A Hippocampus for Linear Attention: An Exact Memory for What the Recurrent State Forgets

Linear attention reduces the quadratic cost of softmax attention to $\mathcal{O}(T)$, but its memory state grows as $\mathcal{O}(T)$ in Frobenius norm, causing progressive interference between stored associations. We introduce \textbf{Variational Linear Attention} (VLA), which reframes the memory update as an online regularised least-squares problem with an adaptive penalty matrix maintained via the Sherman-Morrison rank-1 formula. We prove that normalising the write direction to unit length gives the recurrence Jacobian spectral norm exactly $1$ for all sequence lengths and head dimensions (Proposition 2), and that the state norm is self-limiting under bounded inputs (Proposition 1). Empirically, VLA reduces $\|S_t\|_F$ by $109\times$ relative to standard linear attention at $T{=}1{,}000$, achieves near-perfect exact-match accuracy on multi-query associative recall within the effective per-head memory regime ($n_\text{pairs} < d_h$), maintaining substantially higher retrieval performance than DeltaNet and standard linear attention under increasing memory load, and maintains 62% accuracy at the per-head capacity boundary. A Triton-fused kernel achieves $14\times$ speedup over sequential Python and $\mathcal{O}(T)$ scaling, crossing below softmax attention latency at approximately 43,000 tokens.

Linear attention models allow a fixed state size and a fixed amount of compute per token. However, due to their limited state size, linear attention models fall behind in long-context recall compared to softmax-attention-based transformer architectures. Increasing the state size of linear attention improves recall performance but at the cost of higher FLOPs. In this work, we introduce Sparse Delta Memory (SDM), an architecture that scales the hidden state of gated linear RNNs to orders of magnitude higher capacity using a sparse addressing scheme. SDM extends the Gated DeltaNet architecture by replacing the dense key-value outer product with sparse reads and writes to a large explicit memory. We show that, under an isoFLOP constraint and with an identical number of parameters, a higher state memory capacity significantly improves performance on in-context learning and long-context retrieval tasks. Moreover, by learning the initial state of the SDM memory and therefore using it as a parametric memory, we show that the model further improves on a wide range of common-knowledge and reasoning tasks.

Recurrent, attention-free sequence models share a structural weakness: a fading state cannot perform exact recall of something seen once, far in the past. We add to the Kathleen trunk a second memory layer -- a "notebook": a fixed-key holographic (HRR) associative store with a learned local write gate, a self-gating raw read, and write-triggered forgetting -- 25K parameters that attach to the logits of any trunk. (1) Mechanism: on a controlled needle-in-haystack task the notebook reaches 80-82% one-shot recall at 4x the training length, where the bare trunk scores ~4% and a parameter-matched attention head scores 100% inside its training length and 0% beyond it. Addressing is length-invariant by construction; the untrained memory alone recalls at 90% accuracy identically at 512, 2048 and 4096 bytes. Because the store is a linear superposition, two capabilities follow from arithmetic alone: selective unlearning (one subtraction erases one fact to chance, retained facts unharmed) and per-token attribution (counterfactual erasure names the source fact of every correct byte, 100% provenance). (2) Real text: on WikiText-2 bytes the notebook improves prediction of repeated rare words by +0.15-0.27 bits/byte, the gain growing with the distance between mentions and holding zero-shot at 4x training length; write-triggered forgetting eliminates memory pollution at 8x length (first-mention cost +0.33 -> -0.004). (3) Scope and scale: a parameter-matched attention head does generalize on natural-text repetition, so the notebook's claim is exact recall at O(L); on a WikiText-103 ladder (8 to 512 MB) the zero-shot repeat gain rises monotonically. All experiments are pre-registered, seeds reported, and reproducible on a single free-tier GPU.