Watermarking, the practice of embedding imperceptible information into media such as images, videos, audio, and text, is essential for intellectual property protection, content provenance and attribution. The growing complexity of digital ecosystems necessitates watermarks for different uses to be embedded in the same media. However, to detect and decode all watermarks, they need to coexist well with one another. We perform the first study of coexistence of deep image watermarking methods and, contrary to intuition, we find that various open-source watermarks can coexist with only minor impacts on image quality and decoding robustness. The coexistence of watermarks also opens the avenue for ensembling watermarking methods. We show how ensembling can increase the overall message capacity and enable new trade-offs between capacity, accuracy, robustness and image quality, without needing to retrain the base models.

Longitudinal treatment decisions from multivariate time-series data require predicting potential outcomes under future treatment sequences in the presence of timevarying confounding, heterogeneous patient dynamics, and limited domain-specific data. Existing longitudinal causal estimators typically address this problem by training a new model for each cohort or simulator. We introduce Causal Longitudinal Prior-Fitted Networks (CAUSALLONGPFN), a prior-fitted network for time-series causal inference in longitudinal treatment-response data and zero-shot in-context counterfactual outcome prediction. To our knowledge, CAUSALLONGPFN is the first PFN-style model for history-conditional potential-outcome prediction under planned longitudinal treatment sequences, with systematic comparison against established longitudinal causal baselines on branchable counterfactual treatmentresponse benchmarks and factual real-world clinical data. The model is pretrained entirely on synthetic episodes sampled from a broad prior over temporal structural causal models, exposing it to treatment-confounder feedback, latent heterogeneity, nonlinear state evolution, delayed effects, and cumulative treatment responses. At test time, CAUSALLONGPFN remains frozen and is used zero-shot: it conditions on support trajectories, a query history, and a planned future treatment sequence, and returns a predictive distribution over future outcomes without gradient updates or propensity-model fitting. Multi-step predictions are obtained by recursively applying the one-step predictor under the specified treatment sequence. We evaluate the model on branchable cancer, HIV, and warfarin benchmarks with ground-truth counterfactual labels, and on factual-only rolling-origin prediction in MIMIC-III ICU trajectories. CAUSALLONGPFN is competitive with domain-trained longitudinal baselines on counterfactual benchmarks and performs strongly on factual MIMIC-III prediction, suggesting that broad synthetic causal pretraining can provide a frozen, amortized alternative for zero-shot longitudinal treatment-response prediction when repeated domain-specific training is costly or impractical.

In Section 3.2, we proposed the crossdistillation (XD) learning scheme. The distillation objective in Eq (10) is the inner decorrelation minimization between embeddings z and [ z]. In addition to the correlation-based distillation loss, we also investigate the negative logarithm(e.g, To avoid the unbalanced loss magnitude, the distillation loss is introduced as the regularization term controlled by the penalty level γ: L = LSACL(zA,zB)+γLCD (1) LCD = ( [ zA]logzA + [ zB]logzB)/2 (2) We empirically observe that the negative logarithm-based distillation loss failed to outperform the proposed cross-distillation loss LCD with inner-decorrelation minimization. As shown in the ImageNet-100 results below: Method Encoder # of Params (M) Linear Eval Acc.

Liu et al. [2018] first showed that stationary importance sampling methods can be viewed as Rao-Blackwellization of IS estimator, and claimed that the expectation of the likelihood-ratios conditioned on state and action is equal to the distribution ratio, as stated in Property 1. For completeness, we present a proof of Property 1. Recall that d This gives us the expression " This additional marginalization step over time allows us to consider time-independent distribution ratios. Then, using the law of total expectation, we can write the expectation of the second sum in (4) as: " Assumption 1. Plugging in the final expression from (5) back into (4) gives us " Note that in the infinite horizon setting where L!1and for finite n, (6) becomes " Similarly, by generalizing this pattern it can be observed that on unrolling n times, we will get, 1 " 0 X For all experiments, we utilize the domains and algorithm implementations from Caltech OPE Benchmarking Suite (COBS) library by Voloshin et al. [2019]. We include a brief description of each of these domains below, and a full description of each can be found in the work by Voloshin et al. [2019]. Graph Environment The Graph environment is a two-chain environment with 2L states and 2 actions.

The IDs of the 10 datasets used in this work, as well as the number of examples and features, are provided in Table 1 in the main manuscript. All of the datasets correspond to binary classification problems, with varying degrees of class imbalance. While the prediction is always performed in the logarithmic domain, when evaluating the models we transform both the labels and the model predictions back into their original domain. The loss function used for training and evaluation is the standard root mean-squared error (sklearn.metrics.mean_squared_error). We download the raw data programmatically using the Kaggle API, which produces the filetrain.tsv.