Bilevel manifold fitting

The manifold assumption states that high-dimensional ambient data possess a low-dimensional geometric structure, which has promoted many manifold learning models achieving promising performance across wide applications. However, when there exists additive noise or noisy dimensions, it is challenging to estimate the latent manifold since the noisy information tends to mislead the data-driven mapping from the ambient space to the latent space. To address this issue, we formulate a Bilevel Cycle Generative Adversarial Network (BCGAN) that comprises two generative adversarial networks and a specific manifold fitting module. This network can automatically assign masks to the ambient or latent data, learn robust mutual mappings, and generate new synthetic samples. Theoretically, we establish upper bounds on the generalization error for stochastic bilevel minimax problems, revealing the relationship between generalization capability and parameter settings. Experiments on both synthetic and real-world datasets verify the competitiveness and robustness of the proposed approach for manifold fitting with corrupted data.