A Bilevel Cycle Generative Adversarial Network (BCGAN) is formed that comprises two generative adversarial networks and a specific manifold fitting module that establishes upper bounds on the generalization error for stochastic bilevel minimax problems, revealing the relationship between generalization capability and parameter settings.
Abstract
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.
Point cloud denoising is essentially a geometric recovery task that aims to reconstruct the intrinsic structure of a smooth 2D Riemannian manifold embedded in R^3 from noisy, discrete ambient-space samples. Despite the remarkable progress of modern manifold-aware encoders and generative transport models in geometric representation learning, a fundamental objective-geometry mismatch remains underexplored. Theoretically, we identified that this mismatched coupling leads to geometric gradient interference, where conflicting optimization objectives result in structural degradation and point clustering. We introduce Riemannian Orthogonally Decoupled Regularization (RODR) to reformulate the optimization trajectory by disentangling the normal (fitting) and tangential (distribution) components. Guided by a vector-attention and entropy-aware adaptive strategy, RODR effectively preserves high-fidelity geometric details while maintaining sampling uniformity. Experiments demonstrate that RODR reaches performance comparable to state-of-the-art baselines and suggests improved distribution regularity and reduced local aggregation effectively. Our work establishes a generic and interpretable framework for disentangled geometric optimization in point cloud processing.
This paper reveals that samples generated by a well-trained generative model are close to clean ones but far from adversarial ones, and proposes Consistency Model-based Adversarial Purification (CMAP), which optimizes vectors within the latent space of a pre-trained consistency model to generate samples for restoring clean data.
Shuhai Zhang, Jiahao Yang, Hui Luo et al.· IEEE Transactions on Pattern...· 0 citations
In non-stationary streaming environments, simultaneously adapting to complex, non-linear domain shifts via continual learning while mitigating the catastrophic effects of severe, uncalibrated label noise poses a fundamental mathematical challenge. In this paper, we propose \FlatManifold{}, a novel, streamlined robust continual learning framework that utilizes a Nystr\"om manifold flattening map based on the kernel trick and projection onto an orthogonalized Reproducing Kernel Hilbert Space (RKHS). Unlike traditional methods that rely on complex, error-prone sample-filtering pipelines, the proposed approach exploits the intrinsic mathematical robustness of the flattened space itself. By mapping feature distributions onto a fixed orthogonal target topology with a ridge regularizer, the framework naturally smoothes and counteracts the influence of extreme label noise during the optimization process. Concurrently, catastrophic forgetting is prevented via a continual topology brake term that leverages the covariance matrix of past experiences. Extensive evaluation on real-world multi-session robotics datasets demonstrates that even under severe conditions featuring 40\% symmetric label noise, \FlatManifold{} successfully mitigates gradient corruption. Under extreme cross-session domain shifts spanning various seasons and lighting conditions, the proposed framework establishes high generalization capabilities, significantly outperforming standard sequential optimization baselines and proving that structural linearization itself serves as a powerful mathematical barrier against distributed label corruption.
We introduce the Intrinsic Hybrid Latent Diffusion Model (ILDM), a generative framework that integrates probabilistic dimensionality reduction with geometry-aware diffusion on unknown manifolds. While diffusion models (DMs) have achieved state-of-the-art results in high-dimensional data synthesis, they rely on large training datasets and ignore intrinsic geometric structure. Latent diffusion models (LDMs) address the high dimensionality by learning a latent space, but they typically impose a Euclidean structure, failing to capture the underlying manifold geometry, especially problematic in data-sparse regimes. ILDM addresses these limitations by interpreting the latent space as a chart of an unknown Riemannian manifold, with geometry and uncertainty quantified through a probabilistic decoder. The forward process is a hybrid diffusion that switches between Riemannian and Euclidean dynamics based on local uncertainty, where the Riemannian component is governed by a probabilistic metric tensor derived from the decoder. To learn the generative dynamics, we introduce an approximate denoising score matching method tailored to the hybrid diffusion setting, enabling a backward process defined by hybrid Langevin dynamics. Experiments on COIL-100, MNIST, and cardiac MRI datasets demonstrate that ILDM significantly improves generation quality, achieving lower FID and LPIPS scores compared to standard diffusion and latent diffusion models.
Non-negative matrix factorization (NMF) has been widely studied for data representation learning and dimensionality reduction. However, existing NMF-based approaches often suffer from limitations such as inadequate preservation of manifold structure and insufficient exploitation of available label information. To address these issues, this paper proposes a novel NMF-based framework that incorporates supervised information to enhance class discriminability in the learned latent space. In addition, the proposed model explicitly preserves the local geometric structure of the data during factorization, thereby improving the fidelity of the learned representation. By jointly integrating discriminative supervision and manifold preservation, the proposed approach yields more informative embeddings for downstream machine learning tasks. Extensive experiments on image classification and community detection benchmarks demonstrate that the proposed model consistently outperforms several state-of-the-art approaches.
H. Moayed· Acta Universitatis Sapientia...· 0 citations