Parallel Recursive LSTM

Generative sequence modeling faces a fundamental tension between the expressivity of Transformers and the efficiency of linear sequence models. Existing efficient architectures are theoretically bounded by shallow, single-step linear updates, while powerful iterative methods like Test-Time Training (TTT) break hardware parallelism due to two dimensions of serial dependency: token-level state reliance and step-level iteration loops. We propose PRISM (Parallel Residual Iterative Sequence Model) to resolve this tension. PRISM explicitly approximates the expressive gate-residual-direction iteration pattern of TTT in a parallelizable form. We employ a Write-Forget Decoupling strategy that isolates non-linearity within the injection operator. To bypass the serial dependency of explicit solvers, PRISM utilizes a two-stage proxy architecture: a short-convolution anchors the initial residual using local history energy, while a learned predictor estimates the refinement updates directly from the input. This design distills structural patterns associated with iterative correction into a parallelizable feedforward operator. Theoretically, we prove that this formulation achieves Rank-$L$ accumulation, structurally expanding the update scheme beyond the single-step Rank-$1$ bottleneck. Empirically, it achieves comparable performance to explicit optimization methods while achieving \textbf{174x higher throughput}. Codes are available in https://github.com/gpr-prism/prism/.

Transformers process tokens in parallel but are temporally shallow: at position $t$, each layer attends to key-value pairs computed based on the previous layer, yielding a depth capped by the number of layers. Recurrent models offer unbounded temporal depth but suffer from optimization instability and historically underutilize modern accelerators. We introduce the Recurrent Transformer, a simple architectural change where each layer attends to key-value pairs computed off its own activations, yielding layerwise recurrent memory while preserving standard autoregressive decoding cost. We show that the architecture can emulate both (i) a conventional Transformer and (ii) token-to-token recurrent updates under mild assumptions, while avoiding optimization instability. Naively, prefill/training appears bandwidth-bound with effective arithmetic intensity near $1$ because keys and values are revealed sequentially; we give an exact tiling-based algorithm that preserves the mathematical computation while reducing HBM traffic from $Θ(N^2)$ to $Θ(N\log N)$, increasing effective arithmetic intensity to $Θ(N/\log N)$ for sequence length $N$. On 150M and 300M parameter C4 pretraining, Recurrent Transformers improve cross-entropy over a parameter-matched Transformer baseline and achieve the improvement with fewer layers (fixed parameters), suggesting that recurrence can trade depth for width, thus reducing KV cache memory footprint and inference latency.

We study Latent Recurrent Transformer (LRT), a lightweight augmentation of autoregressive transformers that reuses a high-level source-layer hidden state from the previous token as recurrent memory for the next token. Because this state is already computed during ordinary decoding, LRT introduces a cross-token, cross-layer latent pathway while preserving the standard attention mechanism, KV-cache interface, and one model forward per generated token. To pretrain this recurrence without sequentially unrolling the full sequence, we introduce interleaved parallel training: one full-sequence initialization forward constructs a shared buffer, followed by sequential refinement of disjoint position subsets with parallel computation within each subset. This provides every token with recurrent-memory-aware supervision at approximately 2x ideal token compute. Across 1.3B- and 2.1B-parameter nanochat-style backbones and a wide range of training budgets, LRT improves both BPB and CORE under matched effective compute. Additionally, LRT outperforms two-forward PonderLM-2 and matches a three-loop Transformer in BPB, while retaining one-forward-per-token decoding with 9% latency overhead over the standard Transformer.