aiqbench analysis
This research demonstrates that matrix-valued latent tokens in continuous chain-of-thought models exhibit "rank-indifference," meaning the model's loss function does not penalize or reward specific effective ranks during training. The study investigates whether this structural property stems from the readout mechanism itself or is an intrinsic feature of the optimization landscape, finding that various nonlinear readouts still fail to track parallel reasoning paths via rank. These findings are critical for mechanistic interpretability efforts aiming to understand how superposition and parallel computation are encoded in matrix-based latent spaces.
The Gradient Does Not See Rank: Rank-Indifference in Matrix-CODI on ProsQA
arXiv:2609.03090v1 Announce Type: new Abstract: Continuous chain-of-thought models compress reasoning into latent tokens. Matrix-valued variants, which route each latent token through a d x d matrix bottleneck, introduce rank as a single-sample structural observable on the latent matrix Z. If matrix latents carry parallel reasoning paths via superposition, rank should track them, and truncating Z to low rank should hurt accuracy on tasks whose solutions plausibly require multiple components. Across four training regimes of a matrix-CODI model (three on ProsQA, one on GSM8K-Aug below the learning threshold), the rank-k projection ablation curve is flat to within 0.6 percentage points. A three-seed replication yields 81.0 +/- 2.0 percentage points accuracy while the final effective rank of Z spans {4, 12, 13}; the loss does not reward any particular rank. To test whether rank-blindness arises from the flatten-then-project readout alone, we trained four readouts: a bilinear reparametrization, a bilinear-plus-GELU readout nonlinear in Z, an SVD-augmented readout feeding singular values through an MLP, and a quadratic readout in Z Z^T. All four rank-k curves remain flat (Spearman p-values 0.63, 0.14, 0.82, 0.46). The flat curves persist for readouts nonlinear in Z. A linear probe on Z underperforms a raw pretrained hidden state at target prediction (AUC 0.673 vs. 0.846). A negative control on vanilla GPT-2 SFT (no matrix bottleneck, no Z, three seeds, n=500) reproduces a flat rank-k curve under the same intervention paradigm with pooled-mean range 0.20pp, and a random-h sensitivity floor lands at the same accuracy: the rank-k ablation alone conflates rank-blindness with position-irrelevance.
Abstract:Continuous chain-of-thought models compress reasoning into latent tokens. Matrix-valued variants, which route each latent token through a d x d matrix bottleneck, introduce rank as a single-sample structural observable on the latent matrix Z. If matrix latents carry parallel reasoning paths via superposition, rank should track them, and truncating Z to low rank should hurt accuracy on tasks whose solutions plausibly require multiple components. Across four training regimes of a matrix-CODI model (three on ProsQA, one on GSM8K-Aug below the learning threshold), the rank-k projection ablation curve is flat to within 0.6 percentage points. A three-seed replication yields 81.0 +/- 2.0 percentage points accuracy while the final effective rank of Z spans {4, 12, 13}; the loss does not reward any particular rank. To test whether rank-blindness arises from the flatten-then-project readout alone, we trained four readouts: a bilinear reparametrization, a bilinear-plus-GELU readout nonlinear in Z, an SVD-augmented readout feeding singular values through an MLP, and a quadratic readout in Z Z^T. All four rank-k curves remain flat (Spearman p-values 0.63, 0.14, 0.82, 0.46). The flat curves persist for readouts nonlinear in Z. A linear probe on Z underperforms a raw pretrained hidden state at target prediction (AUC 0.673 vs. 0.846). A negative control on vanilla GPT-2 SFT (no matrix bottleneck, no Z, three seeds, n=500) reproduces a flat rank-k curve under the same intervention paradigm with pooled-mean range 0.20pp, and a random-h sensitivity floor lands at the same accuracy: the rank-k ablation alone conflates rank-blindness with position-irrelevance.
Accepted at the ICML 2026 Mechanistic Interpretability Workshop, this https URL. 9 pages. Corrects a data-entry error in the workshop version: the seed-1337 accuracy in the three-seed replication was reported as 80.47% (a control run); the archived value is 78.91%, so the three-seed mean is 81.0 +/- 2.0pp (was 81.5 +/- 1.2pp). All other results are unchanged
Machine Learning (cs.LG)
arXiv:2609.03090 [cs.LG]
(or arXiv:2609.03090v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.03090 arXiv-issued DOI via DataCite (pending registration)
Full text under the publisher's open licence · 323 words
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