In a semi-supervised learning scenario, (possibly noisy) partially observed labels are used as input to train a classifier, in order to assign labels to unclassified samples. In this paper, we construct a complete graph-based binary classifier given only samples' feature vectors and partial labels. Specifically, we first build appropriate similarity graphs with positive and negative edge weights connecting all samples based on inter-node feature distances. By viewing a binary classifier as a piecewise constant graph-signal, we cast classifier learning as a signal restoration problem via a classical maximum a posteriori (MAP) formulation. One unfortunate consequence of negative edge weights is that the graph Laplacian matrix L can be indefinite, and previously proposed graph-signal smoothness prior x T Lx for candidate signal x can lead to pathological solutions. In response, we derive a minimum-norm perturbation matrix ∆ that preserves L's eigen-structure-based on a fast lower-bound computation of L's smallest negative eigenvalue via a novel application of the Haynsworth inertia additivity formula-so that L + ∆ is positive semi-definite, resulting in a stable signal prior. Further, instead of forcing a hard binary decision for each sample, we define the notion of generalized smoothness on graphs that promotes ambiguity in the classifier signal. Finally, we propose an algorithm based on iterative reweighted least squares (IRLS) that solves the posed MAP problem efficiently. Extensive simulation results show that our proposed algorithm outperforms both SVM variants and previous graph-based classifiers using positive-edge graphs noticeably.
Despite generative adversarial networks (GANs) can hallucinate photo-realistic high-resolution (HR) faces from lowresolution (LR) faces, they cannot guarantee preserving the identities of hallucinated HR faces, making the HR faces poorly recognizable. To address this problem, we propose a Siamese GAN (SiGAN) to reconstruct HR faces that visually resemble their corresponding identities. On top of a Siamese network, the proposed SiGAN consists of a pair of two identical generators and one discriminator. We incorporate reconstruction error and identity label information in the loss function of SiGAN in a pairwise manner. By iteratively optimizing the loss functions of the generator pair and discriminator of SiGAN, we cannot only achieve photo-realistic face reconstruction, but also ensures the reconstructed information is useful for identity recognition. Experimental results demonstrate that SiGAN significantly outperforms existing face hallucination GANs in objective face verification performance, while achieving photo-realistic reconstruction. Moreover, for input LR faces from unknown identities who are not included in training, SiGAN can still do a good job.
Recent advent in graph signal processing (GSP) has led to the development of new graph-based transforms and wavelets for image / video coding, where the underlying graph describes inter-pixel correlations. In this paper, we develop a new transform called signed graph Fourier transform (SGFT), where the underlying graph G contains negative edges that describe anti-correlations between pixel pairs. Specifically, we first construct a one-state Markov process that models both inter-pixel correlations and anti-correlations. We then derive the corresponding precision matrix, and show that the loopy graph Laplacian matrix Q of a graph G with a negative edge and two self-loops at its end nodes is approximately equivalent. This proves that the eigenvectors of Q-called SGFTapproximates the optimal Karhunen-Loève Transform (KLT). We show the importance of the self-loops in G to ensure Q is positive semi-definite. We prove that the first eigenvector of Q is piecewise constant (PWC), and thus can well approximate a piecewise smooth (PWS) signal like a depth image. Experimental results show that a block-based coding scheme based on SGFT outperforms a previous scheme using graph transforms with only positive edges for several depth images.
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