Dynamic Mode Decomposition along Depth in Vision Transformers

Vision Transformers (ViTs) are built by stacking independently parameterized blocks, but it remains unclear how much of this depth requires layer specific transformations and how much can be realized through recurrent computation. We study this question with bViT, a single-block recurrent ViT in which one transformer block is applied repeatedly to process an image. This architecture preserves the iterative structure of a deep ViT while removing layer specific block parameterization, providing a controlled setting for studying recurrence in vision. On ImageNet-1K, a 12-step bViT-B achieves accuracy comparable to standard ViT-B under the same training recipe and computational budget, while using an order of magnitude fewer parameters. We observe that recurrent performance improves with representation width, with wider bViTs recovering much more of the performance of standard ViTs than narrow variants. We interpret this behavior as implicit depth multiplexing, where a shared block expresses multiple step-dependent computations through the evolving hidden state. Beyond ImageNet classification, bViT transfers competitively to downstream tasks and enables parameter-efficient fine-tuning. Mechanistic analyses of activations, attention and step-specific pruning show that the shared block changes its effective behavior across recurrent steps rather than simply repeating the same computation. Our results suggest that a large fraction of ViT depth can be implemented through recurrent reuse, provided that the representation space is sufficiently wide.

Vision Transformers trained only on image classification routinely transfer to tasks that demand spatial understanding, yet they receive no spatial supervision during pretraining. We ask where and how robustly such structure is encoded. Probing a frozen ViT-B/16 layerwise for two complementary properties, local patch boundaries (BSDS500) and per-patch depth (NYU Depth V2), reveals a clear hierarchy: boundary structure becomes linearly decodable at layers 5-6 (AP = 0.833), while depth, which requires integrating global cues, peaks two to three layers later at layer 8 (MAE = 0.0875). Both signals collapse at the final classification layer, and random-weight controls confirm the encodings are learned rather than architectural. Causal interventions add specificity: ablating the single direction a linear depth probe reads degrades depth decoding by up to 165%, while ablating any other direction changes it by less than 1%. Targeted activation patching along that direction shows the depth signal is partially re-derived at each layer rather than passively carried in the residual stream, with mid-layer interventions persisting most strongly downstream. The result is that a classification-trained ViT develops an actively maintained spatial hierarchy that mirrors the early-to-late progression observed in the primate visual cortex.

Vision Transformers (ViTs) achieve strong image-recognition performance, but their parameter count grows linearly with depth when each block is independently parameterized. Single-block recurrent ViTs (bViT) remove this growth by repeatedly applying one shared block. Rather than proposing a new architecture, we fix a bViT and provide a controlled empirical characterization of three training and inference regimes under a common CIFAR-100 protocol, asking: (i)~when does recurrence beat independently parameterized depth---at matched FLOPs or at matched parameter memory? (ii)~when a residual recurrent block is trained through an ODE solver, does solver order act as numerical refinement or as an architectural bias? and (iii)~what does robustness beyond the training horizon cost in nominal accuracy? We find that standard ViTs remain preferable when FLOPs are the primary constraint, whereas recurrent ViTs offer a better accuracy--parameter trade-off under memory constraints. Consistent with the standard view of residual networks as Euler discretizations of ODEs, the continuous-time analogue of a residual recurrent block is the state-subtracted vector field $\dot{z}=F_θ(z)-z$; although known in principle, this distinction is easy to violate when the block is wrapped as a black-box vector field, and we qualify the cost at few accuracy points. Because the vector field is learned jointly with the solver, higher-order solvers act as a solver-induced architectural bias rather than a numerical-accuracy improvement, and their gains are not uniform. Finally, stage-wise deep supervision traces an accuracy--robustness frontier: it does not improve nominal accuracy, but degrades gracefully far beyond the training horizon, where naive recurrence collapses to near-random performance.