When Does Training on Synthetic Data Provably Work?

In many real-world computer vision applications, including medical imaging and industrial inspection, binary classification tasks are characterized by a severe scarcity of positive samples. A widely adopted solution is to generate synthetic positive data using image-to-image transformations applied to negative samples. However, a fundamental challenge remains: how can we reliably assess whether such synthetic data will improve downstream model performance? In this work, we propose a geometry-driven metric that predicts the utility of synthetic data without requiring model training. Our approach operates in the embedding space of a pre-trained foundation model and represents the dataset through difference vectors between samples. We evaluate whether the weight vector of a linear classifier can be expressed within the subspace spanned by these variations by measuring the relative projection error. Intuitively, if the variations induced by synthetic data capture task-relevant directions, their span can approximate the classifier, resulting in low projection error. Conversely, poor synthetic data fails to span these directions, leading to higher error. Across multiple datasets and architectures, we show that this metric exhibits strong correlation with downstream classification performance of CNNs trained on mixtures of real negative and synthetic positive data. These findings suggest that the proposed metric serves as a practical and informative tool for evaluating synthetic data quality in data-scarce settings.

Advances in diffusion-based generative models have motivated the use of synthetic image generation to alleviate data scarcity in vision tasks. While this strategy has shown promise in natural image benchmarks such as ImageNet, its effectiveness in sparse, high-variance real-world domains remains unclear. In this work, we focus on domains where images differ substantially from common image datasets and additional data are expensive to obtain. Against non-generative data augmentation baselines, we evaluate the downstream classifier performance improvements yielded by two schools of generative sparse data extension: distribution modeling and sample perturbation. Across five trauma classification tasks using subject-wise train--validation splits, no generative approach consistently outperforms a strong non-generative baseline. Feature-space analysis reveals recurring failure modes: memorization or collapse, distributional drift, and generation of visually plausible but simplified canonical instances that are easier to classify than real data.

Digital twins (DTs) and learned world models are increasingly used to generate synthetic data that augment the scarce real datasets available for training artificial intelligence (AI) models in engineering systems. Owing to the inevitable simulation-to-reality (sim-to-real) gap, however, augmentation may fail to improve the performance of the trained model on the real data distribution. This paper addresses the resulting decision problem: Given a real dataset, a candidate synthetic dataset, and a fixed learning algorithm, decide whether training on the augmented dataset improves the true, population-level performance, while consuming as few real test data points as possible. Two formulations are considered: a direct test on the mean loss difference between the two trained models, and a symmetry-based test on the paired loss difference, which trades a stronger null assumption for faster evidence accumulation. For the latter, we introduce the {adaptive e-process sign-flip test} (aeSFT), a doubly adaptive procedure that adapts both the number of Monte Carlo sign-flip rounds, and hence the computational cost, and the amount of real test data consumed. aeSFT yields anytime-valid Type-I error control, with no need to pre-specify the test-set size. Experiments on a synthetic-data classification task, a DT-aided wireless packet-scheduling task, and a radio-map prediction task show that aeSFT identifies useful synthetic data using substantially fewer real test samples than mean-based sequential testing, matches the power of fixed-sample sign-flip testing and the paired $t$-test, while keeping the false-positive rate below the target level.