We introduce FAEclust, a novel functional autoencoder framework for cluster analysis of multi-dimensional functional data, data that are random realizations of vector-valued random functions. Our framework features a universal-approximator encoder that captures complex nonlinear interdependencies among component functions, and a universal-approximator decoder capable of accurately reconstructing both Euclidean and manifold-valued functional data. Stability and robustness are enhanced through innovative regularization strategies applied to functional weights and biases. Additionally, we incorporate a clustering loss into the network's training objective, promoting the learning of latent representations that are conducive to effective clustering. A key innovation is our shape-informed clustering objective, ensuring that the clustering results are resistant to phase variations in the functions. We establish the universal approximation property of our non-linear decoder and validate the effectiveness of our model through extensive experiments.

We analyze task orderings in continual learning for linear regression, assuming joint realizability of training data. We focus on orderings that greedily maximize dissimilarity between consecutive tasks, a concept briefly explored in prior work but still surrounded by open questions. Using tools from the Kaczmarz method literature, we formalize such orderings and develop geometric and algebraic intuitions around them. Empirically, we demonstrate that greedy orderings converge faster than random ones in terms of the average loss across tasks, both for linear regression with random data and for linear probing on CIFAR-100classification tasks. Analytically, in a high-rank regression setting, we prove a loss bound for greedy orderings analogous to that of random ones. However, under general rank, we establish a repetition-dependent separation. Specifically, while prior work showed that for random orderings, with or without replacement, the average loss after k iterations is bounded by O(1/ k)--we prove that single-pass greedy orderings may fail catastrophically, whereas those allowing repetition converge at rate O(1/ 3 k). Overall, we reveal nuances within and between greedy and random orderings.

Existing methods for integrating spatial cues, such as point clouds or depth, either require specialized sensors or fail to effectively exploit depth information for higher-order reasoning. To this end, we propose a novel Spatial Sense and Reasoning method, dubbed SSR, a novel framework that transforms raw depth data into structured, interpretable textual rationales. These textual rationales serve as meaningful intermediate representations to significantly enhance spatial reasoning capabilities. Additionally, we leverage knowledge distillation to compress the generated rationales into compact latent embeddings, which facilitate resourceefficient and plug-and-play integration into existing VLMs without retraining. To enable comprehensive evaluation, we introduce a new dataset named SSR-COT, a million-scale visual-language reasoning dataset enriched with intermediate spatial reasoning annotations, and present SSRBENCH, a comprehensive multi-task benchmark. Extensive experiments on multiple benchmarks demonstrate SSR substantially improves depth utilization and enhances spatial reasoning, thereby advancing VLMs toward more human-like multi-modal understanding.

Large language models (LLMs) raise concerns about content authenticity and integrity because they can generate human-like text at scale. Text watermarks, which embed detectable statistical signals into generated text, offer a provable way to verify content origin. Many detection methods rely on pivotal statistics that are i.i.d.

Parameter-efficient fine-tuning of large language models (LLMs) can exhibit problematic memorization of individual training examples. Empirical privacy auditing (EPA) quantifies this risk by measuring realistic data leakage on membership inference (MI) or reconstruction attacks. A key challenge in EPA is designing ``canary'' examples that are mixed with the privacy-sensitive training data. We propose generating synthetic canaries via high-temperature sampling ($T \geq 0.8$) from LLMs, using prompts tailored to the privacy-sensitive training data. These canaries act as high-influence outliers, ensuring high identifiability and hence strong audits. Further, since the canaries are themselves non-private, they are inspectable and can be inserted with repetition without jeopardizing the privacy of the real data. An important use of models fine-tuned on privacy-sensitive data is the generation of synthetic data. This also comes with privacy risk. We introduce a powerful synthetic data audit based on fine-tuning an auxiliary model on the synthetic data. Auditing the auxiliary model for the original canaries then provides a strong estimate of the privacy leakage through the synthetic data. Finally, leveraging our strong auditing methodologies, we perform a systematic investigation into the interacting effects of model capacity and canary entropy on memorization.

Discrete automated processes in industrial and cyber-physical systems often exhibit a repetitive structure in which successive repetitions follow a common trajectory while differing in duration, amplitude, and fine-scale dynamics. Such \emph{approximately periodic} behavior poses a challenge for Gaussian Processes (GP) modeling: strictly periodic models suppress inter-repetition variability, while non-periodic models fail to capture the strong structural regularities required for generation. In this work, we propose a stochastic generative model for approximately periodic time series. The model is based on a GP whose posterior is modulated by a novel kernel. Our approach decouples intra-repetition structure from inter-repetition variability through a two-stage construction which yields a generative distribution with a identical mean function across repetitions, while allowing smooth variation between repetitions. The modeling choices are supported by an implementation in which realistic synthetic trajectories are generated from toy datasets.

Confidence sequences based on test martingales provide time-uniform uncertainty quantification for the mean of bounded IID observations without parametric distributional assumptions. Their practical efficiency, however, depends strongly on the choice of martingale updates, and many existing constructions do not exploit prior information about plausible data-generating distributions or mean values. We propose a Bayes-assisted framework that uses a Bayesian working predictive model to adaptively construct confidence sequences. For each candidate mean and time point, the predictive distribution selects, among valid one-step martingale factors, the update maximising predictive expected log-growth; validity is therefore preserved even when the prior or working model is misspecified. We prove that if the predictive distribution is Wasserstein-consistent, the resulting procedure is asymptotically log-optimal, matching the per-sample log-growth of an oracle procedure with access to the true distribution. We instantiate the framework using robust predictives based on Dirichlet-process mixtures and Bayesian exponentially tilted empirical likelihood. Experiments on synthetic data, sequential best-arm identification for LLM evaluation, and prediction-powered inference show that informative priors can substantially reduce confidence-sequence width and sampling effort while retaining anytime-valid coverage.

A.1 Preliminary Study2 The basic GPT-2 model1 is trained from scratch on each corpus, which has 12 transformer blocks3 and 12 attention heads with 768 hidden dimensions. The Huggingface transformers [4] and Pytorch4 toolkit [2] are used to train the GPT-2 model in the distributed manner on A100 GPU server. The5 hyper-parameters during training are shown in Table 1.6 Hyper-parameter Value Optimization steps 100K Test interval 10K Dropout rate 0.1 Grad clipping 1.0 Learning rate 5e 5 Batch size 128 Maximum sequence length 256 Warmup steps 10K Learning scheduler Linear decay Random seed 0 Number of GPUs 4 Learning objective Cross-Entropy Loss Table 1: The hyper-parameters during GPT-2 training procedure. Most of the hyper-parameters for our proposed method are the same as that in Table 1 for better8 variable controlling. The specific hyper-parameters for our proposed method are the length of9 repetitive n-gram and its repetition dropout rate p, which are set as 2 and 0.6, respectively.10

Previous work [2, 1] has observed that standard training and greedy decoding usually cause models to generate consecutive repetitive texts. These consecutive repetitive texts are redundant and do not convey new information, which is avoided in human language. There are three types of consecutive repetitions: word-level, phrase-level and sentence-level. The phrase-level means that a phrase consisting of several words is repeated consecutively. The sentence in our paper refers to a sequence split by '.!?' is repeated consecutively 2. We calculate the ratio of consecutive repetition in a sequence x as follows.