes Analysis and compare it with pairwise linear maps across model depths.The space gives no model privileged reference status.Source: §1 · p. 2, Figure 1 · p. 2
- Across four task types, routing among the models does not outperform always using the strongest model.The routing search includes a task chosen because prior literature predicted an architectural difference.Source: §1 · p. 2
2 Related Work
Related work spans weight merging, representational similarity, alignment methods, state-space model limitations, and model routing. This paper uses activation-space alignment to study correspondence across unlike architectures and whether it can inform routing.
2 Related Work
- Weight averaging and permutation alignment can merge some networks with shared parameterization, but do not provide a parameter-level correspondence for models with different widths, depths, and sequence-mixing mechanisms.The paper therefore works on activations instead of weights.Source: §2 · p. 2
- Centered kernel alignment and model stitching compare network representations, while the Platonic Representation Hypothesis argues that independently trained models converge as they scale (Huh et al., 2024).This study tests the idea at small scale, including a non-transformer, and asks whether correspondence can form an explicit common space.Source: §2 · p. 3
- Generalized Procrustes Analysis aligns multiple configurations to their mean, giving every model equal footing through orthogonal maps.The paper contrasts this with pairwise alignment and relative representations, which describe inputs by similarity to anchors.Source: §2 · p. 3
- Associative recall is a documented weakness of selective state-space models because they compress history into a fixed-size state, unlike attention’s ability to index any position (Arora et al., 2023).The paper uses this contrast to design a targeted task.Source: §2 · p. 3
- Prior routing systems learn to select between stronger and weaker models per query, whereas this study asks whether an activation-derived space built without task labels can route among five models..Source: §2 · p. 3
3 Models and Experimental Setup
The experiments use five frozen base models, two author-written probe sets, token-mean activations, and permutation-based measures of representational agreement.
3 Models and Experimental Setup
- Five frozen base models were run in inference mode on one laptop; Mamba ran without accelerated kernels and was about 25× slower per probe than the transformers.Table 1 identifies the five models used throughout.Source: §3 · p. 3, Table 1 · p. 3
- The authors wrote 200 short probes across ten categories, then created a second 200-probe set with the same categories and entirely different content as an independent replication.Both sets were written by the authors.Source: §3 · pp. 3–4
- Representations are token-mean hidden states, usually taken from each model’s final layer; mean pooling avoids several single-token artifacts but may understate Mamba’s category separation.Mamba’s last-token vectors separated categories better than its mean-pooled vectors.Source: §3 · p. 4
- Agreement is measured by Spearman correlation of pairwise cosine distances and by nearest-neighbor matches, with permutation nulls and a 160-probe whole-set analysis excluding code and multilingual text.For 160 probes, nearest-neighbor chance is 0.0063; nulls use 2,000 permutations.Source: §3 · p. 4
4 Stage 1: Per-Model Geometry
PCA projections reveal different depth-wise geometry across the models: transformer middle layers are compressed, Mamba is not, and all models reorganize at their final layer.
4 Stage 1: Per-Model Geometry
- 98–99.8% of variance lies in the first two components across transformer middle-layer plateaus, while Mamba stays at 10–20% before reaching 56% at its final layer.For Qwen, the plateau spans layers 5–20.Source: §4 · p. 4
- All five models reorganize sharply at their own final layer, where each prepares its output.Even the four transformers look unlike one another in their middle layers.Source: §4 · p. 4
- Because middle layers differ across models and final-layer transitions occur at each model’s own depth, matching equal-numbered layers is not meaningful.These patterns motivate analyses that treat depth as a variable.Source: §4 · p. 4
5 Stage 2: Is There Cross-Model Correspondence?
Probe-level correspondence is robust, strongest at the final layer, and weaker—but still substantial—for Mamba-involving pairs. An initial category-centroid signal was marginal and was not used as a foundation for later analyses.
5.1 A first result that did not hold up
- Category-centroid agreement was ρ = 0.914 across ten categories but ρ = 0.808 across eight; random partitions agreed strongly, so the authors did not build on this evidence.Source: §5.1 · pp. 4–5
5.2 Probe-level agreement
- At the final layer, mean Spearman ρ was 0.64 (set 1) and 0.65 (set 2), while nearest-neighbor match was 0.324 and 0.407, significant at every tested depth.The final-layer nearest-neighbor rates were 50× and 65× chance, respectively; agreement was weaker in middle layers and recovered at the final layer.Source: §5.2 · p. 5, Figure 2 · p. 5
5.3 The state-space model is weaker, in degree and not in kind
- Final-layer Mamba-involving pairs averaged 0.230 vs 0.388 (set 1) and 0.306 vs 0.474 (set 2), yet the weakest Mamba–OLMo pair remained 35–43× chance.The gap between Mamba-involving and transformer-only pairs was largest at the final layer and near zero at 75% depth.Source: §5.3 · p. 5, Figure 3 · p. 6
- Pythia, SmolLM2, and OLMo reorganize on entering the final layer and their changes correlate positively, while Mamba’s changes correlate weakly or negatively with other models.The authors describe this pattern without claiming a cause.Source: §5.3 · pp. 5–6
6 Constructing a Shared Space
The paper compares pairwise linear maps with a consensus space built using Generalized Procrustes Analysis (GPA). At the final layer, GPA improves held-out retrieval over pairwise maps, but it loses in the middle layers and its dimension plateau reflects the training-sample limit.
6.2 Generalized Procrustes Analysis
- GPA reduces centered activations to a common dimension, then rotates each model toward an evolving mean consensus; no model is a fixed reference.The shared-space fitting uses one orthogonal matrix per model.Source: §6.2 · pp. 6–7
- At the final layer, GPA mean held-out retrieval accuracy was 0.907 (set 1) and 0.928 (set 2), above pairwise maps’ 0.816 and 0.854.Figure 4 reports the ordered-pair results; the weakest cells involve Mamba and Qwen on set 1.Source: §6.1 · p. 6, Figure 4 · p. 7, §6.2 · p. 7
- At 25–75% depth, pairwise maps reached 0.56–0.67 retrieval accuracy versus GPA’s 0.26–0.40; GPA wins only at the final and lexical embedding layers.The authors therefore construct the shared space at the final layer only.Source: §6.2 · p. 8, Figure 5 · p. 8
- The accuracy plateau from D = 200 reflects the 128-probe folds’ limit of 127 informative PCA components; the released space is fit on all 400 probes.Source: §6.2 · p. 8
7 Stage 3: Where Does the Shared Space Stop Working?
The shared space generalizes beyond its eight fitted prose categories, but its transfer depends on the kind of content: unfamiliar natural-language forms work better than symbolic notation. Adding code and multilingual probes to the fit improves retrieval on those probes with a small cost on the original categories.
Unseen categories
- On unseen code and multilingual probes, never used in fitting, pairwise retrieval accuracy was 0.29 to 0.36—roughly 6 to 14 times chance in that test—far below the in-distribution 0.9 range.Source: §7 · p. 8
Unseen content types
- 0.90 retrieval accuracy for imperative commands and bare lists contrasts with 0.30 for mathematical expressions; symbol-dense notation scored 0.24, while number-dense prose scored 0.81.The authors characterize the boundary as ordinary natural-language vocabulary versus non-standard symbolic notation, based on ten items per content type.Source: §7 · p. 9, Figure 6 · p. 9
Refitting
- 0.45 →0.84 held-out accuracy on code and multilingual probes came with a 0.02 decrease on the eight original categories, from 0.92 →0.90.The released space was refit on all ten categories.Source: §7 · p. 9, Figure 6 · p. 9
8 Stage 4: From Representations to Task Performance
Across four task designs, routing among these models did not outperform always using the strongest model. The apparent correctness signal instead tracked question category, and a category-based router selected the same model for every category.
8.1 A signal that was a confound
- Pythia’s apparent correctness signal tracked question category: it answered 0/24 arithmetic questions correctly and 21/24 factual questions correctly, while factual-only tests did not detect a fine-grained signal.The authors caution that the factual-only tests had few incorrect answers and low power.Source: §8.1 · pp. 9–10, Figure 7 · p. 10
8.2 A router, and four searches for a case where it helps
- 0.979 category-classification accuracy did not produce differentiated routing: the category-based router selected Qwen, tied with SmolLM2, for every category.This matched always using the best model.Source: §8.2 · p. 10
8.3 Closing the routing question
- Across four task designs, no model outside Qwen or SmolLM2 topped any row, so routing showed no demonstrated value over always using the strongest model.The authors describe this as a negative result for this model set and these tasks, while noting that crossovers on other domains cannot be ruled out.Source: Table 2 · p. 10, §8.3 · p. 11
8.2 A router, and four searches for a case where it helps
- Mamba’s associative-recall accuracy was 57.1% for the first item and 28.6% for the last, but the small cells made this primacy pattern a hypothesis for replication, not a finding.Qwen and SmolLM2 were at or near ceiling in every position; OLMo scored 0% on every task, which the authors treat as checkpoint-specific under greedy decoding.Source: §8.2 · p. 11, Figure 8 · p. 11
9 Discussion and Limitations
Different-architecture models show final-layer representational correspondence, but the authors caution that it does not establish a link between Mamba’s alignment and recall performance. The discussion also limits the conclusions by model scale, probe construction, retrieval-pool size, dimensionality, pooling, and task coverage.
9 Discussion and Limitations
- Final-layer correspondence supports an unlabeled shared space, but its category information is not correctness information, and its relation to Mamba’s recall weakness was not tested.The authors report no detectable information about whether a model answers a question correctly and do not claim a causal link between representational agreement and associative recall.Source: §9 · pp. 11–12
- The results cover one run of 1–2B-parameter base checkpoints and four task types, not instruction-tuned or larger models or other routing domains.Most task sets contain 24–48 questions, and some null tests are low-powered.Source: §9 · p. 12
- Both probe sets were author-written, so an independent second set reduces but does not remove concern that writing style inflates agreement.The authors note that probes drawn from natural corpora would remove this particular concern.Source: §9 · p. 12
- Retrieval accuracy was measured in pools of 10 or 32 items, so its absolute values do not generalize to large-database retrieval.Source: §9 · p. 12
- The dimension plateau reflects a sample-size rank limit, leaving the number of dimensions that carry cross-model structure unknown.Source: §9 · p. 12
- Uniform mean pooling may understate Mamba, while the authors do not establish that its lower representational agreement explains its weaker associative recall.Source: §9 · pp. 11–12
10 Conclusion
The conclusion reports real, replicable representational structure across five frozen models that can be aligned into a shared space. It also finds no demonstrated routing advantage for this model set and task scope.
10 Conclusion
- Five frozen models from four organizations, including a state-space model, share representational structure that orthogonal rotations alone can place in one space.Source: §10 · p. 12
- For this model set and these tasks, the shared space does not provide a better way to choose which model to use.Source: §10 · p. 12
- The authors report substantial Claude assistance with experiments, analysis, and drafting, while retaining responsibility for the content.Source: §10 · p. 12