The rapid progress in generative models has resulted in impressive leaps in generation quality, blurring the lines between synthetic and real data. Web-scale datasets are now prone to the inevitable contamination by synthetic data, directly impacting the training of future generated models. Already, some theoretical results on self-consuming generative models (a.k.a., iterative retraining) have emerged in the literature, showcasing that either model collapse or stability could be possible depending on the fraction of generated data used at each retraining step. However, in practice, synthetic data is often subject to human feedback and curated by users before being used and uploaded online. For instance, many interfaces of popular text-to-image generative models, such as Stable Diffusion or Midjourney, produce several variations of an image for a given query which can eventually be curated by the users. In this paper, we theoretically study the impact of data curation on iterated retraining of generative models and show that it can be seen as an implicit preference optimization mechanism .

In this paper, we develop MAE as a unified, modality-agnostic SSL framework.

This allows us to interpret the latent variables as object positions and lets us train the network without supervision.

We propose below the proofs of the results presented in the main text. RKBS developed in (Zhang et al., 2009; Song et al., 2013) to develop the notion of vector-valued (Giles, 1967). " 0, @ j ď n, @ u P U (9) which allows us to say that O P RKBS (Corollary 3.2 of Zhang (2013)) that we recall hereafter: We first define for any linear operator We show our result in the case J=1 and can be directly extended to any cardinality J. Specifically, we tested three expressions: Exp. The two first expressions yield similar result in the ADR experiment at an equal compute cost. We also tried a'branch' and'trunk' networks formulation of the model as in DeepONet (Lu T able S.2: Summary of the architectural hyperparameters used to build the Transducer in the four experiments. 'Depth' corresponds to network number of layers, 'MLP dim' to the dimensionality of the hidden layer As stated, we used for all experiments, the same meta-training procedure. T able S.3: Summary of the meta-learning hyperparameters used to meta-train the Transducer in our four Figure S.1: Examples of sampled functions δ p xq and ν px q used to build operators O We train Tranducers for 200K gradient steps. Flow library (Holl et al., 2020) that allows for batched and differentiable simulations of fluid dynamics Figure S.5: Magnitude of the complex coefficients of the Fourier transform of an exemple pair of input and In order to tackle the high-resolution climate modeling experiment, we take inspiration from Pathak et al. (2022), which combines neural operators with the patch splitting L " 12, in order to match number of trainable parameters.

The supplemental material is organized as follows. Section A provides the results of all the additional synthetic experiments and real data results for various kernel-based methods and the detailed settings. Section B describes the algorithm details we use in Section A. In Section C, we provide some useful lemmas and all the technical proofs of the theoretical results in the main text. In this section, we provide more experiment results, including KRR (Section A.1), KQR for various Section A.7. A.1 Kernel ridge regression For the squared loss, we consider the following two examples. TIRW estimator still performs significantly better. A.2 Kernel quantile regression For the check loss, we consider the following two examples.

Such a phenomenon is illustrated in Figure 1 where the learned predictive function from the source data may significantly differ from the true function.