Low-rank tensor recovery in the presence of sparse but arbitrary errors is an important problem with many practical applications. In this work, we propose a general framework that recovers low-rank tensors, in which the data can be deformed by some unknown transformations and corrupted by arbitrary sparse errors. We give a unified presentation of the surrogate-based formulations that incorporate the features of rectification and alignment simultaneously, and establish worst-case error bounds of the recovered tensor. In this context, the state-of-the-art methods ‘RASL’ and ‘TILT’ can be viewed as two special cases of our work, and yet each only performs part of the function of our method. Subsequently, we study the optimization aspects of the problem in detail by deriving two algorithms, one based on the alternating direction method of multipliers (ADMM) and the other based on proximal gradient. We provide convergence guarantees for the latter algorithm, and demonstrate the performance of the former through in-depth simulations. Finally, we present extensive experimental results on public datasets to demonstrate the effectiveness and efficiency of the proposed framework and algorithms.

Ten years into the revival of deep networks and artificial intelligence, we propose a theoretical framework that sheds light on understanding deep networks

This work proposes a new computational framework for learning a structured generative model for real-world datasets. In particular, we propose to learn a Closed-loop Transcriptionbetween a multi-class, multi-dimensional data distribution and a Linear discriminative representation (CTRL) in the feature space that consists of multiple independent multi-dimensional linear subspaces. In particular, we argue that the optimal encoding and decoding mappings sought can be formulated as a two-player minimax game between the encoder and decoderfor the learned representation. A natural utility function for this game is the so-called rate reduction, a simple information-theoretic measure for distances between mixtures of subspace-like Gaussians in the feature space. Our formulation draws inspiration from closed-loop error feedback from control systems and avoids expensive evaluating and minimizing of approximated distances between arbitrary distributions in either the data space or the feature space. To a large extent, this new formulation unifies the concepts and benefits of Auto-Encoding and GAN and naturally extends them to the settings of learning a both discriminative and generative representation for multi-class and multi-dimensional real-world data. Our extensive experiments on many benchmark imagery datasets demonstrate tremendous potential of this new closed-loop formulation: under fair comparison, visual quality of the learned decoder and classification performance of the encoder is competitive and arguably better than existing methods based on GAN, VAE, or a combination of both. Unlike existing generative models, the so-learned features of the multiple classes are structured instead of hidden: different classes are explicitly mapped onto corresponding independent principal subspaces in the feature space, and diverse visual attributes within each class are modeled by the independent principal components within each subspace.

Reconstruction of object or scene surfaces has tremendous applications in computer vision, computer graphics, and robotics. The topic attracts increased attention with the emerging pipeline of deep learning surface reconstruction, where implicit field functions constructed from deep networks (e.g., multi-layer perceptrons or MLPs) are proposed for generative shape modeling. In this paper, we study a fundamental problem in this context about recovering a surface mesh from an implicit field function whose zero-level set captures the underlying surface. To achieve the goal, existing methods rely on traditional meshing algorithms (e.g., the de-facto standard marching cubes); while promising, they suffer from loss of precision learned in the implicit surface networks, due to the use of discrete space sampling in marching cubes. Given that an MLP with activations of Rectified Linear Unit (ReLU) partitions its input space into a number of linear regions, we are motivated to connect this local linearity with a same property owned by the desired result of polygon mesh. More specifically, we identify from the linear regions, partitioned by an MLP based implicit function, the analytic cells and analytic faces that are associated with the function's zero-level isosurface. We prove that under mild conditions, the identified analytic faces are guaranteed to connect and form a closed, piecewise planar surface . Based on the theorem, we propose an algorithm of analytic marching , which marches among analytic cells to exactly recover the mesh captured by an implicit surface network. We also show that our theory and algorithm are equally applicable to advanced MLPs with shortcut connections and max pooling. Given the parallel nature of analytic marching, we contribute AnalyticMesh , a software package that supports efficient meshing of implicit surface networks via CUDA parallel computing, and mesh simplification for efficient downstream processing. We apply our method to different settings of generative shape modeling using implicit surface networks. Extensive experiments demonstrate our advantages over existing methods in terms of both meshing accuracy and efficiency. Codes are at https://github.com/Karbo123/AnalyticMesh .

The affine rank minimization (ARM) problem arises in many real-world applications. The goal is to recover a low-rank matrix from a small amount of noisy affine measurements. The original problem is NP-hard, and so directly solving the problem is computationally prohibitive. Approximate low-complexity solutions for ARM have recently attracted much research interest. In this paper, we design an iterative algorithm for ARM based on message passing principles. The proposed algorithm is termed turbo-type ARM (TARM), as inspired by the recently developed turbo compressed sensing algorithm for sparse signal recovery. We show that, for right-orthogonally invariant linear (ROIL) operators, a scalar function called state evolution can be established to accurately predict the behaviour of the TARM algorithm. We also show that TARM converges faster than the counterpart algorithms when ROIL operators are used for low-rank matrix recovery. We further extend the TARM algorithm for matrix completion, where the measurement operator corresponds to a random selection matrix. Slight improvement of the matrix completion performance has been demonstrated for the TARM algorithm over the state-of-the-art algorithms.