#01Nov 5, 2026
stat.ML
Unifying Central Flow, Rod Flow, and the Edge Coupling
Eric Regis, Sinho Chewi
At the edge of stability, the iterates of gradient descent oscillate along the sharp directions of the Hessian. This poses a significant challenge for continuous-time models as the usual continuous-time approximation -- the gradient flow -- admits no oscillations. Several continuous-time models have been proposed to capture the behavior at the edge of stability, among them central flow and rod flow. Although both models can be interpreted as performing a coarse-graining in time, the connection between them remains poorly understood. Here we show that both flows can be derived from a third object, the edge coupling: a functional on consecutive iterates whose critical points are the fixed points and period-two orbits of gradient descent. It is revealed that the two flows correspond to coarse-grainings at different time scales. Rod flow coarse-grains over only the fastest time scale, while central flow coarse-grains over both the fast and intermediate time scales. Furthermore, using the edge coupling, we derive rod flows and central flows for momentum methods such as heavy ball and Adam. We verify the derived flows experimentally on representative neural network architectures, where they track the discrete iterates throughout all phases of training -- including the transition from the stable regime to the edge of stability.
#02Nov 5, 2026
cs.LG
User Embeddings are Superpositions of Interpretable Behavioral Modes
Jonathan Kozaczuk, Zheng Dong, Noyan Evirgen and 7 more
User embeddings are central components of modern recommendation, personalization, and behavioral modeling systems. Though these embeddings can compactly encode rich behavioral information, they are inherently dense and opaque, limiting interpretability. To address this gap, we argue that user embeddings can be understood as sparse superpositions of interpretable behavioral modes. This structure is directly predicted by generative models of user behavior for linear embeddings and extends approximately to deep-learning representations by the superposition hypothesis. Superimposed modes can naturally be recovered by sparse dictionary learning methods, such as sparse autoencoders (SAEs), and recovered modes can be automatically named and queried in natural language. We validate mode recovery against known ground truth on synthetic data, where SAEs recover modes better than classical sparse baselines, including on transformer-encoded embeddings. We then apply the method to classical and transformer-based user embeddings from two real-world datasets. We also show how the corresponding natural-language labels enable zero-shot audience retrieval and characterization over embeddings never trained on text. The framework is post-hoc and model-agnostic, requiring no modification to the upstream embedding system.
#03Nov 5, 2026
cs.LG
Muir: Linear Attention with Sensitivity-Allocated Gating and Matched-Filter Readout
Tushaar Gangavarapu, Gil Tabak, Clemens JS Schaefer
Linear attention compresses context into a fixed-size key-value state for linear-time training and constant-memory inference. Gated delta-rule models decay this state and erase its prediction for an incoming key before writing a new value; Gated DeltaNet-2 (GDN-2) generalizes this with channel-wise decay and decoupled, full-rank erase and write gates. However, we show that state updates are far less sensitive to erasure than to writes: key activation and unit normalization concentrate key energy onto a sparse subset of channels, whereas value writes scale all channels, so the erase gate can be parameterized at lower rank. Furthermore, slow-decaying state rows accumulate much larger norms than fast rows, dominating unweighted readout and forcing stronger write-time erasure to suppress interference. Guided by this analysis, we introduce Muir, which fuses all gates into a sensitivity-allocated low-rank projection, re-weights state rows with a matched-filter readout gate, and reinvests the saved parameters in a wider feed-forward network. Because standard chunkwise algorithms become numerically unstable at the large chunk sizes needed to saturate wide matrix units, we design a parallel update for stable, efficient training. Across 166M-1.34B parameters, Muir lowers perplexity by 1.8-2.0% over GDN-2 and 2.5-3.6% over Transformer, raises commonsense accuracy by up to 1.9 points, and improves extractive recall over GDN-2. At 1.34B, Muir outperforms both baselines on multi-hop and multi-needle recall, extrapolates 2,560x from its 4K training length to 10M-token streams without perplexity drift, and maintains near-100% passkey retrieval to 1M tokens with static readout. Muir achieves these gains while matching GDN-2 throughput, reaching its 100B-token validation loss 1.2x faster, and training 2.7x faster than Transformer at 32K.
#04Nov 5, 2026
cs.CV
4D Point Splatting for mmWave Radar Micro-Doppler Differentiable Rendering
Adnan Armouti, Guanyue Qian, Rajalakshmi Nandakumar
Millimeter-wave radar resolves the micrometer displacements of a moving body as micro-Doppler, enabling gait, gesture, activity, pose and vital-sign sensing. Each task relies on a downstream model trained on real data, yet radar datasets lag orders of magnitude behind vision and language. Brute-force collection will not close that gap, and existing synthesis methods (forward-only simulators, learned generators and static-scene inverse renderers) cannot fit a real capture of a moving body, leaving a sim2real gap. We present 4D Point Splatting (4DPS), a physics-based differentiable renderer that fits an explicit moving scene to a real capture. Using the synchronized camera to initialize the room and body, we represent the scene as oriented points carrying material, normal and velocity, and splat each point into range and Doppler bins. Our output is the raw complex return a real radar records, so we apply the same signal processing on the rendered data to obtain any representation for training any SOTA downstream model. We fit 235 sequences on HuPR and 100 on RT-Pose, two datasets with different radars. Our reconstructions set the state of the art, with median correlations of 0.940 and 0.830 on range-azimuth, 0.829 and 0.695 on range-Doppler, and 0.988 and 0.966 on micro-Doppler. Our renders also set the state of the art in downstream sim2real gap. On RT-Pose, a pose model trained on them reaches 21.9 cm MPJPE on real data, against 16.7 cm for real training and 127.6 cm for the RF-Genesis baseline, where both the rendered- and real-trained models learn nearly the same features. Our custom CUDA kernels fit a sequence in 5.7 minutes on HuPR and 12.9 on RT-Pose on one H100, making dataset-scale inverse rendering of mmWave radar possible for the first time. This work can enable synthetic data generation for any human-centered downstream task and model, and scene editing for data augmentation.
#05Nov 5, 2026
cs.LG
Discovering Learning Rules that Shape Others
Hong Jun Lee, Ziyan Wang, Jun Wang and 2 more
In general-sum games, self-interested learners can reach outcomes that are poor for everyone, which has motivated methods that shape the opponent's learning. Methods like Learning with Opponent-Learning Awareness (LOLA), which differentiates through the opponent's learning process, and Model-Free Opponent Shaping (M-FOS), which uses model-free optimisation methods for long-horizon opponent shaping, have enabled mutual cooperation and exploitation of learners. However, these methods require access to the opponent's parameters, policy, rewards or learning rule during deployment. In single-agent reinforcement learning, update rules discovered by meta-learning have achieved state-of-the-art results (e.g. DiscoRL). These rules are separate from the policy they train and read only the agent's own experience. We hereby extend this framework to general-sum settings by introducing Discovered Update Rule for Opponent Shaping (DUOS). To the best of our knowledge, this is the first method to discover an update rule for an agent in general-sum games. DUOS meta-trains the rule to maximise the agent's expected return over its lifetime, and updates an inner network using the agent's own experience. We show that our discovered rule exhibits behaviour consistent with opponent shaping without reading opponent policies or parameters, and ranks 1st in normalised mean score and 1st in self-play among tournament entrants.