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A Unified Semi-Supervised Dimensionality Reduction Framework for Manifold Learning

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 Publication date 2009
and research's language is English




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We present a general framework of semi-supervised dimensionality reduction for manifold learning which naturally generalizes existing supervised and unsupervised learning frameworks which apply the spectral decomposition. Algorithms derived under our framework are able to employ both labeled and unlabeled examples and are able to handle complex problems where data form separate clusters of manifolds. Our framework offers simple views, explains relationships among existing frameworks and provides further extensions which can improve existing algorithms. Furthermore, a new semi-supervised kernelization framework called ``KPCA trick is proposed to handle non-linear problems.



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379 - Xuanqing Liu , Si Si , Xiaojin Zhu 2019
In this paper, we proposed a general framework for data poisoning attacks to graph-based semi-supervised learning (G-SSL). In this framework, we first unify different tasks, goals, and constraints into a single formula for data poisoning attack in G-SSL, then we propose two specialized algorithms to efficiently solve two important cases --- poisoning regression tasks under $ell_2$-norm constraint and classification tasks under $ell_0$-norm constraint. In the former case, we transform it into a non-convex trust region problem and show that our gradient-based algorithm with delicate initialization and update scheme finds the (globally) optimal perturbation. For the latter case, although it is an NP-hard integer programming problem, we propose a probabilistic solver that works much better than the classical greedy method. Lastly, we test our framework on real datasets and evaluate the robustness of G-SSL algorithms. For instance, on the MNIST binary classification problem (50000 training data with 50 labeled), flipping two labeled data is enough to make the model perform like random guess (around 50% error).
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Dimension reduction (DR) aims to learn low-dimensional representations of high-dimensional data with the preservation of essential information. In the context of manifold learning, we define that the representation after information-lossless DR preserves the topological and geometric properties of data manifolds formally, and propose a novel two-stage DR method, called invertible manifold learning (inv-ML) to bridge the gap between theoretical information-lossless and practical DR. The first stage includes a homeomorphic sparse coordinate transformation to learn low-dimensional representations without destroying topology and a local isometry constraint to preserve local geometry. In the second stage, a linear compression is implemented for the trade-off between the target dimension and the incurred information loss in excessive DR scenarios. Experiments are conducted on seven datasets with a neural network implementation of inv-ML, called i-ML-Enc. Empirically, i-ML-Enc achieves invertible DR in comparison with typical existing methods as well as reveals the characteristics of the learned manifolds. Through latent space interpolation on real-world datasets, we find that the reliability of tangent space approximated by the local neighborhood is the key to the success of manifold-based DR algorithms.
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