Zhaoqiang Liu

Zhaoqiang Liu, Jiulong Liu, Subhroshekhar Ghosh et al.

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order $\sqrt{\frac{k\log L}{m}}$, where $m$ is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

Zhaoqiang Liu, Jun Han

In this paper, we propose projected gradient descent (PGD) algorithms for signal estimation from noisy nonlinear measurements. We assume that the unknown $p$-dimensional signal lies near the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. In particular, we consider two cases when the nonlinear link function is either unknown or known. For unknown nonlinearity, similarly to \cite{liu2020generalized}, we make the assumption of sub-Gaussian observations and propose a linear least-squares estimator. We show that when there is no representation error and the sensing vectors are Gaussian, roughly $O(k \log L)$ samples suffice to ensure that a PGD algorithm converges linearly to a point achieving the optimal statistical rate using arbitrary initialization. For known nonlinearity, we assume monotonicity as in \cite{yang2016sparse}, and make much weaker assumptions on the sensing vectors and allow for representation error. We propose a nonlinear least-squares estimator that is guaranteed to enjoy an optimal statistical rate. A corresponding PGD algorithm is provided and is shown to also converge linearly to the estimator using arbitrary initialization. In addition, we present experimental results on image datasets to demonstrate the performance of our PGD algorithms.

Jiulong Liu, Zhaoqiang Liu

In this paper, we aim to estimate the direction of an underlying signal from its nonlinear observations following the semi-parametric single index model (SIM). Unlike conventional compressed sensing where the signal is assumed to be sparse, we assume that the signal lies in the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. This is mainly motivated by the tremendous success of deep generative models in various real applications. Our reconstruction method is non-iterative (though approximating the projection step may use an iterative procedure) and highly efficient, and it is shown to attain the near-optimal statistical rate of order $\sqrt{(k \log L)/m}$, where $m$ is the number of measurements. We consider two specific instances of the SIM, namely noisy $1$-bit and cubic measurement models, and perform experiments on image datasets to demonstrate the efficacy of our method. In particular, for the noisy $1$-bit measurement model, we show that our non-iterative method significantly outperforms a state-of-the-art iterative method in terms of both accuracy and efficiency.

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.

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.

Zhaoqiang Liu, Wen Li, Junren Chen

Generalized eigenvalue problems (GEPs) find applications in various fields of science and engineering. For example, principal component analysis, Fisher's discriminant analysis, and canonical correlation analysis are specific instances of GEPs and are widely used in statistical data processing. In this work, we study GEPs under generative priors, assuming that the underlying leading generalized eigenvector lies within the range of a Lipschitz continuous generative model. Under appropriate conditions, we show that any optimal solution to the corresponding optimization problems attains the optimal statistical rate. Moreover, from a computational perspective, we propose an iterative algorithm called the Projected Rayleigh Flow Method (PRFM) to approximate the optimal solution. We theoretically demonstrate that under suitable assumptions, PRFM converges linearly to an estimated vector that achieves the optimal statistical rate. Numerical results are provided to demonstrate the effectiveness of the proposed method.

Anqi Tang, Youming Chen, Shuchen Xue et al.

Diffusion models (DMs) have demonstrated remarkable ability to generate diverse and high-quality images by efficiently modeling complex data distributions. They have also been explored as powerful generative priors for signal recovery, resulting in a substantial improvement in the quality of reconstructed signals. However, existing research on signal recovery with diffusion models either focuses on specific reconstruction problems or is unable to handle nonlinear measurement models with discontinuous or unknown link functions. In this work, we focus on using DMs to achieve accurate recovery from semi-parametric single index models, which encompass a variety of popular nonlinear models that may have {\em discontinuous} and {\em unknown} link functions. We propose an efficient reconstruction method that only requires one round of unconditional sampling and (partial) inversion of DMs. Theoretical analysis on the effectiveness of the proposed methods has been established under appropriate conditions. We perform numerical experiments on image datasets for different nonlinear measurement models. We observe that compared to competing methods, our approach can yield more accurate reconstructions while utilizing significantly fewer neural function evaluations.

Yang Zheng, Wen Li, Zhaoqiang Liu

Inverse problems (IPs) involve reconstructing signals from noisy observations. Recently, diffusion models (DMs) have emerged as a powerful framework for solving IPs, achieving remarkable reconstruction performance. However, existing DM-based methods frequently encounter issues such as heavy computational demands and suboptimal convergence. In this work, building upon the idea of the recent work DMPlug, we propose two novel methods, DMILO and DMILO-PGD, to address these challenges. Our first method, DMILO, employs intermediate layer optimization (ILO) to alleviate the memory burden inherent in DMPlug. Additionally, by introducing sparse deviations, we expand the range of DMs, enabling the exploration of underlying signals that may lie outside the range of the diffusion model. We further propose DMILO-PGD, which integrates ILO with projected gradient descent (PGD), thereby reducing the risk of suboptimal convergence. We provide an intuitive theoretical analysis of our approaches under appropriate conditions and validate their superiority through extensive experiments on diverse image datasets, encompassing both linear and nonlinear IPs. Our results demonstrate significant performance gains over state-of-the-art methods, highlighting the effectiveness of DMILO and DMILO-PGD in addressing common challenges in DM-based IP solvers.

Yijian Wang, Tongxian Guo, Zhaoqiang Liu

The fuzzy job shop scheduling problem (FJSSP) emerges as an innovative extension to the job shop scheduling problem (JSSP), incorporating a layer of uncertainty that aligns the problem more closely with the complexities of real-world manufacturing environments. This improvement increases the computational complexity of deriving the solution while improving its applicability. In the domain of deterministic scheduling, neural combinatorial optimization (NCO) has recently demonstrated remarkable efficacy. However, its application to the realm of fuzzy scheduling has been relatively unexplored. This paper aims to bridge this gap by investigating the feasibility of employing neural networks to assimilate and process fuzzy information for the resolution of FJSSP, thereby leveraging the advancements in NCO to enhance fuzzy scheduling methodologies. To achieve this, we approach the FJSSP as a generative task and introduce an expectation-maximization algorithm-based autoregressive model (EMARM) to address it. During training, our model alternates between generating scheduling schemes from given instances (E-step) and adjusting the autoregressive model weights based on these generated schemes (M-step). This novel methodology effectively navigates around the substantial hurdle of obtaining ground-truth labels, which is a prevalent issue in NCO frameworks. In testing, the experimental results demonstrate the superior capability of EMARM in addressing the FJSSP, showcasing its effectiveness and potential for practical applications in fuzzy scheduling.

Zhaoqiang Liu, Subhroshekhar Ghosh, Jun Han et al.

In 1-bit compressive sensing, each measurement is quantized to a single bit, namely the sign of a linear function of an unknown vector, and the goal is to accurately recover the vector. While it is most popular to assume a standard Gaussian sensing matrix for 1-bit compressive sensing, using structured sensing matrices such as partial Gaussian circulant matrices is of significant practical importance due to their faster matrix operations. In this paper, we provide recovery guarantees for a correlation-based optimization algorithm for robust 1-bit compressive sensing with randomly signed partial Gaussian circulant matrices and generative models. Under suitable assumptions, we match guarantees that were previously only known to hold for i.i.d.~Gaussian matrices that require significantly more computation. We make use of a practical iterative algorithm, and perform numerical experiments on image datasets to corroborate our theoretical results.

Zhaoqiang Liu, Subhroshekhar Ghosh, Jonathan Scarlett

Compressive phase retrieval is a popular variant of the standard compressive sensing problem in which the measurements only contain magnitude information. In this paper, motivated by recent advances in deep generative models, we provide recovery guarantees with near-optimal sample complexity for phase retrieval with generative priors. We first show that when using i.i.d. Gaussian measurements and an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs, roughly $O(k \log L)$ samples suffice to guarantee that any signal minimizing an amplitude-based empirical loss function is close to the true signal. Attaining this sample complexity with a practical algorithm remains a difficult challenge, and finding a good initialization for gradient-based methods has been observed to pose a major bottleneck. To partially address this, we further show that roughly $O(k \log L)$ samples ensure sufficient closeness between the underlying signal and any {\em globally optimal} solution to an optimization problem designed for spectral initialization (though finding such a solution may still be challenging). We also adapt this result to sparse phase retrieval, and show that $O(s \log n)$ samples are sufficient for a similar guarantee when the underlying signal is $s$-sparse and $n$-dimensional, matching an information-theoretic lower bound. While these guarantees do not directly correspond to a practical algorithm, we propose a practical spectral initialization method motivated by our findings, and experimentally observe performance gains over various existing spectral initialization methods for sparse phase retrieval.

Zhaoqiang Liu, Jonathan Scarlett

In this paper, we study the problem of signal estimation from noisy non-linear measurements when the unknown $n$-dimensional signal is in the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We make the assumption of sub-Gaussian measurements, which is satisfied by a wide range of measurement models, such as linear, logistic, 1-bit, and other quantized models. In addition, we consider the impact of adversarial corruptions on these measurements. Our analysis is based on a generalized Lasso approach (Plan and Vershynin, 2016). We first provide a non-uniform recovery guarantee, which states that under i.i.d.~Gaussian measurements, roughly $O\left(\frac{k}{ε^2}\log L\right)$ samples suffice for recovery with an $\ell_2$-error of $ε$, and that this scheme is robust to adversarial noise. Then, we apply this result to neural network generative models, and discuss various extensions to other models and non-i.i.d.~measurements. Moreover, we show that our result can be extended to the uniform recovery guarantee under the assumption of a so-called local embedding property, which is satisfied by the 1-bit and censored Tobit models.

Zhaoqiang Liu, Selwyn Gomes, Avtansh Tiwari et al.

The goal of standard 1-bit compressive sensing is to accurately recover an unknown sparse vector from binary-valued measurements, each indicating the sign of a linear function of the vector. Motivated by recent advances in compressive sensing with generative models, where a generative modeling assumption replaces the usual sparsity assumption, we study the problem of 1-bit compressive sensing with generative models. We first consider noiseless 1-bit measurements, and provide sample complexity bounds for approximate recovery under i.i.d.~Gaussian measurements and a Lipschitz continuous generative prior, as well as a near-matching algorithm-independent lower bound. Moreover, we demonstrate that the Binary $ε$-Stable Embedding property, which characterizes the robustness of the reconstruction to measurement errors and noise, also holds for 1-bit compressive sensing with Lipschitz continuous generative models with sufficiently many Gaussian measurements. In addition, we apply our results to neural network generative models, and provide a proof-of-concept numerical experiment demonstrating significant improvements over sparsity-based approaches.

Zhaoqiang Liu, Jonathan Scarlett

It has recently been shown that for compressive sensing, significantly fewer measurements may be required if the sparsity assumption is replaced by the assumption the unknown vector lies near the range of a suitably-chosen generative model. In particular, in (Bora {\em et al.}, 2017) it was shown roughly $O(k\log L)$ random Gaussian measurements suffice for accurate recovery when the generative model is an $L$-Lipschitz function with bounded $k$-dimensional inputs, and $O(kd \log w)$ measurements suffice when the generative model is a $k$-input ReLU network with depth $d$ and width $w$. In this paper, we establish corresponding algorithm-independent lower bounds on the sample complexity using tools from minimax statistical analysis. In accordance with the above upper bounds, our results are summarized as follows: (i) We construct an $L$-Lipschitz generative model capable of generating group-sparse signals, and show that the resulting necessary number of measurements is $Ω(k \log L)$; (ii) Using similar ideas, we construct ReLU networks with high depth and/or high depth for which the necessary number of measurements scales as $Ω\big( kd \frac{\log w}{\log n}\big)$ (with output dimension $n$), and in some cases $Ω(kd \log w)$. As a result, we establish that the scaling laws derived in (Bora {\em et al.}, 2017) are optimal or near-optimal in the absence of further assumptions.

Zhaoqiang Liu

Nonnegative matrix factorization (NMF) has been widely used in machine learning and signal processing because of its non-subtractive, part-based property which enhances interpretability. It is often assumed that the latent dimensionality (or the number of components) is given. Despite the large amount of algorithms designed for NMF, there is little literature about automatic model selection for NMF with theoretical guarantees. In this paper, we propose an algorithm that first calculates an empirical second-order moment from the empirical fourth-order cumulant tensor, and then estimates the latent dimensionality by recovering the support union (the index set of non-zero rows) of a matrix related to the empirical second-order moment. By assuming a generative model of the data with additional mild conditions, our algorithm provably detects the true latent dimensionality. We show on synthetic examples that our proposed algorithm is able to find an approximately correct number of components.

Zhaoqiang Liu, Vincent Y. F. Tan

The learning of mixture models can be viewed as a clustering problem. Indeed, given data samples independently generated from a mixture of distributions, we often would like to find the {\it correct target clustering} of the samples according to which component distribution they were generated from. For a clustering problem, practitioners often choose to use the simple $k$-means algorithm. $k$-means attempts to find an {\it optimal clustering} that minimizes the sum-of-squares distance between each point and its cluster center. In this paper, we consider fundamental (i.e., information-theoretic) limits of the solutions (clusterings) obtained by optimizing the sum-of-squares distance. In particular, we provide sufficient conditions for the closeness of any optimal clustering and the correct target clustering assuming that the data samples are generated from a mixture of spherical Gaussian distributions. We also generalize our results to log-concave distributions. Moreover, we show that under similar or even weaker conditions on the mixture model, any optimal clustering for the samples with reduced dimensionality is also close to the correct target clustering. These results provide intuition for the informativeness of $k$-means (with and without dimensionality reduction) as an algorithm for learning mixture models.

Zhaoqiang Liu, Vincent Y. F. Tan

We propose a geometric assumption on nonnegative data matrices such that under this assumption, we are able to provide upper bounds (both deterministic and probabilistic) on the relative error of nonnegative matrix factorization (NMF). The algorithm we propose first uses the geometric assumption to obtain an exact clustering of the columns of the data matrix; subsequently, it employs several rank-one NMFs to obtain the final decomposition. When applied to data matrices generated from our statistical model, we observe that our proposed algorithm produces factor matrices with comparable relative errors vis-à-vis classical NMF algorithms but with much faster speeds. On face image and hyperspectral imaging datasets, we demonstrate that our algorithm provides an excellent initialization for applying other NMF algorithms at a low computational cost. Finally, we show on face and text datasets that the combinations of our algorithm and several classical NMF algorithms outperform other algorithms in terms of clustering performance.