Kyungwon Jeong

Kyungwon Jeong, Won-Gi Paeng, Honggyo Suh

Weight matrices in deep networks exhibit geometric continuity -- principal singular vectors of adjacent layers point in similar directions. While this property has been widely observed, its origin remains unexplained. Through experiments on toy MLPs and small transformers, we identify two mechanisms: residual connections create cross-layer gradient coherence that aligns weight updates across layers, and symmetry-breaking nonlinearities constrain all layers to a shared coordinate frame, preventing the rotation drift that would otherwise destabilize weight structure. Crucially, a nonlinear but rotation-preserving activation fails to retain continuity, isolating symmetry breaking -- not nonlinearity itself -- as the active ingredient. Activation and normalization play distinct roles: activation concentrates continuity in the leading singular direction, while normalization distributes it across multiple directions. In transformers, continuity is projection-specific: Q, K, Gate, and Up (which read from the residual stream) develop input-space ($\mathbf{v}_1$) continuity; O and Down (which write to it) develop output-space ($\mathbf{u}_1$) continuity; V alone, lacking an adjacent nonlinearity, develops only low continuity.

Won-Gi Paeng, Daesuk Kwon, Kyungwon Jeong et al.

In this work, we present a generalized formulation of the Transformer algorithm by reinterpreting its core mechanisms within the framework of Path Integral formalism. In this perspective, the attention mechanism is recast as a process that integrates all possible transition paths leading to future token states, with temporal evolution governed by the Feed-Forward Network. By systematically mapping each component of the Transformer to its counterpart in the Path Integral formulation, we obtain a more compact and efficient representation, in which the contextual information of a sequence is condensed into memory-like segments. These segments are recurrently processed across Transformer layers, enabling more effective long-term information retention. We validate the effectiveness of this approach through the Passkey retrieval task and a summarization task, demonstrating that the proposed method preserves historical information while exhibiting memory usage that scales linearly with sequence length. This contrasts with the non-linear memory growth typically observed in standard attention mechanisms. We expect that this quantum-inspired generalization of the Transformer architecture will open new avenues for enhancing both the efficiency and expressiveness of future Transformer models.