Whether distinct neural architectures develop common collective dynamics remains an open question. Recent analysis of Transformer language models revealed a nearly flat, weakly infrared-enhanced time-scale density of states (TDOS) associated with near-marginal long-memory dynamics. Here we test whether a closely related organization emerges in Mamba, whose selective state-space dynamics provides a fundamentally different microscopic mechanism. Mamba allows relaxation dynamics to be resolved at three levels: the intrinsic spectrum of the learned state-space generator, its input-conditioned selective rescaling, and the collective TDOS of the complete block measured from its Jacobian. These spectra are not identical: selective dynamics and the remaining block transformations substantially reorganize the microscopic relaxation hierarchy. Nevertheless, the full block develops a reproducible slow-mode continuum whose infrared sector becomes progressively better resolved with increasing sequence length. Cumulative analysis yields $ρ(λ)\simλ^β$, with the long-sequence Mamba exponent stabilizing near $β_{\rm M}\simeq-0.17$. The corresponding memory dynamics follows $K(t)\sim t^{-(1+β)}$, close to the marginal $1/t$ regime. Despite fundamentally different microscopic dynamics, Transformer full-block spectra exhibit closely related infrared organization, with representative exponents of order $β_{\rm Tr}\sim-0.1$. These results separate explicit state-space memory from collective infrared organization and show that distinct sequence architectures can develop closely related near-marginal slow-mode dynamics. They extend infrared collective organization beyond Transformers and provide an independent test of the dynamical structure described by Cognitive Field Theory.
While Root Mean Square Normalization has become the de facto standard for accelerating modern sequence models, its reliance on the quadratic accumulation of independent scalars ($\sum x^2$) inherently triggers outlier-induced numerical instability, gradient starvation, and anisotropic phase distortion. We introduce Mean Root Square Normalization (MRSNorm). By structurally pairing channels into 2D phasors, MRSNorm mathematically inverts the traditional scaling paradigm: it computes the localized $L_2$ magnitudes (Root Square) before aggregating them via a global $L_1$ average (Mean). This operational inversion strictly constrains activations to a phasor manifold, preserving conformal invariance. By sharing a single affine weight across phasor components, MRSNorm halves the total number of learnable parameters, proving that unconstrained spatial scaling in standard norms is a harmful redundancy. We analytically demonstrate that this geometric constraint yields a built-in, trigonometric gradient clipper governed by the Pythagorean identity, unconditionally equalizing the local gradient norm to ensure Gradient Homogeneity. Empirical evaluations on a ResNet with CIFAR-100 show that despite halved parameters, MRSNorm provides critical structural stability under rigorous stress tests. Under extreme hyperparameter settings where standard normalizations suffer from gradient divergence, MRSNorm successfully prevents numerical explosion and secures stable optimization trajectories. Our findings propose a fundamental paradigm shift toward phasor-based deep representation learning. The implementation of MRSNorm is available at Appendix C.
Mary Letey, Yue M. Lu, Cengiz Pehlevan +1stat.ML cs.LG stat.CO
Modern sequence models have a striking capacity for in-context learning (ICL); they can perform new tasks based only on examples given in the prompt. Understanding how this ability emerges requires theory that captures important properties of natural data. Linear regression has served as a useful sandbox for ICL theory, but existing work has largely focused on prompts with independent examples. In this work, we extend this setting to sequentially correlated data, a basic feature of real sequences. We present a solvable model based on linear attention and test our predictions on realistic transformer architectures. We identify two distinct effects: First, when the query token is independent of the context, within-context correlations induce an effective context length: correlated prompts behave like shorter i.i.d. prompts. Second, when the query is also correlated with its context, test error is reduced, particularly for softmax attention when compared to linear attention. These results suggest that correlated prompts alter not only the effective sample size of in-context learning, but also which attention architectures are best matched to the task.