Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

High-order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non-convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, \emph{high-order Lloyd algorithm} (HLloyd), and \emph{high-order spectral clustering} (HSC), for high-order clustering. The convergence guarantees and statistical optimality are established for the proposed procedure under a mild sub-Gaussian noise assumption. Under the Gaussian tensor block model, we completely characterize the statistical-computational trade-off for achieving high-order exact clustering based on three different signal-to-noise ratio regimes. The analysis relies on new techniques of high-order spectral perturbation analysis and a "singular-value-gap-free" error bound in tensor estimation, which are substantially different from the matrix spectral analyses in the literature. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets.Comment: 65 page

Face Hallucination via Deep Neural Networks.

We firstly address aligned low-resolution (LR) face images (i.e. 16X16 pixels) by designing a discriminative generative network, named URDGN. URDGN is composed of two networks: a generative model and a discriminative model. We introduce a pixel-wise L2 regularization term to the generative model and exploit the feedback of the discriminative network to make the upsampled face images more similar to real ones. We present an end-to-end transformative discriminative neural network (TDN) devised for super-resolving unaligned tiny face images. TDN embeds spatial transformation layers to enforce local receptive fields to line-up with similar spatial supports. To upsample noisy unaligned LR face images, we propose decoder-encoder-decoder networks. A transformative discriminative decoder network is employed to upsample and denoise LR inputs simultaneously. Then we project the intermediate HR faces to aligned and noise-free LR faces by a transformative encoder network. Finally, high-quality hallucinated HR images are generated by our second decoder. Furthermore, we present an end-to-end multiscale transformative discriminative neural network (MTDN) to super-resolve unaligned LR face images of different resolutions in a unified framework. We propose a method that explicitly incorporates structural information of faces into the face super-resolution process by using a multi-task convolutional neural network (CNN). Our method not only uses low-level information (i.e. intensity similarity), but also middle-level information (i.e. face structure) to further explore spatial constraints of facial components from LR inputs images. We demonstrate that supplementing residual images or feature maps with additional facial attribute information can significantly reduce the ambiguity in face super-resolution. To explore this idea, we develop an attribute-embedded upsampling network. In this manner, our method is able to super-resolve LR faces by a large upscaling factor while reducing the uncertainty of one-to-many mappings remarkably. We further push the boundaries of hallucinating a tiny, non-frontal face image to understand how much of this is possible by leveraging the availability of large datasets and deep networks. To this end, we introduce a novel Transformative Adversarial Neural Network (TANN) to jointly frontalize very LR out-of-plane rotated face images (including profile views) and aggressively super-resolve them by 8X, regardless of their original poses and without using any 3D information. Besides recovering an HR face images from an LR version, this thesis also addresses the task of restoring realistic faces from stylized portrait images, which can also be regarded as face hallucination