Michael K. Ng

Li Yang, Serena Morigi, Michael K. Ng et al.

Sparse signal recovery from under-determined systems presents significant challenges when using conventional L_0 and L_1 penalties, primarily due to computational complexity and estimation bias. This paper introduces a truncated Huber penalty, a non-convex metric that effectively bridges the gap between unbiased sparse recovery and differentiable optimization. The proposed penalty applies quadratic regularization to small entries while truncating large magnitudes, avoiding non-differentiable points at optima. Theoretical analysis demonstrates that, for an appropriately chosen threshold, any s-sparse solution recoverable via conventional penalties remains a local optimum under the truncated Huber function. This property allows the exact and robust recovery theories developed for other penalty regularization functions to be directly extended to the truncated Huber function. To solve the optimization problem, we develop a block coordinate descent (BCD) algorithm with finite-step convergence guarantees under spark conditions. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed approach. Furthermore, we extend the truncated Huber-penalized model to the gradient domain, illustrating its applicability in signal denoising and image smoothing.

William T. Ng, K. Siu, Albert C. Cheung et al.

A reliable and efficient representation of multivariate time series is crucial in various downstream machine learning tasks. In multivariate time series forecasting, each variable depends on its historical values and there are inter-dependencies among variables as well. Models have to be designed to capture both intra- and inter-relationships among the time series. To move towards this goal, we propose the Time Series Attention Transformer (TSAT) for multivariate time series representation learning. Using TSAT, we represent both temporal information and inter-dependencies of multivariate time series in terms of edge-enhanced dynamic graphs. The intra-series correlations are represented by nodes in a dynamic graph; a self-attention mechanism is modified to capture the inter-series correlations by using the super-empirical mode decomposition (SMD) module. We applied the embedded dynamic graphs to times series forecasting problems, including two real-world datasets and two benchmark datasets. Extensive experiments show that TSAT clearly outerperforms six state-of-the-art baseline methods in various forecasting horizons. We further visualize the embedded dynamic graphs to illustrate the graph representation power of TSAT. We share our code at https://github.com/RadiantResearch/TSAT.

Xue-lei Lin, Pin Lyu, Michael K. Ng et al.

In this paper, a second order finite difference scheme is investigated for time-dependent one-side space fractional diffusion equations with variable coefficients. The existing schemes for the equation with variable coefficients have temporal convergence rate no better than second order and spatial convergence rate no better than first order, theoretically. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the diffusion coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.

Yisi Luo, Xile Zhao, Zhemin Li et al.

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can only represent data on finite meshgrid due to their intrinsical discrete nature, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR), which can continuously represent data beyond meshgrid with infinite resolution. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in the LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.

Junren Chen, Michael K. Ng, Jonathan Scarlett

Phase-only compressed sensing (PO-CS) concerns the recovery of sparse signals from the phases of complex measurements. Recent results show that sparse signals in the standard sphere $\mathbb{S}^{n-1}$ can be exactly recovered from complex Gaussian phases by a linearization procedure, which recasts PO-CS as linear compressed sensing and then applies (quadratically constrained) basis pursuit to obtain $\mathbf{x}^\sharp$. This paper focuses on the instance optimality and robustness of $\mathbf{x}^{\sharp}$. First, we strengthen the nonuniform instance optimality of Jacques and Feuillen (2021) to a uniform one over the entire signal space. We show the existence of some universal constant $C$ such that $\|\mathbf{x}^\sharp-\mathbf{x}\|_2\le Cs^{-1/2}σ_{\ell_1}(\mathbf{x},Σ^n_s)$ holds for all $\mathbf{x}$ in the unit Euclidean sphere, where $σ_{\ell_1}(\mathbf{x},Σ^n_s)$ is the $\ell_1$ distance of $\mathbf{x}$ to its closest $s$-sparse signal. This is achieved by showing the new sensing matrices corresponding to all approximately sparse signals simultaneously satisfy RIP. Second, we investigate the estimator's robustness to noise and corruption. We show that dense noise with entries bounded by some small $τ_0$, appearing either prior or posterior to retaining the phases, increments $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ by $O(τ_0)$. This is near-optimal (up to log factors) for any algorithm. On the other hand, adversarial corruption, which changes an arbitrary $ζ_0$-fraction of the measurements to any phase-only values, increments $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ by $O(\sqrt{ζ_0\log(1/ζ_0)})$. The developments are then combined to yield a robust instance optimal guarantee that resembles the standard one in linear compressed sensing.

Yimin Fu, Songbo Wang, Feiyan Wu et al.

The accurate target-background separation in infrared small target detection (IRSTD) highly depends on the discriminability of extracted representations. However, most existing methods are confined to domain-consistent settings, while overlooking whether such discriminability can generalize to unseen domains. In practice, distribution shifts between training and testing data are inevitable due to variations in observational conditions and environmental factors. Meanwhile, the intrinsic indistinctiveness of infrared small targets aggravates overfitting to domain-specific patterns. Consequently, the detection performance of models trained on source domains can be severely degraded when deployed in unseen domains. To address this challenge, we propose a spatial-spectral collaborative perception network (S$^2$CPNet) for cross-domain IRSTD. Moving beyond conventional spatial learning pipelines, we rethink IRSTD representations from a frequency perspective and reveal inconsistencies in spectral phase as the primary manifestation of domain discrepancies. Based on this insight, we develop a phase rectification module (PRM) to derive generalizable target awareness. Then, we employ an orthogonal attention mechanism (OAM) in skip connections to preserve positional information while refining informative representations. Moreover, the bias toward domain-specific patterns is further mitigated through selective style recomposition (SSR). Extensive experiments have been conducted on three IRSTD datasets, and the proposed method consistently achieves state-of-the-art performance under diverse cross-domain settings.

Yihang Gao, Ka Chun Cheung, Michael K. Ng

Physics-informed neural networks (PINNs) have attracted significant attention for solving partial differential equations (PDEs) in recent years because they alleviate the curse of dimensionality that appears in traditional methods. However, the most disadvantage of PINNs is that one neural network corresponds to one PDE. In practice, we usually need to solve a class of PDEs, not just one. With the explosive growth of deep learning, many useful techniques in general deep learning tasks are also suitable for PINNs. Transfer learning methods may reduce the cost for PINNs in solving a class of PDEs. In this paper, we proposed a transfer learning method of PINNs via keeping singular vectors and optimizing singular values (namely SVD-PINNs). Numerical experiments on high dimensional PDEs (10-d linear parabolic equations and 10-d Allen-Cahn equations) show that SVD-PINNs work for solving a class of PDEs with different but close right-hand-side functions.

Michael Ng, Hanrui Wu, Andy Yip

Graph neural networks have been shown to be very effective in utilizing pairwise relationships across samples. Recently, there have been several successful proposals to generalize graph neural networks to hypergraph neural networks to exploit more complex relationships. In particular, the hypergraph collaborative networks yield superior results compared to other hypergraph neural networks for various semi-supervised learning tasks. The collaborative network can provide high quality vertex embeddings and hyperedge embeddings together by formulating them as a joint optimization problem and by using their consistency in reconstructing the given hypergraph. In this paper, we aim to establish the algorithmic stability of the core layer of the collaborative network and provide generalization guarantees. The analysis sheds light on the design of hypergraph filters in collaborative networks, for instance, how the data and hypergraph filters should be scaled to achieve uniform stability of the learning process. Some experimental results on real-world datasets are presented to illustrate the theory.

Junjun Pan, Michael K. Ng

Polarization is a unique characteristic of transverse wave and is represented by Stokes parameters. Analysis of polarization states can reveal valuable information about the sources. In this paper, we propose a separable low-rank quaternion linear mixing model to polarized signals: we assume each column of the source factor matrix equals a column of polarized data matrix and refer to the corresponding problem as separable quaternion matrix factorization (SQMF). We discuss some properties of the matrix that can be decomposed by SQMF. To determine the source factor matrix in quaternion space, we propose a heuristic algorithm called quaternion successive projection algorithm (QSPA) inspired by the successive projection algorithm. To guarantee the effectiveness of QSPA, a new normalization operator is proposed for the quaternion matrix. We use a block coordinate descent algorithm to compute nonnegative factor activation matrix in real number space. We test our method on the applications of polarization image representation and spectro-polarimetric imaging unmixing to verify its effectiveness.

Junren Chen, Michael K. Ng

A covariance matrix estimator using two bits per entry was recently developed by Dirksen, Maly and Rauhut [Annals of Statistics, 50(6), pp. 3538-3562]. The estimator achieves near minimax rate for general sub-Gaussian distributions, but also suffers from two downsides: theoretically, there is an essential gap on operator norm error between their estimator and sample covariance when the diagonal of the covariance matrix is dominated by only a few entries; practically, its performance heavily relies on the dithering scale, which needs to be tuned according to some unknown parameters. In this work, we propose a new 2-bit covariance matrix estimator that simultaneously addresses both issues. Unlike the sign quantizer associated with uniform dither in Dirksen et al., we adopt a triangular dither prior to a 2-bit quantizer inspired by the multi-bit uniform quantizer. By employing dithering scales varying across entries, our estimator enjoys an improved operator norm error rate that depends on the effective rank of the underlying covariance matrix rather than the ambient dimension, thus closing the theoretical gap. Moreover, our proposed method eliminates the need of any tuning parameter, as the dithering scales are entirely determined by the data. Experimental results under Gaussian samples are provided to showcase the impressive numerical performance of our estimator. Remarkably, by halving the dithering scales, our estimator oftentimes achieves operator norm errors less than twice of the errors of sample covariance.

Junren Chen, Yueqi Wang, Michael K. Ng

Low-rank multivariate regression (LRMR) is an important statistical learning model that combines highly correlated tasks as a multiresponse regression problem with low-rank priori on the coefficient matrix. In this paper, we study quantized LRMR, a practical setting where the responses and/or the covariates are discretized to finite precision. We focus on the estimation of the underlying coefficient matrix. To make consistent estimator that could achieve arbitrarily small error possible, we employ uniform quantization with random dithering, i.e., we add appropriate random noise to the data before quantization. Specifically, uniform dither and triangular dither are used for responses and covariates, respectively. Based on the quantized data, we propose the constrained Lasso and regularized Lasso estimators, and derive the non-asymptotic error bounds. With the aid of dithering, the estimators achieve minimax optimal rate, while quantization only slightly worsens the multiplicative factor in the error rate. Moreover, we extend our results to a low-rank regression model with matrix responses. We corroborate and demonstrate our theoretical results via simulations on synthetic data or image restoration.

Junren Chen, Michael K. Ng

In this paper, we study the estimation performance of empirical $\ell_2$ risk minimization (ERM) in noisy (standard) phase retrieval (NPR) given by $y_k = |α_k^*x_0|^2+η_k$, or noisy generalized phase retrieval (NGPR) formulated as $y_k = x_0^*A_kx_0 + η_k$, where $x_0\in\mathbb{K}^d$ is the desired signal, $n$ is the sample size, $η= (η_1,...,η_n)^\top$ is the noise vector. We establish new error bounds under different noise patterns, and our proofs are valid for both $\mathbb{K}=\mathbb{R}$ and $\mathbb{K}=\mathbb{C}$. In NPR under arbitrary noise vector $η$, we derive a new error bound $O\big(\|η\|_\infty\sqrt{\frac{d}{n}} + \frac{|\mathbf{1}^\topη|}{n}\big)$, which is tighter than the currently known one $O\big(\frac{\|η\|}{\sqrt{n}}\big)$ in many cases. In NGPR, we show $O\big(\|η\|\frac{\sqrt{d}}{n}\big)$ for arbitrary $η$. In both problems, the bounds for arbitrary noise immediately give rise to $\tilde{O}(\sqrt{\frac{d}{n}})$ for sub-Gaussian or sub-exponential random noise, with some conventional but inessential assumptions (e.g., independent or zero-mean condition) removed or weakened. In addition, we make a first attempt to ERM under heavy-tailed random noise assumed to have bounded $l$-th moment. To achieve a trade-off between bias and variance, we truncate the responses and propose a corresponding robust ERM estimator, which is shown to possess the guarantee $\tilde{O}\big(\big[\sqrt{\frac{d}{n}}\big]^{1-1/l}\big)$ in both NPR, NGPR. All the error bounds straightforwardly extend to the more general problems of rank-$r$ matrix recovery, and these results deliver a conclusion that the full-rank frame $\{A_k\}_{k=1}^n$ in NGPR is more robust to biased noise than the rank-1 frame $\{α_kα_k^*\}_{k=1}^n$ in NPR. Extensive experimental results are presented to illustrate our theoretical findings.

Yihang Gao, Huafeng Liu, Michael K. Ng et al.

Wide applications of differentiable two-player sequential games (e.g., image generation by GANs) have raised much interest and attention of researchers to study efficient and fast algorithms. Most of the existing algorithms are developed based on nice properties of simultaneous games, i.e., convex-concave payoff functions, but are not applicable in solving sequential games with different settings. Some conventional gradient descent ascent algorithms theoretically and numerically fail to find the local Nash equilibrium of the simultaneous game or the local minimax (i.e., local Stackelberg equilibrium) of the sequential game. In this paper, we propose the HessianFR, an efficient Hessian-based Follow-the-Ridge algorithm with theoretical guarantees. Furthermore, the convergence of the stochastic algorithm and the approximation of Hessian inverse are exploited to improve algorithm efficiency. A series of experiments of training generative adversarial networks (GANs) have been conducted on both synthetic and real-world large-scale image datasets (e.g. MNIST, CIFAR-10 and CelebA). The experimental results demonstrate that the proposed HessianFR outperforms baselines in terms of convergence and image generation quality.

Junren Chen, Jonathan Scarlett, Michael K. Ng et al.

In generative compressed sensing (GCS), we want to recover a signal $\mathbf{x}^* \in \mathbb{R}^n$ from $m$ measurements ($m\ll n$) using a generative prior $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$, where $G$ is typically an $L$-Lipschitz continuous generative model and $\mathbb{B}_2^k(r)$ represents the radius-$r$ $\ell_2$-ball in $\mathbb{R}^k$. Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $\mathbf{x}^*$ rather than for all $\mathbf{x}^*$ simultaneously. In this paper, we build a unified framework to derive uniform recovery guarantees for nonlinear GCS where the observation model is nonlinear and possibly discontinuous or unknown. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. Specifically, using a single realization of the sensing ensemble and generalized Lasso, {\em all} $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$ can be recovered up to an $\ell_2$-error at most $ε$ using roughly $\tilde{O}({k}/{ε^2})$ samples, with omitted logarithmic factors typically being dominated by $\log L$. Notably, this almost coincides with existing non-uniform guarantees up to logarithmic factors, hence the uniformity costs very little. As part of our technical contributions, we introduce the Lipschitz approximation to handle discontinuous observation models. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy. Experimental results are presented to corroborate our theory.

Yu Guo, Guoqing Chen, Tieyong Zeng et al.

Color image restoration methods typically represent images as vectors in Euclidean space or combinations of three monochrome channels. However, they often overlook the correlation between these channels, leading to color distortion and artifacts in the reconstructed image. To address this, we present Quaternion Nuclear Norm Minus Frobenius Norm Minimization (QNMF), a novel approach for color image reconstruction. QNMF utilizes quaternion algebra to capture the relationships among RGB channels comprehensively. By employing a regularization technique that involves nuclear norm minus Frobenius norm, QNMF approximates the underlying low-rank structure of quaternion-encoded color images. Theoretical proofs are provided to ensure the method's mathematical integrity. Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.

Yihang Gao, Michael K. Ng

The cubic regularization method (CR) and its adaptive version (ARC) are popular Newton-type methods in solving unconstrained non-convex optimization problems, due to its global convergence to local minima under mild conditions. The main aim of this paper is to develop a momentum-accelerated adaptive cubic regularization method (ARCm) to improve the convergent performance. With the proper choice of momentum step size, we show the global convergence of ARCm and the local convergence can also be guaranteed under the \KL property. Such global and local convergence can also be established when inexact solvers with low computational costs are employed in the iteration procedure. Numerical results for non-convex logistic regression and robust linear regression models are reported to demonstrate that the proposed ARCm significantly outperforms state-of-the-art cubic regularization methods (e.g., CR, momentum-based CR, ARC) and the trust region method. In particular, the number of iterations required by ARCm is less than 10\% to 50\% required by the most competitive method (ARC) in the experiments.

Yihang Gao, Man-Chung Yue, Michael K. Ng

The cubic regularization method (CR) is a popular algorithm for unconstrained non-convex optimization. At each iteration, CR solves a cubically regularized quadratic problem, called the cubic regularization subproblem (CRS). One way to solve the CRS relies on solving the secular equation, whose computational bottleneck lies in the computation of all eigenvalues of the Hessian matrix. In this paper, we propose and analyze a novel CRS solver based on an approximate secular equation, which requires only some of the Hessian eigenvalues and is therefore much more efficient. Two approximate secular equations (ASEs) are developed. For both ASEs, we first study the existence and uniqueness of their roots and then establish an upper bound on the gap between the root and that of the standard secular equation. Such an upper bound can in turn be used to bound the distance from the approximate CRS solution based ASEs to the true CRS solution, thus offering a theoretical guarantee for our CRS solver. A desirable feature of our CRS solver is that it requires only matrix-vector multiplication but not matrix inversion, which makes it particularly suitable for high-dimensional applications of unconstrained non-convex optimization, such as low-rank recovery and deep learning. Numerical experiments with synthetic and real data-sets are conducted to investigate the practical performance of the proposed CRS solver. Experimental results show that the proposed solver outperforms two state-of-the-art methods.

Junren Chen, Michael K. Ng, Zhaoqiang Liu

The problem of recovering a signal $\boldsymbol x\in \mathbb{R}^n$ from a quadratic system $\{y_i=\boldsymbol x^\top\boldsymbol A_i\boldsymbol x,\ i=1,\ldots,m\}$ with full-rank matrices $\boldsymbol A_i$ frequently arises in applications such as unassigned distance geometry and sub-wavelength imaging. With i.i.d. standard Gaussian matrices $\boldsymbol A_i$, this paper addresses the high-dimensional case where $m\ll n$ by incorporating prior knowledge of $\boldsymbol x$. First, we consider a $k$-sparse $\boldsymbol x$ and introduce the thresholded Wirtinger flow (TWF) algorithm that does not require the sparsity level $k$. TWF comprises two steps: the spectral initialization that identifies a point sufficiently close to $\boldsymbol x$ (up to a sign flip) when $m=O(k^2\log n)$, and the thresholded gradient descent which, when provided a good initialization, produces a sequence linearly converging to $\boldsymbol x$ with $m=O(k\log n)$ measurements. Second, we explore the generative prior, assuming that $x$ lies in the range of an $L$-Lipschitz continuous generative model with $k$-dimensional inputs in an $\ell_2$-ball of radius $r$. With an estimate correlated with the signal, we develop the projected gradient descent (PGD) algorithm that also comprises two steps: the projected power method that provides an initial vector with $O\big(\sqrt{\frac{k \log L}{m}}\big)$ $\ell_2$-error given $m=O(k\log(Lnr))$ measurements, and the projected gradient descent that refines the $\ell_2$-error to $O(δ)$ at a geometric rate when $m=O(k\log\frac{Lrn}{δ^2})$. Experimental results corroborate our theoretical findings and show that: (i) our approach for the sparse case notably outperforms the existing provable algorithm sparse power factorization; (ii) leveraging the generative prior allows for precise image recovery in the MNIST dataset from a small number of quadratic measurements.

Xiongjun Zhang, Michael K. Ng

Tensor decomposition is a powerful tool for extracting physically meaningful latent factors from multi-dimensional nonnegative data, and has been an increasing interest in a variety of fields such as image processing, machine learning, and computer vision. In this paper, we propose a sparse nonnegative Tucker decomposition and completion method for the recovery of underlying nonnegative data under noisy observations. Here the underlying nonnegative data tensor is decomposed into a core tensor and several factor matrices with all entries being nonnegative and the factor matrices being sparse. The loss function is derived by the maximum likelihood estimation of the noisy observations, and the $\ell_0$ norm is employed to enhance the sparsity of the factor matrices. We establish the error bound of the estimator of the proposed model under generic noise scenarios, which is then specified to the observations with additive Gaussian noise, additive Laplace noise, and Poisson observations, respectively. Our theoretical results are better than those by existing tensor-based or matrix-based methods. Moreover, the minimax lower bounds are shown to be matched with the derived upper bounds up to logarithmic factors. Numerical examples on both synthetic and real-world data sets demonstrate the superiority of the proposed method for nonnegative tensor data completion.

Jifei Miao, Junjun Pan, Michael K. Ng

Reduced biquaternion (RB), as a four-dimensional algebra highly suitable for representing color pixels, has recently garnered significant attention from numerous scholars. In this paper, for color image processing problems, we introduce a concept of the non-negative RB matrix and then use the multiplication properties of RB to propose a non-negative RB matrix factorization (NRBMF) model. The NRBMF model is introduced to address the challenge of reasonably establishing a non-negative quaternion matrix factorization model, which is primarily hindered by the multiplication properties of traditional quaternions. Furthermore, this paper transforms the problem of solving the NRBMF model into an RB alternating non-negative least squares (RB-ANNLS) problem. Then, by introducing a method to compute the gradient of the real function with RB matrix variables, we solve the RB-ANNLS optimization problem using the RB projected gradient algorithm and conduct a convergence analysis of the algorithm. Finally, we validate the effectiveness and superiority of the proposed NRBMF model in color face recognition.

Junjun Pan, Valentin Leplat, Michael Ng et al.

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.

Chuanyang Zheng, Yihang Gao, Han Shi et al.

Positional encoding plays a crucial role in transformers, significantly impacting model performance and length generalization. Prior research has introduced absolute positional encoding (APE) and relative positional encoding (RPE) to distinguish token positions in given sequences. However, both APE and RPE remain fixed after model training regardless of input data, limiting their adaptability and flexibility. Hence, we expect that the desired positional encoding should be data-adaptive and can be dynamically adjusted with the given attention. In this paper, we propose a Data-Adaptive Positional Encoding (DAPE) method, which dynamically and semantically adjusts based on input context and learned fixed priors. Experimental validation on real-world datasets (Arxiv, Books3, and CHE) demonstrates that DAPE enhances model performances in terms of trained length and length generalization, where the improvements are statistically significant. The model visualization suggests that our model can keep both local and anti-local information. Finally, we successfully train the model on sequence length 128 and achieve better performance at evaluation sequence length 8192, compared with other static positional encoding methods, revealing the benefit of the adaptive positional encoding method.

Zhu Han, Ce Zhang, Lianru Gao et al.

Cross-scene image classification aims to transfer prior knowledge of ground materials to annotate regions with different distributions and reduce hand-crafted cost in the field of remote sensing. However, existing approaches focus on single-source domain generalization to unseen target domains, and are easily confused by large real-world domain shifts due to the limited training information and insufficient diversity modeling capacity. To address this gap, we propose a novel multi-source collaborative domain generalization framework (MS-CDG) based on homogeneity and heterogeneity characteristics of multi-source remote sensing data, which considers data-aware adversarial augmentation and model-aware multi-level diversification simultaneously to enhance cross-scene generalization performance. The data-aware adversarial augmentation adopts an adversary neural network with semantic guide to generate MS samples by adaptively learning realistic channel and distribution changes across domains. In views of cross-domain and intra-domain modeling, the model-aware diversification transforms the shared spatial-channel features of MS data into the class-wise prototype and kernel mixture module, to address domain discrepancies and cluster different classes effectively. Finally, the joint classification of original and augmented MS samples is employed by introducing a distribution consistency alignment to increase model diversity and ensure better domain-invariant representation learning. Extensive experiments on three public MS remote sensing datasets demonstrate the superior performance of the proposed method when benchmarked with the state-of-the-art methods.

Zhigang Jia, Yuelian Xiang, Meixiang Zhao et al.

The cross-channel deblurring problem in color image processing is difficult to solve due to the complex coupling and structural blurring of color pixels. Until now, there are few efficient algorithms that can reduce color artifacts in deblurring process. To solve this challenging problem, we present a novel cross-space total variation (CSTV) regularization model for color image deblurring by introducing a quaternion blur operator and a cross-color space regularization functional. The existence and uniqueness of the solution are proved and a new L-curve method is proposed to find a balance of regularization terms on different color spaces. The Euler-Lagrange equation is derived to show that CSTV has taken into account the coupling of all color channels and the local smoothing within each color channel. A quaternion operator splitting method is firstly proposed to enhance the ability of color artifacts reduction of the CSTV regularization model. This strategy also applies to the well-known color deblurring models. Numerical experiments on color image databases illustrate the efficiency and effectiveness of the new model and algorithms. The color images restored by them successfully maintain the color and spatial information and are of higher quality in terms of PSNR, SSIM, MSE and CIEde2000 than the restorations of the-state-of-the-art methods.

Yihang Gao, Michael K. Ng, Vincent Y. F. Tan

Kolmogorov--Arnold networks (KANs) have demonstrated their potential as an alternative to multi-layer perceptions (MLPs) in various domains, especially for science-related tasks. However, transfer learning of KANs remains a relatively unexplored area. In this paper, inspired by Tucker decomposition of tensors and evidence on the low tensor-rank structure in KAN parameter updates, we develop low tensor-rank adaptation (LoTRA) for fine-tuning KANs. We study the expressiveness of LoTRA based on Tucker decomposition approximations. Furthermore, we provide a theoretical analysis to select the learning rates for each LoTRA component to enable efficient training. Our analysis also shows that using identical learning rates across all components leads to inefficient training, highlighting the need for an adaptive learning rate strategy. Beyond theoretical insights, we explore the application of LoTRA for efficiently solving various partial differential equations (PDEs) by fine-tuning KANs. Additionally, we propose Slim KANs that incorporate the inherent low-tensor-rank properties of KAN parameter tensors to reduce model size while maintaining superior performance. Experimental results validate the efficacy of the proposed learning rate selection strategy and demonstrate the effectiveness of LoTRA for transfer learning of KANs in solving PDEs. Further evaluations on Slim KANs for function representation and image classification tasks highlight the expressiveness of LoTRA and the potential for parameter reduction through low tensor-rank decomposition.

Xuhang Chen, Michael Kwok-Po Ng, Kim-Fung Tsang et al.

Brain network analysis plays a crucial role in diagnosing and monitoring neurodegenerative disorders such as Alzheimer's disease (AD). Existing approaches for constructing structural brain networks from diffusion tensor imaging (DTI) often rely on specialized toolkits that suffer from inherent limitations: operator subjectivity, labor-intensive workflows, and restricted capacity to capture complex topological features and disease-specific biomarkers. To overcome these challenges and advance computational neuroimaging instrumentation, ConnectomeDiffuser is proposed as a novel diffusion-based framework for automated end-to-end brain network construction from DTI. The proposed model combines three key components: (1) a Template Network that extracts topological features from 3D DTI scans using Riemannian geometric principles, (2) a diffusion model that generates comprehensive brain networks with enhanced topological fidelity, and (3) a Graph Convolutional Network classifier that incorporates disease-specific markers to improve diagnostic accuracy. ConnectomeDiffuser demonstrates superior performance by capturing a broader range of structural connectivity and pathology-related information, enabling more sensitive analysis of individual variations in brain networks. Experimental validation on datasets representing two distinct neurodegenerative conditions demonstrates significant performance improvements over other brain network methods. This work contributes to the advancement of instrumentation in the context of neurological disorders, providing clinicians and researchers with a robust, generalizable measurement framework that facilitates more accurate diagnosis, deeper mechanistic understanding, and improved therapeutic monitoring of neurodegenerative diseases such as AD.

Yaru Liu, Yiqi Gu, Michael K. Ng

In this paper, we develop a new optimization framework for the least squares learning problem via fully connected neural networks or physics-informed neural networks. The gradient descent sometimes behaves inefficiently in deep learning because of the high non-convexity of loss functions and the vanishing gradient issue. Our idea is to introduce auxiliary variables to separate the layers of the deep neural networks and reformulate the loss functions for ease of optimization. We design the self-adaptive weights to preserve the consistency between the reformulated loss and the original mean squared loss, which guarantees that optimizing the new loss helps optimize the original problem. Numerical experiments are presented to verify the consistency and show the effectiveness and robustness of our models over gradient descent.

Xiaoling Zhou, Ou Wu, Michael K. Ng et al.

Data valuation has garnered increasing attention in recent years, given the critical role of high-quality data in various applications. Among diverse data valuation approaches, Shapley value-based methods are predominant due to their strong theoretical grounding. However, the exact computation of Shapley values is often computationally prohibitive, prompting the development of numerous approximation techniques. Despite notable advancements, existing methods generally neglect the incorporation of value distribution information and fail to account for dynamic data conditions, thereby compromising their performance and application potential. In this paper, we highlight the crucial role of both global and local statistical properties of value distributions in the context of data valuation for machine learning. First, we conduct a comprehensive analysis of these distributions across various simulated and real-world datasets, uncovering valuable insights and key patterns. Second, we propose an enhanced data valuation method that fuses the explored distribution characteristics into two regularization terms to refine Shapley value estimation. The proposed regularizers can be seamlessly incorporated into various existing data valuation methods. Third, we introduce a novel approach for dynamic data valuation that infers updated data values without recomputing Shapley values, thereby significantly improving computational efficiency. Extensive experiments have been conducted across a range of tasks, including Shapley value estimation, value-based data addition and removal, mislabeled data detection, and dynamic data valuation. The results showcase the consistent effectiveness and efficiency of our proposed methodologies, affirming the significant potential of global and local value distributions in data valuation.

Chuan Wang, Xi-le Zhao, Zhilong Han et al.

Matrix operations (e.g., inversion and singular value decomposition (SVD)) are fundamental in science and engineering. In many emerging real-world applications (such as wireless communication and signal processing), these operations must be performed repeatedly over matrices with parameters varying continuously. However, conventional methods tackle each matrix operation independently, underexploring the inherent low-rankness and continuity along the parameter dimension, resulting in significantly redundant computation. To address this challenge, we propose \textbf{\textit{Neural Matrix Computation Framework} (NeuMatC)}, which elegantly tackles general parametric matrix operation tasks by leveraging the underlying low-rankness and continuity along the parameter dimension. Specifically, NeuMatC unsupervisedly learns a low-rank and continuous mapping from parameters to their corresponding matrix operation results. Once trained, NeuMatC enables efficient computations at arbitrary parameters using only a few basic operations (e.g., matrix multiplications and nonlinear activations), significantly reducing redundant computations. Experimental results on both synthetic and real-world datasets demonstrate the promising performance of NeuMatC, exemplified by over $3\times$ speedup in parametric inversion and $10\times$ speedup in parametric SVD compared to the widely used NumPy baseline in wireless communication, while maintaining acceptable accuracy.

Yihang Gao, Michael K. Ng, Michael W. Mahoney et al.

We study online statistical inference for the solutions of stochastic optimization problems with equality and inequality constraints. Such problems are prevalent in statistics and machine learning, encompassing constrained $M$-estimation, physics-informed models, safe reinforcement learning, and algorithmic fairness. We develop a stochastic sequential quadratic programming (SSQP) method to solve these problems, where the step direction is computed by sequentially performing a quadratic approximation of the objective and a linear approximation of the constraints. Despite having access to unbiased estimates of population gradients, a key challenge in constrained stochastic problems lies in dealing with the bias in the step direction. As such, we apply a momentum-style gradient moving-average technique within SSQP to debias the step. We show that our method achieves global almost-sure convergence and exhibits local asymptotic normality with an optimal primal-dual limiting covariance matrix in the sense of Hájek and Le Cam. In addition, we provide a plug-in covariance matrix estimator for practical inference. To our knowledge, the proposed SSQP method is the first fully online method that attains primal-dual asymptotic minimax optimality without relying on projection operators onto the constraint set, which are generally intractable for nonlinear problems. Through extensive experiments on benchmark nonlinear problems, as well as on constrained generalized linear models and portfolio allocation problems using both synthetic and real data, we demonstrate superior performance of our method, showing that the method and its asymptotic behavior not only solve constrained stochastic problems efficiently but also provide valid and practical online inference in real-world applications.

Yu Guo, Guoqing Chen, Tieyong Zeng et al.

Recovering hidden structures from incomplete or noisy data remains a pervasive challenge across many fields, particularly where multi-dimensional data representation is essential. Quaternion matrices, with their ability to naturally model multi-dimensional data, offer a promising framework for this problem. This paper introduces the quaternion nuclear norm over the Frobenius norm (QNOF) as a novel nonconvex approximation for the rank of quaternion matrices. QNOF is parameter-free and scale-invariant. Utilizing quaternion singular value decomposition, we prove that solving the QNOF can be simplified to solving the singular value $L_1/L_2$ problem. Additionally, we extend the QNOF to robust quaternion matrix completion, employing the alternating direction multiplier method to derive solutions that guarantee weak convergence under mild conditions. Extensive numerical experiments validate the proposed model's superiority, consistently outperforming state-of-the-art quaternion methods.

Wei Liu, Xin Liu, Michael K. Ng et al.

Semi-supervised clustering is a basic problem in various applications. Most existing methods require knowledge of the ideal cluster number, which is often difficult to obtain in practice. Besides, satisfying the must-link constraints is another major challenge for these methods. In this work, we view the semi-supervised clustering task as a partitioning problem on a graph associated with the given dataset, where the similarity matrix includes a scaling parameter to reflect the must-link constraints. Utilizing a relaxation technique, we formulate the graph partitioning problem into a continuous optimization model that does not require the exact cluster number, but only an overestimate of it. We then propose a block coordinate descent algorithm to efficiently solve this model, and establish its convergence result. Based on the obtained solution, we can construct the clusters that theoretically meet the must-link constraints under mild assumptions. Furthermore, we verify the effectiveness and efficiency of our proposed method through comprehensive numerical experiments.

Sijia Xia, Michael K. Ng, Xiongjun Zhang

In this paper, we study the problem of low-rank tensor learning, where only a few of training samples are observed and the underlying tensor has a low-rank structure. The existing methods are based on the sum of nuclear norms of unfolding matrices of a tensor, which may be suboptimal. In order to explore the low-rankness of the underlying tensor effectively, we propose a nonconvex model based on transformed tensor nuclear norm for low-rank tensor learning. Specifically, a family of nonconvex functions are employed onto the singular values of all frontal slices of a tensor in the transformed domain to characterize the low-rankness of the underlying tensor. An error bound between the stationary point of the nonconvex model and the underlying tensor is established under restricted strong convexity on the loss function (such as least squares loss and logistic regression) and suitable regularity conditions on the nonconvex penalty function. By reformulating the nonconvex function into the difference of two convex functions, a proximal majorization-minimization (PMM) algorithm is designed to solve the resulting model. Then the global convergence and convergence rate of PMM are established under very mild conditions. Numerical experiments are conducted on tensor completion and binary classification to demonstrate the effectiveness of the proposed method over other state-of-the-art methods.

Zhantao Ma, Michael K. Ng

Multispectral images (MSI) contain light information in different wavelengths of objects, which convey spectral-spatial information and help improve the performance of various image processing tasks. Numerous techniques have been created to extend the application of total variation regularization in restoring multispectral images, for example, based on channel coupling and adaptive total variation regularization. The primary contribution of this paper is to propose and develop a new multispectral total variation regularization in a generalized opponent transformation domain instead of the original multispectral image domain. Here opponent transformations for multispectral images are generalized from a well-known opponent transformation for color images. We will explore the properties of generalized opponent transformation total variation (GOTTV) regularization and the corresponding optimization formula for multispectral image restoration. To evaluate the effectiveness of the new GOTTV method, we provide numerical examples that showcase its superior performance compared to existing multispectral image total variation methods, using criteria such as MPSNR and MSSIM.

Junren Chen, Cheng-Long Wang, Michael K. Ng et al.

In this paper, we propose a uniformly dithered 1-bit quantization scheme for high-dimensional statistical estimation. The scheme contains truncation, dithering, and quantization as typical steps. As canonical examples, the quantization scheme is applied to the estimation problems of sparse covariance matrix estimation, sparse linear regression (i.e., compressed sensing), and matrix completion. We study both sub-Gaussian and heavy-tailed regimes, where the underlying distribution of heavy-tailed data is assumed to have bounded moments of some order. We propose new estimators based on 1-bit quantized data. In sub-Gaussian regime, our estimators achieve near minimax rates, indicating that our quantization scheme costs very little. In heavy-tailed regime, while the rates of our estimators become essentially slower, these results are either the first ones in an 1-bit quantized and heavy-tailed setting, or already improve on existing comparable results from some respect. Under the observations in our setting, the rates are almost tight in compressed sensing and matrix completion. Our 1-bit compressed sensing results feature general sensing vector that is sub-Gaussian or even heavy-tailed. We also first investigate a novel setting where both the covariate and response are quantized. In addition, our approach to 1-bit matrix completion does not rely on likelihood and represent the first method robust to pre-quantization noise with unknown distribution. Experimental results on synthetic data are presented to support our theoretical analysis.

Junren Chen, Michael K. Ng

In this paper, we study color image inpainting as a pure quaternion matrix completion problem. In the literature, the theoretical guarantee for quaternion matrix completion is not well-established. Our main aim is to propose a new minimization problem with an objective combining nuclear norm and a quadratic loss weighted among three channels. To fill the theoretical vacancy, we obtain the error bound in both clean and corrupted regimes, which relies on some new results of quaternion matrices. A general Gaussian noise is considered in robust completion where all observations are corrupted. Motivated by the error bound, we propose to handle unbalanced or correlated noise via a cross-channel weight in the quadratic loss, with the main purpose of rebalancing noise level, or removing noise correlation. Extensive experimental results on synthetic and color image data are presented to confirm and demonstrate our theoretical findings.

Wen Yu, Baiying Lei, Yanyan Shen et al.

The diagnosis of early stages of Alzheimer's disease (AD) is essential for timely treatment to slow further deterioration. Visualizing the morphological features for the early stages of AD is of great clinical value. In this work, a novel Multidirectional Perception Generative Adversarial Network (MP-GAN) is proposed to visualize the morphological features indicating the severity of AD for patients of different stages. Specifically, by introducing a novel multidirectional mapping mechanism into the model, the proposed MP-GAN can capture the salient global features efficiently. Thus, by utilizing the class-discriminative map from the generator, the proposed model can clearly delineate the subtle lesions via MR image transformations between the source domain and the pre-defined target domain. Besides, by integrating the adversarial loss, classification loss, cycle consistency loss and \emph{L}1 penalty, a single generator in MP-GAN can learn the class-discriminative maps for multiple-classes. Extensive experimental results on Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset demonstrate that MP-GAN achieves superior performance compared with the existing methods. The lesions visualized by MP-GAN are also consistent with what clinicians observe.

Yun-Yang Liu, Xi-Le Zhao, Guang-Jing Song et al.

The robust tensor completion (RTC) problem, which aims to reconstruct a low-rank tensor from partially observed tensor contaminated by a sparse tensor, has received increasing attention. In this paper, by leveraging the superior expression of the fully-connected tensor network (FCTN) decomposition, we propose a $\textbf{FCTN}$-based $\textbf{r}$obust $\textbf{c}$onvex optimization model (RC-FCTN) for the RTC problem. Then, we rigorously establish the exact recovery guarantee for the RC-FCTN. For solving the constrained optimization model RC-FCTN, we develop an alternating direction method of multipliers (ADMM)-based algorithm, which enjoys the global convergence guarantee. Moreover, we suggest a $\textbf{FCTN}$-based $\textbf{r}$obust $\textbf{n}$on$\textbf{c}$onvex optimization model (RNC-FCTN) for the RTC problem. A proximal alternating minimization (PAM)-based algorithm is developed to solve the proposed RNC-FCTN. Meanwhile, we theoretically derive the convergence of the PAM-based algorithm. Comprehensive numerical experiments in several applications, such as video completion and video background subtraction, demonstrate that proposed methods are superior to several state-of-the-art methods.

Lina Zhuang, Michael K. Ng

Hyperspectral imaging with high spectral resolution plays an important role in finding objects, identifying materials, or detecting processes. The decrease of the widths of spectral bands leads to a decrease in the signal-to-noise ratio (SNR) of measurements. The decreased SNR reduces the reliability of measured features or information extracted from HSIs. Furthermore, the image degradations linked with various mechanisms also result in different types of noise, such as Gaussian noise, impulse noise, deadlines, and stripes. This paper introduces a fast and parameter-free hyperspectral image mixed noise removal method (termed FastHyMix), which characterizes the complex distribution of mixed noise by using a Gaussian mixture model and exploits two main characteristics of hyperspectral data, namely low-rankness in the spectral domain and high correlation in the spatial domain. The Gaussian mixture model enables us to make a good estimation of Gaussian noise intensity and the location of sparse noise. The proposed method takes advantage of the low-rankness using subspace representation and the spatial correlation of HSIs by adding a powerful deep image prior, which is extracted from a neural denoising network. An exhaustive array of experiments and comparisons with state-of-the-art denoisers were carried out. The experimental results show significant improvement in both synthetic and real datasets. A MATLAB demo of this work will be available at https://github.com/LinaZhuang for the sake of reproducibility.

Junjun Pan, Michael K. Ng

Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M=P_1SP_2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We discuss some mathematics properties of CoS-NMF, and present the relationships with other related matrix factorizations such as CUR decomposition, generalized separable NMF(GS-NMF), and bi-orthogonal tri-factorization (BiOR-NM3F). An optimization model for CoS-NMF is proposed and alternated fast gradient method is employed to solve the model. Numerical experiments on synthetic datasets, document datasets and facial databases are conducted to verify the effectiveness of our CoS-NMF model. Compared to state-of-the-art methods, CoS-NMF model performs very well in co-clustering task, and preserves a good approximation to the input data matrix as well.

Yihang Gao, Michael K. Ng

In this paper, we study a physics-informed algorithm for Wasserstein Generative Adversarial Networks (WGANs) for uncertainty quantification in solutions of partial differential equations. By using groupsort activation functions in adversarial network discriminators, network generators are utilized to learn the uncertainty in solutions of partial differential equations observed from the initial/boundary data. Under mild assumptions, we show that the generalization error of the computed generator converges to the approximation error of the network with high probability, when the number of samples are sufficiently taken. According to our established error bound, we also find that our physics-informed WGANs have higher requirement for the capacity of discriminators than that of generators. Numerical results on synthetic examples of partial differential equations are reported to validate our theoretical results and demonstrate how uncertainty quantification can be obtained for solutions of partial differential equations and the distributions of initial/boundary data. However, the quality or the accuracy of the uncertainty quantification theory in all the points in the interior is still the theoretical vacancy, and required for further research.

Yi-Si Luo, Xi-Le Zhao, Tai-Xiang Jiang et al.

In this paper, we study multi-dimensional image recovery. Recently, transform-based tensor nuclear norm minimization methods are considered to capture low-rank tensor structures to recover third-order tensors in multi-dimensional image processing applications. The main characteristic of such methods is to perform the linear transform along the third mode of third-order tensors, and then compute tensor nuclear norm minimization on the transformed tensor so that the underlying low-rank tensors can be recovered. The main aim of this paper is to propose a nonlinear multilayer neural network to learn a nonlinear transform via the observed tensor data under self-supervision. The proposed network makes use of low-rank representation of transformed tensors and data-fitting between the observed tensor and the reconstructed tensor to construct the nonlinear transformation. Extensive experimental results on tensor completion, background subtraction, robust tensor completion, and snapshot compressive imaging are presented to demonstrate that the performance of the proposed method is better than that of state-of-the-art methods.

Yihang Gao, Michael K. Ng, Mingjie Zhou

Studied here are Wasserstein generative adversarial networks (WGANs) with GroupSort neural networks as their discriminators. It is shown that the error bound of the approximation for the target distribution depends on the width and depth (capacity) of the generators and discriminators and the number of samples in training. A quantified generalization bound is established for the Wasserstein distance between the generated and target distributions. According to the theoretical results, WGANs have a higher requirement for the capacity of discriminators than that of generators, which is consistent with some existing results. More importantly, the results with overly deep and wide (high-capacity) generators may be worse than those with low-capacity generators if discriminators are insufficiently strong. Numerical results obtained using Swiss roll and MNIST datasets confirm the theoretical results.

Guang-Jing Song, Michael K. Ng, Xiongjun Zhang

One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study $n_1 \times n_2 \times n_3$ third-order tensor completion based on transformed tensor singular value decomposition, and provide a bound on the number of required sample entries. Our approach is to make use of the multi-rank of the underlying tensor instead of its tubal rank in the bound. In numerical experiments on synthetic and imaging data sets, we demonstrate the effectiveness of our proposed bound for the number of sample entries. Moreover, our theoretical results are valid to any unitary transformation applied to $n_3$-dimension under transformed tensor singular value decomposition.

Jing Guo, Shuping Wang, Chen Luo et al.

Non-local self-similarity based low rank algorithms are the state-of-the-art methods for image denoising. In this paper, a new method is proposed by solving two issues: how to improve similar patches matching accuracy and build an appropriate low rank matrix approximation model for Gaussian noise. For the first issue, similar patches can be found locally or globally. Local patch matching is to find similar patches in a large neighborhood which can alleviate noise effect, but the number of patches may be insufficient. Global patch matching is to determine enough similar patches but the error rate of patch matching may be higher. Based on this, we first use local patch matching method to reduce noise and then use Gaussian patch mixture model to achieve global patch matching. The second issue is that there is no low rank matrix approximation model to adapt to Gaussian noise. We build a new model according to the characteristics of Gaussian noise, then prove that there is a globally optimal solution of the model. By solving the two issues, experimental results are reported to show that the proposed approach outperforms the state-of-the-art denoising methods includes several deep learning ones in both PSNR / SSIM values and visual quality.

Zhigang Jia, Qiyu Jin, Michael K. Ng et al.

The image nonlocal self-similarity (NSS) prior refers to the fact that a local patch often has many nonlocal similar patches to it across the image and has been widely applied in many recently proposed machining learning algorithms for image processing. However, there is no theoretical analysis on its working principle in the literature. In this paper, we discover a potential causality between NSS and low-rank property of color images, which is also available to grey images. A new patch group based NSS prior scheme is proposed to learn explicit NSS models of natural color images. The numerical low-rank property of patched matrices is also rigorously proved. The NSS-based QMC algorithm computes an optimal low-rank approximation to the high-rank color image, resulting in high PSNR and SSIM measures and particularly the better visual quality. A new tensor NSS-based QMC method is also presented to solve the color video inpainting problem based on quaternion tensor representation. The numerical experiments on color images and videos indicate the advantages of NSS-based QMC over the state-of-the-art methods.

Tai-Xiang Jiang, Xi-Le Zhao, Hao Zhang et al.

In this paper, we propose a novel tensor learning and coding model for third-order data completion. Our model is to learn a data-adaptive dictionary from the given observations, and determine the coding coefficients of third-order tensor tubes. In the completion process, we minimize the low-rankness of each tensor slice containing the coding coefficients. By comparison with the traditional pre-defined transform basis, the advantages of the proposed model are that (i) the dictionary can be learned based on the given data observations so that the basis can be more adaptively and accurately constructed, and (ii) the low-rankness of the coding coefficients can allow the linear combination of dictionary features more effectively. Also we develop a multi-block proximal alternating minimization algorithm for solving such tensor learning and coding model, and show that the sequence generated by the algorithm can globally converge to a critical point. Extensive experimental results for real data sets such as videos, hyperspectral images, and traffic data are reported to demonstrate these advantages and show the performance of the proposed tensor learning and coding method is significantly better than the other tensor completion methods in terms of several evaluation metrics.

Guangjing Song, Michael K. Ng, Tai-Xiang Jiang

In this paper, we develop a new alternating projection method to compute nonnegative low rank matrix approximation for nonnegative matrices. In the nonnegative low rank matrix approximation method, the projection onto the manifold of fixed rank matrices can be expensive as the singular value decomposition is required. We propose to use the tangent space of the point in the manifold to approximate the projection onto the manifold in order to reduce the computational cost. We show that the sequence generated by the alternating projections onto the tangent spaces of the fixed rank matrices manifold and the nonnegative matrix manifold, converge linearly to a point in the intersection of the two manifolds where the convergent point is sufficiently close to optimal solutions. This convergence result based inexact projection onto the manifold is new and is not studied in the literature. Numerical examples in data clustering, pattern recognition and hyperspectral data analysis are given to demonstrate that the performance of the proposed method is better than that of nonnegative matrix factorization methods in terms of computational time and accuracy.

Wen Yu, Baiying Lei, Michael K. Ng et al.

It is of great significance to apply deep learning for the early diagnosis of Alzheimer's Disease (AD). In this work, a novel tensorizing GAN with high-order pooling is proposed to assess Mild Cognitive Impairment (MCI) and AD. By tensorizing a three-player cooperative game based framework, the proposed model can benefit from the structural information of the brain. By incorporating the high-order pooling scheme into the classifier, the proposed model can make full use of the second-order statistics of the holistic Magnetic Resonance Imaging (MRI) images. To the best of our knowledge, the proposed Tensor-train, High-pooling and Semi-supervised learning based GAN (THS-GAN) is the first work to deal with classification on MRI images for AD diagnosis. Extensive experimental results on Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset are reported to demonstrate that the proposed THS-GAN achieves superior performance compared with existing methods, and to show that both tensor-train and high-order pooling can enhance classification performance. The visualization of generated samples also shows that the proposed model can generate plausible samples for semi-supervised learning purpose.

Tai-Xiang Jiang, Michael K. Ng, Junjun Pan et al.

The main aim of this paper is to develop a new algorithm for computing nonnegative low rank tensor approximation for nonnegative tensors that arise in many multi-dimensional imaging applications. Nonnegativity is one of the important property as each pixel value refers to nonzero light intensity in image data acquisition. Our approach is different from classical nonnegative tensor factorization (NTF) which requires each factorized matrix and/or tensor to be nonnegative. In this paper, we determine a nonnegative low Tucker rank tensor to approximate a given nonnegative tensor. We propose an alternating projections algorithm for computing such nonnegative low rank tensor approximation, which is referred to as NLRT. The convergence of the proposed manifold projection method is established. Experimental results for synthetic data and multi-dimensional images are presented to demonstrate the performance of NLRT is better than state-of-the-art NTF methods.