What Does Layer-Importance Reveal About Transformers and State-Space Models?

Recent state-space models (SSMs) such as Mamba achieve language modeling performance comparable to transformers despite relying on fundamentally different architectures. This raises an important question: how do these structural differences influence the geometry and functional nature of their internal representations? We study this question through a multi-scale analysis of representations in transformers, SSMs, and hybrid architecture. First, we find that SSMs distribute their representational information evenly across all dimensions, whereas transformer representations are heavily dominated by a single principal direction. By evaluating hybrid architectures, we observe that the representation space becomes increasingly skewed toward a single dominant direction after each attention layer. Next, we explore how the different geometric spread of representations impacts representational capacity through compressibility. Surprisingly, we find that despite their contrasting geometric structures, both architectures exhibit tightly matched effective capacities. We further investigate whether this skewed geometry affects how concepts are encoded. Using rank-constrained probes, we demonstrate that both architectures encode concepts in subspaces of surprisingly similar dimensionality. Furthermore, we demonstrate that the transformers' dominant principal direction does not inherently encode more conceptual information. Finally, we zoom in and examine the alignment between manifolds, either by analyzing representations of specific topics or by looking at the nearest neighborhoods of tokens, and find that they are highly aligned. Ultimately, our analysis suggests that while transformers and SSMs induce different usage of latent space, they display a striking functional convergence at the level of local semantic manifolds.

Combining attention's global retrieval with the sequential importance signal of state space models (SSMs) is the open challenge of hybrid language modeling. Transformers see everywhere but cannot prioritize; SSMs know what matters but cannot revisit. Existing hybrids -- Jamba (block level) and Hymba (head level) -- place the two in separate compartments, so neither informs the other during the attention computation itself. We propose SISA (SSM-Informed Softmax Attention), which adds an SSM-derived importance term directly inside the attention score and realizes the full operation as a single SDPA call on augmented query/key vectors -- no recurrent state, no custom kernel. At 152M / 5B tokens, SISA reaches LAMBADA-greedy 17.3% (vs. Transformer 13.9 and Mamba-3 15.5) and attains NIAH 100% from step 1K, 7x faster than Transformer's retrieval convergence; at 369M, Mamba-3 leads LAMBADA while SISA preserves perfect NIAH and stock-SDPA execution. SISA thus defines a third design axis for SSM-attention hybrids -- score-level fusion -- beyond the block-level and head-level paradigms that have dominated the field.

We study how depth, finite precision, state dimension, and chain-of-thought (CoT) affect the expressive power of multi-layer state-space models (SSMs). For the explicit-table $K$-function-composition problem, a canonical benchmark for sequential information propagation, we prove that any $L$-layer SSM solving $(L+3)$-function composition must satisfy $d^2p=Ω(N/L^3)$, where $d$ is the state dimension and $p$ is the per-scalar precision. Conversely, $K$-function composition is solved exactly by a $(K+1)$-layer generalized SSM with $d=1$ and $p=Θ(\log N)$. This gives a worst-case depth hierarchy for this formal problem family. We then distinguish post-input reasoning, in which all thought tokens are generated after the input, from input-interleaved reasoning, in which thought tokens may be inserted while the input stream is being read. Post-input reasoning does not circumvent our communication-based lower-bound pipeline, whereas input-interleaved reasoning admits bidirectional simulations with general deterministic one-pass streaming algorithms at the granularity of persistent memory. Finally, width and precision are not interchangeable under exact step-preserving simulation in the base affine-state model, but become interchangeable through the streaming-memory characterization once input-interleaved reasoning is allowed.