Federated Learning (FL) is a distributed machine learning (ML) paradigm with collaboration among multiple clients without sharing data. FL is challenging under data heterogeneity and partial client participation. Learning sparse models is useful for communication and computational efficiency in FL, but it is especially difficult in the small-sample high-dimensional regime (d >> N) where optimization can yield parameter configurations that fail to generalize to unseen test data. While magnitude-based pruning doesn't account for uncertainty exploration in the parameter space, a formulation with probabilistic gates and an L0 constraint allows sampling from competing sparse configurations during training. In this work, we study entropy regularization of gate distributions as a mechanism to maintain uncertainty in sparse federated optimization by preventing early commitment to sparse support. We examine its impact under data heterogeneity, client participation heterogeneity, and sparsity. Experiments on synthetic and real-world benchmarks show consistent improvements over federated iterative hard thresholding (Fed-IHT) and pruning after dense federated averaging (FedAvg) training, both in statistical performance on test data and in sparsity recovery accuracy.

Safety classifiers deployed in production operate under a stationarity assumption that fails silently: when input distributions drift, accuracy degrades with no error signal until ground-truth labels arrive. We present an online monitor that detects distributional shift in classifier scores via a sliding-window KS statistic with empirically calibrated alarm thresholds. In a pre-registered factorial evaluation (4 classifiers $\times$ 5 shift conditions $\times$ 20 seeds $\times$ 2 window sizes; 800 cells), the monitor achieves 86.6% valid detection (mean latency 39.5 steps) across synthetic-onset, real-jailbreak, and adversarial regimes; a classifier $\times$ shift interaction ($η^2 = 0.185$) shows that monitoring must be tuned per classifier. Attempting to recover post-detection coverage via weighted conformal prediction exposes a failure mode: density-ratio estimation collapses for generative classifiers because logistic regression separates source from target perfectly in 3584-4096-dimensional embedding space, clipping all importance weights to zero; projecting to $\leq 32$ dimensions restores coverage. We then extend the framework to gradient-based evasion and give the first threat-model characterisation of score-disagreement monitoring as a canary. We falsify three assumptions: that architectural diversity drives the signal (false, $η^2 = 0.011$), that it is generic out-of-distribution detection (false, GCG-specific, $p < 10^{-12}$), and that an adaptive attacker can suppress it (false while the canary is confident). We derive the exact security boundary, a confidence-gated equilibrium at which a monitor-aware attacker stalls at gap $= 1/(2λ)$, and provide a calibration-free scan martingale achieving false-alarm rate $\leq 1\%$ across all classifiers with no per-model tuning.

Modern LLM workflows move coordinate-indexed objects across checkpoints: steering vectors, sparse autoencoders, top-$k$ neuron sets, attribution lists, and merge alignments. This is only well posed after fixing the model's residual-stream gauge, which we show is architecture-dependent: LayerNorm residual charts have permutation gauge $S_d$ (up to a global sign flip), while RMSNorm charts with generic per-channel gain have signed-permutation gauge $B_d = S_d \ltimes \{\pm 1\}^d$. Permutation-only alignment is therefore symmetry-incomplete for RMSNorm models. We introduce sign-marginalized Hungarian matching and prove a sharp failure mode: with decorrelated coordinates, raw signed-correlation matching has a structural permutation-accuracy ceiling at the positive-sign fraction of the true gauge, which sign-marginalization removes. We then make coordinate-preserving transport, not function-level merging, the primary object: composing saved-checkpoint local $B_d$ gauges along same-base fine-tuning trajectories recovers 91.1% of cross-run coordinates at 1500 steps versus 60.3% for endpoint matching, and the gain is not explained by merely routing through the base. The recovered gauge transfers tools that permutation-only alignment breaks: TinyLlama SAE reconstruction has NMSE 0.004 under $B_d$ versus 1.08 under $S_d$; Qwen sentiment steering preserves 95.8% of its effect versus 17.2%; refusal steering reverses sign under $S_d$; coordinate-preserving merges behave the same way. The same covariance governs stateful training: signed transport of AdamW state preserves the resumed trajectory, while permutation-only state follows a different one from a functionally identical checkpoint. Finally, gauge-sweep audits show index-level interpretability claims are reproducible only relative to an explicit gauge.

Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.

How some people's brains make an extraordinary recovery from stroke A well-known actor who had experienced a stroke was treated by stroke specialist Sandor Nardai. The actor had been left with aphasia, or an impaired ability to speak - brutal for anyone, but "probably the most devastating thing that could happen to an actor", says Nardai. After three months of recovery, though, the actor was able to say some words. After a year, he voiced a commercial. Remarkably, he eventually got well enough to return to live theatre, says Nardai, who is at Semmelweis University in Hungary.

A cascade over a network refers to the diffusion process where behavior changes occurring in one part of an interconnected population lead to a series of sequential changes throughout the entire population. In recent years, there has been a surge in interest and efforts to understand and model cascade mechanisms since they motivate many significant research topics across different disciplines. The propagation structure of cascades is governed by underlying diffusion networks that are often hidden. Inferring diffusion networks thus enables interventions in cascading process to maximize information propagation and provides insights into the Granger causality of interaction mechanisms among individuals. In this project, we propose a novel double network mixture model for inferring latent diffusion network in presence of strong cascade heterogeneity. The new model represents cascade pathways as a distributional mixture over diffusion networks that capture different cascading patterns at the population level. We develop a data-driven optimization method to infer diffusion networks using only visible temporal cascade records, avoiding the need to model complex and heterogeneous individual states. Both statistical and computational guarantees are established for the proposed method. We apply the proposed model to analyze research topic cascades in social sciences across U.S. universities and uncover the latent research topic diffusion network among top U.S. social science programs.

Real-world datasets collected from sensors or human inputs are prone to noise and errors, posing significant challenges for applying offline reinforcement learning (RL). While existing methods have made progress in addressing corrupted actions and rewards, they remain insufficient for handling corruption in high-dimensional state spaces and for cases where multiple elements in the dataset are corrupted simultaneously. Diffusion models, known for their strong denoising capabilities, offer a promising direction for this problem--but their tendency to overfit noisy samples limits their direct applicability. To overcome this, we propose Ambient Diffusion-Guided Dataset Recovery (ADG), a novel approach that pioneers the use of diffusion models to tackle data corruption in offline RL. First, we introduce Ambient Denoising Diffusion Probabilistic Models (DDPM) from approximated distributions, which enable learning on partially corrupted datasets with theoretical guarantees.

We introduce Cycle-Sync, a robust and global framework for estimating camera poses (both rotations and locations). Our core innovation is a location solver that adapts message-passing least squares (MPLS)--originally developed for group synchronization--to camera location estimation. We modify MPLS to emphasize cycle-consistent information, redefine cycle consistencies using estimated distances from previous iterations, and incorporate a Welsch-type robust loss. We establish the strongest known deterministic exact-recovery guarantee for camera location estimation, showing that cycle consistency alone--without access to inter-camera distances--suffices to achieve the lowest sample complexity currently known. To further enhance robustness, we introduce a plug-and-play outlier rejection module inspired by robust subspace recovery, and we fully integrate cycle consistency into MPLS for rotation synchronization. Our global approach avoids the need for bundle adjustment. Experiments on synthetic and real datasets show that Cycle-Sync consistently outperforms leading pose estimators, including full structure-frommotion pipelines with bundle adjustment.

Building on a two-parameter Weibull framework for diagnosing transformer weight distributions, we study why the Weibull weight-scale parameter $λ$ grows, overshoots, and then relaxes during AdamW training. We derive a leading-order three-force decomposition of the squared weight norm from the AdamW update: an alignment force measuring the correlation between weights and the adaptive update direction, an injection force from adaptive step magnitude, and a decay force from decoupled weight decay. On self-trained Pythia-70M models with ground-truth optimizer moments, alignment dominates the rise phase, contributing 88-94% of the absolute force budget across four random seeds and remaining robust to super-weight removal. Near saturation, alignment and decay approach balance, explaining the transition from weight-scale growth to relaxation. These force dynamics directly govern the squared-norm component underlying $λ(t)$; the remaining RMS-to-Weibull reconstruction offset is measurable and decomposes into bridge and integration components, totaling approximately 5-6% in densely sampled regions. To extend the analysis to real models where optimizer moments are unavailable, we introduce a spline displacement method that recovers the alignment force from sparse checkpoints with approximately 92-94% accuracy, about twice the naive two-point baseline. We further observe that the peak value of $λ(t)$ varies with training-data coherence in our experiments, suggesting a data-dependent component of weight-scale growth that we leave to a controlled follow-up study. Code and data are available at https://github.com/tiexinding/NPM-Weibull-public.

Clustering is a fundamental problem in statistics and machine learning. We propose the first one-bit clustering method for two-component sub-Gaussian mixture models. The method uses only one bit per entry of each sample obtained via a dithered quantizer. Under a mild non-spikiness condition on the cluster centers, we show that a variant of Lloyd's algorithm achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to that in the unquantized setting. This result further implies exact recovery under an explicit separation condition, which exceeds the optimal threshold for unquantized data by only a logarithmic factor. When the dimension $p$ is sufficiently large, the non-spikiness condition can be enforced by applying a random rotation using a Haar distributed matrix prior to quantization. In particular, it holds with high probability when $p \gtrsim 1$ for partial recovery and $p \gtrsim \log n \log\log n$ for exact recovery, where $n$ is the sample size. We also establish a minimax lower bound, showing that the misclassification rate and separation condition exhibit sharp constants in general. Numerical results are provided to corroborate the theory and demonstrate the efficacy of the proposed method.