Recently, there has been a renewed interest in using linear RNNs for efficient sequence modeling.

We introduce the setting of continuous index learning, in which a function of many variables varies only along a small number of directions at each point. For efficient estimation, it is beneficial for a learning algorithm to adapt, near each point $x$, to the subspace that captures the local variability of the function $f$. We pose this task as kernel adaptation along a manifold with noise, and introduce Local EGOP learning, a recursive algorithm that utilizes the Expected Gradient Outer Product (EGOP) quadratic form as both a metric and inverse-covariance of our target distribution. We prove that Local EGOP learning adapts to the regularity of the function of interest, showing that under a supervised noisy manifold hypothesis, intrinsic dimensional learning rates are achieved for arbitrarily high-dimensional noise. Empirically, we compare our algorithm to the feature learning capabilities of deep learning. Additionally, we demonstrate improved regression quality compared to two-layer neural networks in the continuous single-index setting.

Deep convolutional artificial neural networks (ANNs) are the leading class of candidate models of the mechanisms of visual processing in the primate ventral stream. While initially inspired by brain anatomy, over the past years, these ANNs have evolved from a simple eight-layer architecture in AlexNet to extremely deep and branching architectures, demonstrating increasingly better object categorization performance, yet bringing into question how brain-like they still are. In particular, typical deep models from the machine learning community are often hard to map onto the brain's anatomy due to their vast number of layers and missing biologically-important connections, such as recurrence. Here we demonstrate that better anatomical alignment to the brain and high performance on machine learning as well as neuroscience measures do not have to be in contradiction. We developed CORnet-S, a shallow ANN with four anatomically mapped areas and recurrent connectivity, guided by Brain-Score, a new large-scale composite of neural and behavioral benchmarks for quantifying the functional fidelity of models of the primate ventral visual stream. Despite being significantly shallower than most models, CORnet-S is the top model on Brain-Score and outperforms similarly compact models on ImageNet. Moreover, our extensive analyses of CORnet-S circuitry variants reveal that recurrence is the main predictive factor of both Brain-Score and ImageNet top-1 performance. Finally, we report that the temporal evolution of the CORnet-S IT neural population resembles the actual monkey IT population dynamics. Taken together, these results establish CORnet-S, a compact, recurrent ANN, as the current best model of the primate ventral visual stream.

Attention and self-attention mechanisms, are now central to state-of-the-art deep learning on sequential tasks. However, most recent progress hinges on heuristic approaches with limited understanding of attention's role in model optimization and computation, and rely on considerable memory and computational resources that scale poorly. In this work, we present a formal analysis of how self-attention affects gradient propagation in recurrent networks, and prove that it mitigates the problem of vanishing gradients when trying to capture long-term dependencies by establishing concrete bounds for gradient norms. Building on these results, we propose a relevancy screening mechanism, inspired by the cognitive process of memory consolidation, that allows for a scalable use of sparse self-attention with recurrence. While providing guarantees to avoid vanishing gradients, we use simple numerical experiments to demonstrate the tradeoffs in performance and computational resources by efficiently balancing attention and recurrence. Based on our results, we propose a concrete direction of research to improve scalability of attentive networks.

Attention mechanisms have become a standard tool for sequence modeling tasks, in particular by stacking self-attention layers over the entire input sequence as in the Transformer architecture. In this work we introduce a novel attention procedure called staircase attention that, unlike self-attention, operates across the sequence (in time) recurrently processing the input by adding another step of processing. A step in the staircase comprises of backward tokens (encoding the sequence so far seen) and forward tokens (ingesting a new part of the sequence). Thus our model can trade off performance and compute, by increasing the amount of recurrence through time and depth. Staircase attention is shown to be able to solve tasks that involve tracking that conventional Transformers cannot, due to this recurrence. Further, it is shown to provide improved modeling power for the same size model (number of parameters) compared to self-attentive Transformers on large language modeling and dialogue tasks, yielding significant perplexity gains.

Hirschberg's algorithm (1975) reduces the space complexity for the longest common subsequence problem from $O(N^2)$ to $O(N)$ via recursive midpoint bisection on a grid dynamic program (DP). We show that the underlying idea generalizes to a broad class of dynamic programs with local dependencies on directed acyclic graphs (DP DAGs). Modeling a DP as deterministic time evolution over a topologically ordered DAG with frontier width $ω$ and bounded in-degree, and assuming a max-type semiring with deterministic tie breaking, we prove that in a standard offline random-access model any such DP admits deterministic traceback in space $O(ω\log T + (\log T)^{O(1)})$ cells over a fixed finite alphabet, where $T$ is the number of states. Our construction replaces backward dynamic programs by forward-only recomputation and organizes the time order into a height-compressed recursion tree whose nodes expose small "middle frontiers'' across which every optimal path must pass. The framework yields near-optimal traceback bounds for asymmetric and banded sequence alignment, one-dimensional recurrences, and dynamic-programming formulations on graphs of bounded pathwidth. We also show that an $Ω(ω)$ space term (in bits) is unavoidable in forward single-pass models and discuss conjectured $\sqrt{T}$-type barriers in streaming settings, supporting the view that space-efficient traceback is a structural property of width-bounded DP DAGs rather than a peculiarity of grid-based algorithms.

In this paper, we examine the convergence landscape of multi-agent learning under uncertainty. Specifically, we analyze two stochastic models of regularized learning in continuous games -- one in continuous and one in discrete time with the aim of characterizing the long-run behavior of the induced sequence of play. In stark contrast to deterministic, full-information models of learning (or models with a vanishing learning rate), we show that the resulting dynamics do not converge in general. In lieu of this, we ask instead which actions are played more often in the long run, and by how much. We show that, in strongly monotone games, the dynamics of regularized learning may wander away from equilibrium infinitely often, but they always return to its vicinity in finite time (which we estimate), and their long-run distribution is sharply concentrated around a neighborhood thereof. We quantify the degree of this concentration, and we show that these favorable properties may all break down if the underlying game is not strongly monotone -- underscoring in this way the limits of regularized learning in the presence of persistent randomness and uncertainty.

We precisely characterize the expressivity of computable Recurrent Graph Neural Networks (recurrent GNNs). We prove that recurrent GNNs with finite-precision parameters, sum aggregation, and ReLU activation, can compute any graph algorithm that respects the natural message-passing invariance induced by the Color Refinement (or Weisfeiler-Leman) algorithm. While it is well known that the expressive power of GNNs is limited by this invariance [Morris et al., AAAI 2019; Xu et al., ICLR 2019], we establish that recurrent GNNs can actually match this limit. This is in contrast to non-recurrent GNNs, which have the power of Weisfeiler-Leman only in a very weak, "non-uniform", sense where each graph size requires a different GNN to compute with. Our construction introduces only a polynomial overhead in both time and space. Furthermore, we show that by incorporating random initialization, for connected graphs recurrent GNNs can express all graph algorithms. In particular, any polynomial-time graph algorithm can be emulated on connected graphs in polynomial time by a recurrent GNN with random initialization.

Human activity recognition (HAR) from inertial sensors is essential for ubiquitous computing, mobile health, and ambient intelligence. Conventional deep models such as Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), and transformers have advanced HAR but remain limited by vanishing or exloding gradients, high computational cost, and difficulty in capturing long-range dependencies. Structured state-space models (SSMs) like Mamba address these challenges with linear complexity and effective temporal modeling, yet they are restricted to first-order dynamics without stable longterm memory mechanisms. We introduce Momentum Mamba, a momentum-augmented SSM that incorporates second-order dynamics to improve stability of information flow across time steps, robustness, and long-sequence modeling. Two extensions further expand its capacity: Complex Momentum Mamba for frequency-selective memory scaling. Experiments on multiple HAR benchmarks demonstrate consistent gains over vanilla Mamba and Transformer baselines in accuracy, robustness, and convergence speed. With only moderate increases in training cost, momentum-augmented SSMs offer a favorable accuracy-efficiency balance, establishing them as a scalable paradigm for HAR and a promising principal framework for broader sequence modeling applications.