arXiv subjects

Zhaoqiang Liu

Publications and source records attributed to Zhaoqiang Liu.

35 records · Page 2Linked to original sources

arXiv2024

The Surprising Effectiveness of Skip-Tuning in Diffusion Sampling

With the incorporation of the UNet architecture, diffusion probabilistic models have become a dominant force in image generation tasks. One key design in UNet is the skip connections between the encoder and decoder blocks. Although skip connections have been shown to improve training stability and model performance, we reveal that such shortcuts can be a limiting factor for the complexity of the transformation. As the sampling steps decrease, the generation process and the role of the UNet get closer to the push-forward transformations from Gaussian distribution to the target, posing a challenge for the network's complexity. To address this challenge, we propose Skip-Tuning, a simple yet surprisingly effective training-free tuning method on the skip connections. Our method can achieve 100% FID improvement for pretrained EDM on ImageNet 64 with only 19 NFEs (1.75), breaking the limit of ODE samplers regardless of sampling steps. Surprisingly, the improvement persists when we increase the number of sampling steps and can even surpass the best result from EDM-2 (1.58) with only 39 NFEs (1.57). Comprehensive exploratory experiments are conducted to shed light on the surprising effectiveness. We observe that while Skip-Tuning increases the score-matching losses in the pixel space, the losses in the feature space are reduced, particularly at intermediate noise levels, which coincide with the most effective range accounting for image quality improvement.

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arXiv2024

Uniform Recovery Guarantees for Quantized Corrupted Sensing Using Structured or Generative Priors

This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of all signals and corruptions using a single draw of the sub-Gaussian sensing matrix and uniform dither. For signal and corruption with structured priors (e.g., sparsity, low-rankness), our uniform error rate for constrained Lasso typically coincides with the non-uniform one [Sun, Cui and Liu, 2022] up to logarithmic factors. By contrast, our uniform error rate for unconstrained Lasso exhibits worse dependence on the structured parameters due to regularization parameters larger than the ones for non-uniform recovery. For signal and corruption living in the ranges of some Lipschitz continuous generative models (referred to as generative priors), we achieve uniform recovery via constrained Lasso with a measurement number proportional to the latent dimensions of the generative models. Our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves inner product. As a by-product, our QPE result refines the one in [Xu and Jacques, 2020] under sub-Gaussian random matrix, and in this specific instance we are able to sharpen the uniform error decaying rate (for the projected-back projection estimator with signals in some convex symmetric set) presented therein from $O(m^{-1/16})$ to $O(m^{-1/8})$.

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arXiv2023

A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing

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.

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arXiv2023

DiffFit: Unlocking Transferability of Large Diffusion Models via Simple Parameter-Efficient Fine-Tuning

Diffusion models have proven to be highly effective in generating high-quality images. However, adapting large pre-trained diffusion models to new domains remains an open challenge, which is critical for real-world applications. This paper proposes DiffFit, a parameter-efficient strategy to fine-tune large pre-trained diffusion models that enable fast adaptation to new domains. DiffFit is embarrassingly simple that only fine-tunes the bias term and newly-added scaling factors in specific layers, yet resulting in significant training speed-up and reduced model storage costs. Compared with full fine-tuning, DiffFit achieves 2$\times$ training speed-up and only needs to store approximately 0.12\% of the total model parameters. Intuitive theoretical analysis has been provided to justify the efficacy of scaling factors on fast adaptation. On 8 downstream datasets, DiffFit achieves superior or competitive performances compared to the full fine-tuning while being more efficient. Remarkably, we show that DiffFit can adapt a pre-trained low-resolution generative model to a high-resolution one by adding minimal cost. Among diffusion-based methods, DiffFit sets a new state-of-the-art FID of 3.02 on ImageNet 512$\times$512 benchmark by fine-tuning only 25 epochs from a public pre-trained ImageNet 256$\times$256 checkpoint while being 30$\times$ more training efficient than the closest competitor.

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arXiv2023

DDP: Diffusion Model for Dense Visual Prediction

We propose a simple, efficient, yet powerful framework for dense visual predictions based on the conditional diffusion pipeline. Our approach follows a "noise-to-map" generative paradigm for prediction by progressively removing noise from a random Gaussian distribution, guided by the image. The method, called DDP, efficiently extends the denoising diffusion process into the modern perception pipeline. Without task-specific design and architecture customization, DDP is easy to generalize to most dense prediction tasks, e.g., semantic segmentation and depth estimation. In addition, DDP shows attractive properties such as dynamic inference and uncertainty awareness, in contrast to previous single-step discriminative methods. We show top results on three representative tasks with six diverse benchmarks, without tricks, DDP achieves state-of-the-art or competitive performance on each task compared to the specialist counterparts. For example, semantic segmentation (83.9 mIoU on Cityscapes), BEV map segmentation (70.6 mIoU on nuScenes), and depth estimation (0.05 REL on KITTI). We hope that our approach will serve as a solid baseline and facilitate future research

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arXiv2022

Misspecified Phase Retrieval with Generative Priors

In this paper, we study phase retrieval under model misspecification and generative priors. In particular, we aim to estimate an $n$-dimensional signal $\mathbf{x}$ from $m$ i.i.d.~realizations of the single index model $y = f(\mathbf{a}^T\mathbf{x})$, where $f$ is an unknown and possibly random nonlinear link function and $\mathbf{a} \in \mathbb{R}^n$ is a standard Gaussian vector. We make the assumption $\mathrm{Cov}[y,(\mathbf{a}^T\mathbf{x})^2] \ne 0$, which corresponds to the misspecified phase retrieval problem. In addition, the underlying signal $\mathbf{x}$ is assumed to lie in the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We propose a two-step approach, for which the first step plays the role of spectral initialization and the second step refines the estimated vector produced by the first step iteratively. We show that both steps enjoy a statistical rate of order $\sqrt{(k\log L)\cdot (\log m)/m}$ under suitable conditions. Experiments on image datasets are performed to demonstrate that our approach performs on par with or even significantly outperforms several competing methods.

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arXiv2022

Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors

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.

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arXiv2022

Generative Principal Component Analysis

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.

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arXiv2022

The Informativeness of K -Means for Learning Mixture Models

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.

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arXiv2022

Non-Iterative Recovery from Nonlinear Observations using Generative Models

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.

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arXiv2021

Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors

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.

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arXiv2021

Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

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.

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arXiv2020

The Generalized Lasso with Nonlinear Observations and Generative Priors

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.

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arXiv2020

Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priors

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.

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arXiv2020

Information-Theoretic Lower Bounds for Compressive Sensing with Generative Models

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.

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arXiv2018

Model Selection for Nonnegative Matrix Factorization by Support Union Recovery

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.

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arXiv2017

Rank-One NMF-Based Initialization for NMF and Relative Error Bounds under a Geometric Assumption

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.

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