Recurrent Memory

Transformers use the same forward computation stream to both predict the next token and store useful state for future token predictions. We formulate the \emph{state-prediction separation hypothesis}: disentangling the two roles yields better language modeling performance. We design a Transformer variant that uses separate computation streams to separate the two functions, and conduct pretraining experiments across various scales. Our experiments show that state-prediction separation consistently offers better data-- and compute--efficiency, improving validation loss and outperforming standard Transformers by 2--3 percentage points on average on downstream tasks. We also conduct extensive empirical analysis that rules out potential confounders and demonstrates the fundamental difference in the gradients our design entails.

Selective state space models (SSMs), such as Mamba, achieve strong per-token expressivity by making the time discretization step $\Delta$ a learned function of the input. However, in doing so, $\Delta$ ceases to represent a physical sampling interval, limiting its irregular time series modeling capability. Continuous-time SSMs, such as S5, preserve the physical meaning of $\Delta$ and handle irregular timestamps natively, but their dynamics remain linear time-invariant (LTI), limiting per-token expressivity. We propose \textbf{TIDES}, a selective SSM variant that reconciles selective and continuous architectures by moving input-dependence off the step size and onto the diagonal state matrix. As a result, $\Delta$ retains its physical meaning, tied to the state discretization, allowing the model to handle irregular timestamps natively without sacrificing the per-token expressivity that makes selective SSMs effective. We show this on a novel \emph{Fading Flash} experimental benchmark, a compact controlled diagnostic for sequence models that jointly tests input-dependence and extrapolation to out-of-distribution $\Delta$ values, and isolates the distinct failure modes of current state-of-the-art architectures that TIDES avoids by construction. On large-scale benchmarks, TIDES sets the new state-of-the-art average rank on UEA time-series classification and the Physiome-ODE regression benchmark.

Mixture-of-experts (MoE) is a common approach for increasing parameter capacity, but applying MoE to state space model (SSM) token mixers can multiply the cost of the recurrent state update. We study how to introduce expert specialization into selective SSMs while preserving computational efficiency. We show that MoE--SSM can refer to two designs: (1) MoE over separated SSMs, which maintains multiple state trajectories and thus scales compute with the number of experts; and (2) MoE-parameterized SSM, which mixes experts in parameter space, maintains a single state trajectory, and evaluates the recurrence once. Our method, Switch Mamba (Swimba), follows the second design by routing over expert-produced SSM streams. Theoretically, we establish well-definedness and stability for MoE-parameterized SSMs and characterize the relationship between the two designs. Empirically, we evaluate Swimba on standard benchmark tasks and measure real-time throughput and latency. Under matched FLOPs, Swimba achieves slightly better average performance than the baseline, with a small slowdown in real-time latency and throughput. Overall, these results suggest that parameter-space MoE can increase SSM capacity while keeping the dominant recurrence cost fixed.

Recurrent models trained from scratch have recently become competitive on ARC-style reasoning tasks, but the usual framing around small recurrent backbones overlooks two important parts of the system: task-conditioned memory and synthetic augmentation data. We study this regime through CHARM, a compact hybrid ARC model that combines recurrent reasoning with structured task memory, synthetic data, and inference-time aggregation. In existing approaches, task-conditioned memory supplies a large hidden source of capacity, reaching more than 30x the size of the recurrent backbone. We introduce a compositional sparse embedding for task conditioning that reduces learned task-memory parameters by over 90% while improving pass@2 in controlled ARC-AGI-1 ablations. We also find that matched synthetic data is the largest measured driver of ARC-AGI-2 gains. For the recurrent backbone, recurrent depth helps only when balanced with learning horizon. Combining these ingredients, our system reaches 80.5% pass@2 on ARC-AGI-1 and 39.3% pass@2 on ARC-AGI-2 public evaluation.

Inferring the evolution of high-dimensional and multi-modal (e.g., spatio-temporal) physical fields from irregular sparse measurements in real time is a fundamental challenge in science and engineering. Existing approaches, including diffusion-based generative models and functional tensor methods, typically operate in offline settings, depend on full temporal observations, or incur substantial inference cost. We propose StreamPhy, an end-to-end framework that enables efficient and accurate streaming inference of full-field physical dynamics from incoming irregular sparse measurements. The framework integrates a data-adaptive observation encoder that is robust to arbitrary observation patterns, a structured state-space model that supports memory-efficient online updates across irregular time intervals, and an expressive Functional Tensor Feature-wise Linear Modulation (FT-FiLM) decoder for continuous-field generation. We prove that FT-FiLM is more expressive than the functional Tucker model, admitting a richer function class for handling complex dynamics. Experiments on three representative physical systems under challenging sampling patterns show that StreamPhy consistently outperforms state-of-the-art baselines, with at least 48\% improvement in accuracy and up to 20--100x faster inference than diffusion-based methods.

Reservoir computing turns sequence learning into linear regression on fixed dynamical features, but those features come from a single highly dependent trajectory, so the true amount of usable data is often unclear. We develop a transport-based learning theory for contractive echo state networks driven by stochastic inputs by viewing the reservoir as a Markov process and proving a Wasserstein contraction that guarantees a unique stationary state law and explicit mixing rates. Using contraction-to-concentration tools, we obtain finite-sample deviation bounds for time-averaged features and empirical covariance estimates, and translate them into excess-risk and stability guarantees for ridge-trained linear readouts. The theory yields an explicit effective-sample-size principle and exposes a sharp trade-off: making reservoirs more “critical” can increase dynamical memory while slowing mixing and raising the data requirements for reliable training. Overall, the results complement echo-state and capacity analyses by providing verifiable design rules that link stability parameters to mixing time, generalization, and forecasting robustness under independent or weakly dependent inputs.

Flow matching learns continuous-time transports from a simple source distribution to a target data distribution by training a velocity field on intermediate states along interpolating trajectories. However, because the model observes only the projected data-space state $(x_t,t)$, different transport trajectories can induce conflicting velocity directions at nearby intermediate states, leading to curved transport paths that limit the fidelity of low-NFE sampling. To address this limitation, we propose \emph{State Augmented Flows} (SAF), a lightweight framework that lifts flow matching into a transient augmented state space. SAF transports an augmented state $(x_t,z_t)$, where auxiliary coordinates provide additional contextual information during transport while vanishing at the terminal time. This enlarges the state observed by the velocity model without changing the final sample space, and is compatible with existing coupling strategies and flow-matching architectures. Across MNIST, CIFAR-10, and ImageNet-32, SAF consistently improves over the rectified-flow baseline, with the largest gains in the low-NFE regime. These results suggest that augmenting the transported state provides a new and complementary mechanism for improving flow-based generative modeling.

Two-photon calcium imaging is a standard tool for recording large neural populations in vivo, yet inferring spikes accurately across the growing diversity of calcium indicators remains an open problem. Existing supervised methods achieve reasonable in-domain accuracy but generalize poorly to unseen indicators, because different indicators induce distinct fluorescence kinetics and signal statistics while existing architectures remain relatively simple generic temporal regressors without dynamics-matched inductive bias. We propose SpikeSSL, a universal spike inference framework whose temporal backbone is a bank of bidirectional IIR state-space layers broadly motivated by calcium dynamics. A multi-modal conditioning encoder maps indicator identity, sampling rate, and trace-level signal statistics into a global conditioning vector that modulates the backbone via Adaptive Layer Normalization, while a heteroscedastic variance head provides calibrated per-frame uncertainty. On a benchmark with five fixed evaluation splits built from 33 public ground-truth datasets, SpikeSSL achieves state-of-the-art performance in both in-domain and zero-shot leave-one-indicator-out settings. We also develop a biophysical simulation pipeline capable of generating paired fluorescence-spike traces with systematically varied kinetic parameters, spike statistics, response nonlinearities, baseline drift, and noise. Using this pipeline, we synthesize approximately 11{,}000 simulated traces. Augmenting training with these data effectively closes the cross-indicator domain gap and improves zero-shot generalization.

Implicit neural representations (INRs) model signals as continuous functions from coordinates to values, but their performance depends strongly on how spectral capacity is allocated. Yet most Quantum INR (QINR) architectures treat spectral components as independent outputs of separate quantum blocks. We introduce the Spectral Quantum Memory Model (SQMM), a memory coupled quantum INR in which sequential hybrid data reuploading layers (bands) communicate through a shared coherent quantum memory register. The construction gives a controlled way to share spectral information across quantum modules while preserving the finite Fourier structure of a band. When memory read and write layers are input independent, we prove that the memory state forms an operator valued Fourier series whose support grows by Minkowski summation of local band spectra. Downstream bands then condition their Fourier coefficients on the spectral content stored by earlier bands. This theory motivates a spectral memory alignment loss that encourages memory to encode the spectrum expressed by the deployed prefix of the model. Across audio and image reconstruction benchmarks, SQMM improves over QINR and INR baselines, achieving state of the art results.

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.

Inorganic crystals are periodic, highly symmetric arrangements of atoms in three-dimensional space. Their structures are constrained by the symmetry operations of a crystallographic \emph{space group} and restricted to lie in specific affine subspaces known as \emph{Wyckoff positions}. The frequency an atom appears in the crystal and its rough positioning are determined by its Wyckoff positions. Most generative models that predict atomic coordinates overlook these symmetry constraints, leading to unrealistically high populations of proposed crystals exhibiting limited symmetry. We introduce Space Group Conditional Flow Matching (SGFM), a novel generative framework that samples significantly closer to the target population of highly symmetric, stable crystals. We achieve this by conditioning the entire generation process on a given space group and set of Wyckoff positions; specifically, we define a conditionally symmetric noise distribution and a group-conditioned, equivariant, parametric vector field that restricts the motion of atoms to their initial Wyckoff position. Our form of group-conditioned equivariance is achieved using an efficient reformulation of \emph{group averaging} tailored for symmetric crystals. It reduces the computational overhead of symmetrization to a negligible level. SGFM achieves state-of-the-art performance in \textit{De Novo Generation} on the MP-$20$ benchmark.

Hyperspectral single-source domain generalization aims to train a robust model capable of overcoming domain shifts using only one available source domain for cross-scene hyperspectral image (HSI) analysis. Existing methods typically adopt data augmentation to broaden decision boundaries or utilize style transfer techniques for test-time alignment. However, these studies either fail to completely bridge the domain gap or inadvertently disrupt the discriminative spectral information, leading to suboptimal generalization and degraded classification performance on the target domain. To overcome these limitations, we propose a novel method, termed the domain memory retention framework, which optimizes domain memory during training to explicitly encode the semantic structure of the source domain. During inference, the target features are projected to the source distribution by leveraging this memory, thereby skipping domain shifts and improving generalization performance across unseen scenes. Specifically, the memory is formulated as a codebook, consisting of multiple semantic codewords. We further develop a semantic richness constraint and a perturbation robustness constraint to enhance representation diversity while empowering domain invariance for the memory. Extensive experiments conducted on three remote sensing datasets demonstrate that the proposed method outperforms state-of-the-art methods.

Selective state-space models such as Mamba replace fixed linear state transitions by input-dependent transitions, yielding recurrent sequence models that retain linear-time inference while adapting their memory dynamics to the current token. We show that even a simple form of this selective memory update has universal approximation power. Specifically, for every compact input domain, finite horizon, and continuous causal sequence-to-sequence map, a clocked diagonal selective state-space scan with strictly contractive gates and a linear-in-state readout uniformly approximates the target map. The proof is constructive: input-dependent diagonal propagation generates tensor-product partition features over prefixes, and the linear-in-state readout combines these features with sampled target values. Thus the high-order interactions required for universality are produced inside the selective scan itself, rather than by attention, nonlinear hidden-state dynamics, or a universal multilayer perceptron applied after a sequence encoder. These results identify selectivity as a mathematically sufficient mechanism for expressive linear-time state-space sequence modeling. Selectivity lets the model decide, token by token, which parts of the past should be preserved, faded, or combined with the present. The universal approximation theorem shows that this adaptive use of memory is sufficient to reproduce any well-behaved causal sequence rule over a fixed horizon.

We introduce Skill Learning from Video Memory (SLVMBench), the first benchmark that jointly evaluates whether video large language models (video-LLMs) can learn skills from long video memory and apply them to real-time tasks. SLVMBench presents models with 2–3 hour video streams that contain a tutorial video embedded in a stream of arbitrary irrelevant videos, resembling real-world human learning practices. Video-LLMs are asked to apply the acquired skill to answer real-time questions about an ongoing video. Unlike long-video understanding benchmarks that emphasize passive comprehension and skill-learning benchmarks that rely on short, immediate demonstrations, SLVMBench tests the full pipeline of memorizing and extracting procedural knowledge, as well as transferring it to real-time tasks. Moreover, rigorous human annotations feature sub-second-level temporal calibration, manually engineered questions eliminating common-sense guessing, and collated tutorials to ensure coverage of the required skills. Evaluations on state-of-the-art proprietary and open-source video LLMs show that video-LLMs struggle substantially with learning and applying skill knowledge from videos. Moreover, performance degrades markedly when the skill knowledge is placed within a long video memory. These results reveal a key limitation of existing video LLMs and position SLVMBench as the first benchmark for studying real-time skill acquisition and application from long-context video memory.

Streaming 3D reconstruction under a strict constant-memory budget hinges on how the recurrent state is updated as the stream evolves. We profile TTT3R-style per-token gates across five benchmarks and discover a structural bottleneck: the gate is intrinsically bounded in magnitude (median $0.31$; never exceeding $0.6$) and nearly frame-invariant, yielding an effective memory horizon of only $\sim$3 frames per state token, which serves as the structural origin of long-sequence drift. We trace this to a missing axis: existing inference-time methods modulate updates only at the per-token, intra-frame level, while the orthogonal frame-level question of \emph{how strongly each frame should contribute to the state} has been silently treated as content-independent. We close this gap with a scalar frame-level gate $\alpha_t \in (0, 1]$ derived in closed form from frame-to-frame changes of internal features---a continuous relaxation of classical SLAM keyframe selection that requires no parameters, no training, and no extra forward pass. Across six benchmarks spanning camera pose, video depth, and 3D reconstruction at sequence lengths up to $4,541$ frames, our gate cuts ATE by $51\%$ on long TUM-RGBD pose sequences, reduces AbsRel by $12.8\%$ on Bonn video depth, and on KITTI long-sequence pose estimation surpasses LongStream and Keyframe-VO which are specially constructed for long sequences via retraining and policy learning, while retaining strictly constant memory at zero training cost.

The theory of state tracking in recurrent architectures has predominantly focused on expressive capacity: whether a fixed architecture can theoretically realize a set of symbolic transition rules. We argue that equally important is error control, the dynamics governing hidden-state drift along the directions that distinguish symbolic states. We prove that affine recurrent networks, a class of models encompassing State-Space Models and Linear Attention, cannot correct errors along state-separating subspaces once they preserve state representations. Consequently, practical affine trackers do not learn robust state tracking; rather, they learn finite horizon solutions governed by accumulated state-relevant error. We characterize the mechanics of this failure, showing that tracking remains readable only while the accumulating within-class spread remains small relative to the initial between-class separation. We demonstrate empirically on group state-tracking tasks that this breakdown is predictable: tracking collapses when the distinguishability ratio crosses the readability threshold of the trained decoder. Across trained models, the point of this crossing predicts the horizon at which downstream accuracy fails. These results establish that robust state tracking is determined not only by an architecture's theoretical expressivity but crucially by its error control.

Long-horizon language agents must operate under limited runtime memory, yet existing memory mechanisms often organize experience around descriptive criteria—relevance, salience, or summary quality. For an agent, however, memory is valuable not because it faithfully describes the past, but because it preserves the distinctions between histories that must remain separated under a fixed budget to support good decisions. We cast this as a decision-centric rate-distortion problem, measuring memory quality by the loss in achievable decision quality induced by compression. This yields an exact forgetting boundary for what can be safely forgotten, and a memory-distortion frontier characterizing the optimal tradeoff between memory budget and decision quality. Motivated by this decision-centric view of memory, we propose **DeMem**, an online memory learner that refines its partition only when data certify that a shared state would induce decision conflict, and prove near-minimax regret guarantees. On both controlled synthetic diagnostics and long-horizon conversational benchmarks, DeMem yields consistent gains under the same runtime budget, supporting the principle that memory should preserve the distinctions that matter for decisions, not descriptions.

Sequential recommenders need long histories, but full-prefix attention scales with user lifetime. We argue that an effective memory mechanism should provide space invariance, semantic fidelity, and lifelong evolvability, which existing fixed-window or capped-memory methods do not jointly satisfy. We propose RecMem, a recurrent compression framework that folds history segments into M forget-gated memory slots. RecMem has a history-length-independent rollout error bound, and its fixed-capacity memory is trained by next-item prediction to preserve useful signals. On MerRec, RecMem uses 5.6x fewer decoder-side tokens than full attention while retaining 88–97% of Recall@10–200. Compared with a compressed truncation baseline at similar decoder cost, RecMem improves Recall@50+ by 4–6 percentage points by compressing the full history rather than discarding old interactions. It also remains stable over 1,800 test steps.

Long-context recall in linear-time sequence models highlights a tradeoff in how they write to memory. State-based linear models, such as state-space models (SSMs) and linear Transformers, write densely, updating the entire state for each newly arrived token, which leads to interference and makes specific past tokens hard to recover. Sliding-window attention (SWA) exhibits the opposite behavior: it writes sparsely by storing explicit token representations, but only within a fixed window, so recall drops once the relevant token is evicted. Interpolating between these models, we introduce Raven, a linear-time sequence model that maintains a fixed set of memory slots and, at each step, decays and updates only a selected subset via learned, input-dependent routing. This lets Raven mitigate SWA's position-based overwriting and hard eviction while reducing interference from dense state updates in SSMs, thereby preserving long-range content much more effectively. Across recall-intensive benchmarks, Raven is competitive with or outperforms prior linear-time baselines, achieving strong long-context recall where both SWA and SSMs sharply degrade. It remains effective when extrapolating to context lengths as large as 16x its training length, with similar gains in hybrid architectures.

Temporal integration gives continuous-time recurrent networks memory, but in deep stacks it also delays bottom-up signals and attenuates top-down errors. We ask whether this failure mode can be addressed by making the bottom-up input to each layer prospective, replacing instantaneous layer inputs with local look-ahead signals that compensate for integration-induced lag. We develop Recursive Quadrature Filters (RQFs), a biologically motivated class of complex-valued temporal filters that remain equivalent to single-channel diagonal state-space models (SSMs). This equivalence makes RQF layers trainable with the same parallel scan and convolutional algorithms used for diagonal SSMs. At the implementation level, this prospective input amounts to a lightweight two-tap input update, making it a drop-in modification for RQFs and SSMs. For the resulting discrete-time network, we prove that spatial-only backpropagation in a deep RQF network with instantaneous bottom-up inputs yields error signals that decay geometrically with depth, whereas prospective-input coding restores order-one gradient flow. We evaluate prospective-input coding on the Speech Commands dataset using standalone RQF recurrent stacks across two feature representations, mel-frequency cepstral coefficients (MFCCs) and raw audio, and local and non-local credit-assignment strategies. Under spatial-only backpropagation, prospective-input coding improves validation accuracy from 65.7% to 84.9% on MFCC features and from 46.4% to 61.0% on raw audio. Under full backpropagation through time, the gains persist, improving accuracy from 89.4% to 93.2% on MFCC features and from 80.1% to 82.7% on raw audio. Taken together, our results identify prospective-input coding as a local, broadly applicable mechanism for improving credit assignment in multi-layer recurrent networks with a continuous-time substrate.