Prompt tuning has emerged as a powerful parameter-efficient fine-tuning technique, allowing large pretrained Transformers to adapt to downstream tasks by optimizing a small set of prompt embeddings. Despite its empirical success, the extent to which prompt tuning can memorize data remains poorly understood. In this paper, we provide both theoretical and empirical analyses of data memorization ability of prompt-tuned Transformers. Building on recent theoretical frameworks, we derive an upper bound on the required prompt length for exact memorization of finite datasets and establish a trade-off between prompt length and the number of autoregressive generation steps. Specifically, we show that a constant-size Transformer can memorize ninput-output pairs with prompts of length O( nN), where N denotes the sequence length. Empirical results further demonstrate that prompt-tuned, randomly initialized Transformers are able to effectively memorize finite datasets. These models also capture the intrinsic low-rank structure of the data, leading to a reduction in the required prompt length. Finally, we analyze how the initialization of the Transformer backbone affects the performance of prompt tuning. Our findings provide new insights into the expressivity, efficiency, and underlying mechanisms of prompt tuning, bridging theoretical memorization limits with observed empirical behaviors.

Memorization in diffusion models is often treated as a global property of the model or dataset. In practice, however, a single diffusion model can simultaneously generate both memorized and novel samples. Which training samples are most likely to be memorized? In this work, we show that memorization is governed by \emph{local data coverage}. Leveraging the connection between diffusion models and kernel density estimation (KDE), we derive a theoretical criterion that predicts whether a point is memorized based on the density of training data in its neighborhood and the size of the training dataset. In the high-dimensional limit, this leads to a sharp, local transition: regions of low coverage are dominated by isolated training samples, which are memorized, while dense regions support interpolation and generalization. We validate these predictions empirically, showing that memorization increases with local sparsity and that diffusion models exhibit a coexistence of memorized and novel samples within the same model. Extending this framework to multi-class settings, we further show that classes with higher intra-class sparsity (and thus lower local coverage) are more strongly memorized. Our results provide a local view of memorization in diffusion models, explaining when and where memorization occurs in terms of data geometry.

Prompt tuning has emerged as a powerful parameter-efficient fine-tuning technique, allowing large pretrained Transformers to adapt to downstream tasks by optimizing a small set of prompt embeddings. Despite its empirical success, the extent to which prompt tuning can memorize data remains poorly understood. In this paper, we provide both theoretical and empirical analyses of data memorization ability of prompt-tuned Transformers. Building on recent theoretical frameworks, we derive an upper bound on the required prompt length for exact memorization of finite datasets and establish a trade-off between prompt length and the number of autoregressive generation steps. Specifically, we show that a constant-size Transformer can memorize $n$ input-output pairs with prompts of length $\tilde{O}(\sqrt{nN})$, where $N$ denotes the sequence length. Empirical results further demonstrate that prompt-tuned, randomly initialized Transformers are able to effectively memorize finite datasets. These models also capture the intrinsic low-rank structure of the data, leading to a reduction in the required prompt length. Finally, we analyze how the initialization of the Transformer backbone affects the performance of prompt tuning. Our findings provide new insights into the expressivity, efficiency, and underlying mechanisms of prompt tuning, bridging theoretical memorization limits with observed empirical behaviors.

Highly over-parameterized models can simultaneously memorize noisy labels and generalize well, yet how these behaviors coexist remains poorly understood. In this work, we investigate the underlying mechanisms of this coexistence using modular arithmetic tasks under heavy label noise. Through extensive experiments on two-layer neural networks, we find that larger models tend to generalize better under appropriate optimization and model configurations, while noisy labels are memorized faster than clean data. Over-parameterized models internally form a generalization structure, but its expression in the output is suppressed by the need to fit noisy labels. Remarkably, even with 80\% label noise, near-perfect test accuracy can be achieved by extracting this internal structure using frequency-based methods. We further propose a task-agnostic method to partition networks into generalization and memorization components. Although this subnetwork improves generalization, it is limited compared with frequency-based extraction, indicating that the generalization structure is distributed across neurons and motivating the development of new tools to retrieve generalizable knowledge from over-parameterized networks.

Diffusion models are central to modern generative modeling, and understanding how they balance memorization and generalization is critical for reliable deployment. Recent work has shown that memorization in diffusion models is shaped by training dynamics, with generalization and memorization emerging at different stages of training. However, deployed diffusion models are often further distilled, introducing an additional training phase whose impact on memorization is not well understood. In this work, we analyze how distillation reshapes memorization behavior in diffusion models, taking consistency distillation as a representative framework. Empirically, we show that when applied to a teacher model that has memorized data, consistency distillation significantly reduces transferred memorization in the student while preserving, and sometimes improving, sample quality. To explain this behavior, we provide a theoretical analysis using a random feature neural network model [Bonnaire et al., 2025], showing that consistency distillation suppresses unstable feature directions associated with memorization while preserving stable, generalizable modes. Our findings suggest that distillation can serve not only as an acceleration tool, but also as a mechanism for improving the memorization-generalization trade-off.

A key capability of modern neural networks is their capacity to simultaneously learn underlying rules and memorize specific facts or exceptions. Yet, theoretical understanding of this dual capability remains limited. We introduce the Rules-and-Facts (RAF) model, a minimal solvable setting that enables precise characterization of this phenomenon by bridging two classical lines of work in the statistical physics of learning: the teacher-student framework for generalization and Gardner-style capacity analysis for memorization. In the RAF model, a fraction $1 - \varepsilon$ of training labels is generated by a structured teacher rule, while a fraction $\varepsilon$ consists of unstructured facts with random labels. We characterize when the learner can simultaneously recover the underlying rule - allowing generalization to new data - and memorize the unstructured examples. Our results quantify how overparameterization enables the simultaneous realization of these two objectives: sufficient excess capacity supports memorization, while regularization and the choice of kernel or nonlinearity control the allocation of capacity between rule learning and memorization. The RAF model provides a theoretical foundation for understanding how modern neural networks can infer structure while storing rare or non-compressible information.

The advancement of machine learning (ML) techniques relies on large amounts of training data. However, data collection also poses a significant risk of exposing private information.

Memorization happens through the entire SSL encoder .