Search Results for author: Risheng Liu

Found 78 papers, 32 papers with code

Learn from the Past: A Proxy based Adversarial Defense Framework to Boost Robustness

no code implementations19 Oct 2023 Yaohua Liu, Jiaxin Gao, Zhu Liu, Xianghao Jiao, Xin Fan, Risheng Liu

In light of the vulnerability of deep learning models to adversarial samples and the ensuing security issues, a range of methods, including Adversarial Training (AT) as a prominent representative, aimed at enhancing model robustness against various adversarial attacks, have seen rapid development.

Adversarial Defense

Diving into Darkness: A Dual-Modulated Framework for High-Fidelity Super-Resolution in Ultra-Dark Environments

no code implementations11 Sep 2023 Jiaxin Gao, Ziyu Yue, Yaohua Liu, Sihan Xie, Xin Fan, Risheng Liu

Super-resolution tasks oriented to images captured in ultra-dark environments is a practical yet challenging problem that has received little attention.


Fearless Luminance Adaptation: A Macro-Micro-Hierarchical Transformer for Exposure Correction

no code implementations2 Sep 2023 Gehui Li, JinYuan Liu, Long Ma, Zhiying Jiang, Xin Fan, Risheng Liu

To overcome these limitations, we propose a Macro-Micro-Hierarchical transformer, which consists of a macro attention to capture long-range dependencies, a micro attention to extract local features, and a hierarchical structure for coarse-to-fine correction.

Face Recognition Semantic Segmentation

Learning Heavily-Degraded Prior for Underwater Object Detection

1 code implementation24 Aug 2023 Chenping Fu, Xin Fan, Jiewen Xiao, Wanqi Yuan, Risheng Liu, Zhongxuan Luo

Therefore, we propose a residual feature transference module (RFTM) to learn a mapping between deep representations of the heavily degraded patches of DFUI- and underwater- images, and make the mapping as a heavily degraded prior (HDP) for underwater detection.

object-detection Object Detection

PAIF: Perception-Aware Infrared-Visible Image Fusion for Attack-Tolerant Semantic Segmentation

1 code implementation8 Aug 2023 Zhu Liu, JinYuan Liu, Benzhuang Zhang, Long Ma, Xin Fan, Risheng Liu

We first conduct systematic analyses about the components of image fusion, investigating the correlation with segmentation robustness under adversarial perturbations.

Infrared And Visible Image Fusion Segmentation +2

Bilevel Generative Learning for Low-Light Vision

1 code implementation7 Aug 2023 Yingchi Liu, Zhu Liu, Long Ma, JinYuan Liu, Xin Fan, Zhongxuan Luo, Risheng Liu

In this study, we propose a generic low-light vision solution by introducing a generative block to convert data from the RAW to the RGB domain.

Bilevel Optimization

WaterFlow: Heuristic Normalizing Flow for Underwater Image Enhancement and Beyond

no code implementations2 Aug 2023 Zengxi Zhang, Zhiying Jiang, JinYuan Liu, Xin Fan, Risheng Liu

Underwater images suffer from light refraction and absorption, which impairs visibility and interferes the subsequent applications.

Image Enhancement

Multi-Spectral Image Stitching via Spatial Graph Reasoning

no code implementations31 Jul 2023 Zhiying Jiang, Zengxi Zhang, JinYuan Liu, Xin Fan, Risheng Liu

Multi-spectral image stitching leverages the complementarity between infrared and visible images to generate a robust and reliable wide field-of-view (FOV) scene.

Image Stitching

Learning with Constraint Learning: New Perspective, Solution Strategy and Various Applications

no code implementations28 Jul 2023 Risheng Liu, Jiaxin Gao, Xuan Liu, Xin Fan

The complexity of learning problems, such as Generative Adversarial Network (GAN) and its variants, multi-task and meta-learning, hyper-parameter learning, and a variety of real-world vision applications, demands a deeper understanding of their underlying coupling mechanisms.


Joint Perceptual Learning for Enhancement and Object Detection in Underwater Scenarios

no code implementations7 Jul 2023 Chenping Fu, Wanqi Yuan, Jiewen Xiao, Risheng Liu, Xin Fan

Based on these factual opinions, we propose a bilevel optimization formulation for jointly learning underwater object detection and image enhancement, and then unroll to a dual perception network (DPNet) for the two tasks.

Bilevel Optimization Image Enhancement +3

Bilevel Fast Scene Adaptation for Low-Light Image Enhancement

1 code implementation2 Jun 2023 Long Ma, Dian Jin, Nan An, JinYuan Liu, Xin Fan, Risheng Liu

A bilevel learning framework is constructed to endow the scene-irrelevant generality of the encoder towards diverse scenes (i. e., freezing the encoder in the adaptation and testing phases).

Denoising Hyperparameter Optimization +1

Contrastive Learning Based Recursive Dynamic Multi-Scale Network for Image Deraining

no code implementations29 May 2023 Zhiying Jiang, Risheng Liu, Shuzhou Yang, Zengxi Zhang, Xin Fan

Extensive experiments on synthetic benchmarks and real-world images demonstrate that the proposed RDMC delivers strong performance on the depiction of rain streaks and outperforms the state-of-the-art methods.

Contrastive Learning object-detection +3

PEARL: Preprocessing Enhanced Adversarial Robust Learning of Image Deraining for Semantic Segmentation

no code implementations25 May 2023 Xianghao Jiao, Yaohua Liu, Jiaxin Gao, Xinyuan Chu, Risheng Liu, Xin Fan

In light of the significant progress made in the development and application of semantic segmentation tasks, there has been increasing attention towards improving the robustness of segmentation models against natural degradation factors (e. g., rain streaks) or artificially attack factors (e. g., adversarial attack).

Adversarial Attack Rain Removal +2

A Task-guided, Implicitly-searched and Meta-initialized Deep Model for Image Fusion

1 code implementation25 May 2023 Risheng Liu, Zhu Liu, JinYuan Liu, Xin Fan, Zhongxuan Luo

Qualitative and quantitative experimental results on different categories of image fusion problems and related downstream tasks (e. g., visual enhancement and semantic understanding) substantiate the flexibility and effectiveness of our TIM.

Motion-Scenario Decoupling for Rat-Aware Video Position Prediction: Strategy and Benchmark

no code implementations17 May 2023 Xiaofeng Liu, Jiaxin Gao, Yaohua Liu, Risheng Liu, Nenggan Zheng

Recently significant progress has been made in human action recognition and behavior prediction using deep learning techniques, leading to improved vision-based semantic understanding.

Action Recognition motion prediction +2

An Interactively Reinforced Paradigm for Joint Infrared-Visible Image Fusion and Saliency Object Detection

1 code implementation17 May 2023 Di Wang, JinYuan Liu, Risheng Liu, Xin Fan

Their common characteristic of seeking complementary cues from different source images motivates us to explore the collaborative relationship between Fusion and Salient object detection tasks on infrared and visible images via an Interactively Reinforced multi-task paradigm for the first time, termed IRFS.

Infrared And Visible Image Fusion object-detection +2

Bi-level Dynamic Learning for Jointly Multi-modality Image Fusion and Beyond

2 code implementations11 May 2023 Zhu Liu, JinYuan Liu, Guanyao Wu, Long Ma, Xin Fan, Risheng Liu

Recently, multi-modality scene perception tasks, e. g., image fusion and scene understanding, have attracted widespread attention for intelligent vision systems.

Scene Understanding

Breaking Modality Disparity: Harmonized Representation for Infrared and Visible Image Registration

no code implementations12 Apr 2023 Zhiying Jiang, Zengxi Zhang, JinYuan Liu, Xin Fan, Risheng Liu

Since the differences in viewing range, resolution and relative position, the multi-modality sensing module composed of infrared and visible cameras needs to be registered so as to have more accurate scene perception.

Image Registration

Hierarchical Optimization-Derived Learning

no code implementations11 Feb 2023 Risheng Liu, Xuan Liu, Shangzhi Zeng, Jin Zhang, Yixuan Zhang

In recent years, by utilizing optimization techniques to formulate the propagation of deep model, a variety of so-called Optimization-Derived Learning (ODL) approaches have been proposed to address diverse learning and vision tasks.

Averaged Method of Multipliers for Bi-Level Optimization without Lower-Level Strong Convexity

1 code implementation7 Feb 2023 Risheng Liu, Yaohua Liu, Wei Yao, Shangzhi Zeng, Jin Zhang

Gradient methods have become mainstream techniques for Bi-Level Optimization (BLO) in learning fields.

Practical Exposure Correction: Great Truths Are Always Simple

no code implementations29 Dec 2022 Long Ma, Tianjiao Ma, Xinwei Xue, Xin Fan, Zhongxuan Luo, Risheng Liu

Improving the visual quality of the given degraded observation by correcting exposure level is a fundamental task in the computer vision community.

Breaking Free from Fusion Rule: A Fully Semantic-driven Infrared and Visible Image Fusion

1 code implementation22 Nov 2022 Yuhui Wu, Zhu Liu, JinYuan Liu, Xin Fan, Risheng Liu

To address these challenges, in this letter, we develop a semantic-level fusion network to sufficiently utilize the semantic guidance, emancipating the experimental designed fusion rules.

Infrared And Visible Image Fusion

Semantic-aware Texture-Structure Feature Collaboration for Underwater Image Enhancement

1 code implementation19 Nov 2022 Di Wang, Long Ma, Risheng Liu, Xin Fan

To address the above limitations, we develop an efficient and compact enhancement network in collaboration with a high-level semantic-aware pretrained model, aiming to exploit its hierarchical feature representation as an auxiliary for the low-level underwater image enhancement.

Image Enhancement object-detection +2

Optimization-Derived Learning with Essential Convergence Analysis of Training and Hyper-training

no code implementations16 Jun 2022 Risheng Liu, Xuan Liu, Shangzhi Zeng, Jin Zhang, Yixuan Zhang

Recently, Optimization-Derived Learning (ODL) has attracted attention from learning and vision areas, which designs learning models from the perspective of optimization.

Image Deconvolution

Unsupervised Misaligned Infrared and Visible Image Fusion via Cross-Modality Image Generation and Registration

1 code implementation24 May 2022 Di Wang, JinYuan Liu, Xin Fan, Risheng Liu

Moreover, to better fuse the registered infrared images and visible images, we present a feature Interaction Fusion Module (IFM) to adaptively select more meaningful features for fusion in the Dual-path Interaction Fusion Network (DIFN).

Image Generation Infrared And Visible Image Fusion +1

Towards Extremely Fast Bilevel Optimization with Self-governed Convergence Guarantees

no code implementations20 May 2022 Risheng Liu, Xuan Liu, Wei Yao, Shangzhi Zeng, Jin Zhang

Gradient methods have become mainstream techniques for Bi-Level Optimization (BLO) in learning and vision fields.

Bilevel Optimization

Revisiting GANs by Best-Response Constraint: Perspective, Methodology, and Application

no code implementations20 May 2022 Risheng Liu, Jiaxin Gao, Xuan Liu, Xin Fan

In past years, the minimax type single-level optimization formulation and its variations have been widely utilized to address Generative Adversarial Networks (GANs).

Toward Fast, Flexible, and Robust Low-Light Image Enhancement

1 code implementation CVPR 2022 Long Ma, Tengyu Ma, Risheng Liu, Xin Fan, Zhongxuan Luo

Existing low-light image enhancement techniques are mostly not only difficult to deal with both visual quality and computational efficiency but also commonly invalid in unknown complex scenarios.

Face Detection Low-Light Image Enhancement +1

Segment, Magnify and Reiterate: Detecting Camouflaged Objects the Hard Way

1 code implementation CVPR 2022 Qi Jia, Shuilian Yao, Yu Liu, Xin Fan, Risheng Liu, Zhongxuan Luo

To tackle camouflaged object detection (COD), we are inspired by humans attention coupled with the coarse-to-fine detection strategy, and thereby propose an iterative refinement framework, coined SegMaR, which integrates Segment, Magnify and Reiterate in a multi-stage detection fashion.

object-detection Object Detection

Self-Augmented Unpaired Image Dehazing via Density and Depth Decomposition

1 code implementation CVPR 2022 Yang Yang, Chaoyue Wang, Risheng Liu, Lin Zhang, Xiaojie Guo, DaCheng Tao

With estimated scene depth, our method is capable of re-rendering hazy images with different thicknesses which further benefits the training of the dehazing network.

Image Dehazing

Learning with Nested Scene Modeling and Cooperative Architecture Search for Low-Light Vision

1 code implementation9 Dec 2021 Risheng Liu, Long Ma, Tengyu Ma, Xin Fan, Zhongxuan Luo

To partially address above issues, we establish Retinex-inspired Unrolling with Architecture Search (RUAS), a general learning framework, which not only can address low-light enhancement task, but also has the flexibility to handle other more challenging downstream vision applications.

Rolling Shutter Correction

Learning Deep Context-Sensitive Decomposition for Low-Light Image Enhancement

1 code implementation9 Dec 2021 Long Ma, Risheng Liu, Jiaao Zhang, Xin Fan, Zhongxuan Luo

Further, by sharing an encoder for these two components, we obtain a more lightweight version (SLiteCSDNet for short).

Low-Light Image Enhancement

Triple-level Model Inferred Collaborative Network Architecture for Video Deraining

no code implementations8 Nov 2021 Pan Mu, Zhu Liu, Yaohua Liu, Risheng Liu, Xin Fan

In this paper, we develop a model-guided triple-level optimization framework to deduce network architecture with cooperating optimization and auto-searching mechanism, named Triple-level Model Inferred Cooperating Searching (TMICS), for dealing with various video rain circumstances.

Optical Flow Estimation Rain Removal

Value-Function-based Sequential Minimization for Bi-level Optimization

1 code implementation11 Oct 2021 Risheng Liu, Xuan Liu, Shangzhi Zeng, Jin Zhang, Yixuan Zhang

We also extend BVFSM to address BLO with additional functional constraints.

Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond

1 code implementation NeurIPS 2021 Risheng Liu, Yaohua Liu, Shangzhi Zeng, Jin Zhang

In particular, by introducing an auxiliary as initialization to guide the optimization dynamics and designing a pessimistic trajectory truncation operation, we construct a reliable approximate version of the original BLO in the absence of LLC hypothesis.

Bilevel Optimization

A Value-Function-based Interior-point Method for Non-convex Bi-level Optimization

no code implementations15 Jun 2021 Risheng Liu, Xuan Liu, Xiaoming Yuan, Shangzhi Zeng, Jin Zhang

Bi-level optimization model is able to capture a wide range of complex learning tasks with practical interest.

A General Descent Aggregation Framework for Gradient-based Bi-level Optimization

1 code implementation16 Feb 2021 Risheng Liu, Pan Mu, Xiaoming Yuan, Shangzhi Zeng, Jin Zhang

In this work, we formulate BLOs from an optimistic bi-level viewpoint and establish a new gradient-based algorithmic framework, named Bi-level Descent Aggregation (BDA), to partially address the above issues.


Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond

1 code implementation27 Jan 2021 Risheng Liu, Jiaxin Gao, Jin Zhang, Deyu Meng, Zhouchen Lin

Bi-Level Optimization (BLO) is originated from the area of economic game theory and then introduced into the optimization community.

Meta-Learning Neural Architecture Search

Optimization-Inspired Learning with Architecture Augmentations and Control Mechanisms for Low-Level Vision

1 code implementation10 Dec 2020 Risheng Liu, Zhu Liu, Pan Mu, Xin Fan, Zhongxuan Luo

Specifically, by introducing a general energy minimization model and formulating its descent direction from different viewpoints (i. e., in a generative manner, based on the discriminative metric and with optimality-based correction), we construct three propagative modules to effectively solve the optimization models with flexible combinations.

BOML: A Modularized Bilevel Optimization Library in Python for Meta Learning

2 code implementations28 Sep 2020 Yaohua Liu, Risheng Liu

learning to learn) has recently emerged as a promising paradigm for a variety of applications.

Bilevel Optimization Meta-Learning

SMPR: Single-Stage Multi-Person Pose Regression

1 code implementation28 Jun 2020 Junqi Lin, Huixin Miao, Junjie Cao, Zhixun Su, Risheng Liu

Existing multi-person pose estimators can be roughly divided into two-stage approaches (top-down and bottom-up approaches) and one-stage approaches.

Multi-Person Pose Estimation regression

A Generic First-Order Algorithmic Framework for Bi-Level Programming Beyond Lower-Level Singleton

no code implementations ICML 2020 Risheng Liu, Pan Mu, Xiaoming Yuan, Shangzhi Zeng, Jin Zhang

In recent years, a variety of gradient-based first-order methods have been developed to solve bi-level optimization problems for learning applications.


Learning Deformable Image Registration from Optimization: Perspective, Modules, Bilevel Training and Beyond

2 code implementations30 Apr 2020 Risheng Liu, Zi Li, Xin Fan, Chenying Zhao, Hao Huang, Zhongxuan Luo

We design a new deep learning based framework to optimize a diffeomorphic model via multi-scale propagation in order to integrate advantages and avoid limitations of these two categories of approaches.

Image Registration Image Segmentation +1

Converged Deep Framework Assembling Principled Modules for CS-MRI

no code implementations29 Oct 2019 Risheng Liu, Yuxi Zhang, Shichao Cheng, Zhongxuan Luo, Xin Fan

Compressed Sensing Magnetic Resonance Imaging (CS-MRI) significantly accelerates MR data acquisition at a sampling rate much lower than the Nyquist criterion.

Investigating Task-driven Latent Feasibility for Nonconvex Image Modeling

no code implementations18 Oct 2019 Risheng Liu, Pan Mu, Jian Chen, Xin Fan, Zhongxuan Luo

Properly modeling latent image distributions plays an important role in a variety of image-related vision problems.

Deblurring Image Deblurring

Investigating Customization Strategies and Convergence Behaviors of Task-specific ADMM

no code implementations24 Sep 2019 Risheng Liu, Pan Mu, Jin Zhang

Alternating Direction Method of Multiplier (ADMM) has been a popular algorithmic framework for separable optimization problems with linear constraints.

Semi-supervised Skin Detection by Network with Mutual Guidance

no code implementations ICCV 2019 Yi He, Jiayuan Shi, Chuan Wang, Haibin Huang, Jiaming Liu, Guanbin Li, Risheng Liu, Jue Wang

In this paper we present a new data-driven method for robust skin detection from a single human portrait image.

Underexposed Image Correction via Hybrid Priors Navigated Deep Propagation

no code implementations17 Jul 2019 Risheng Liu, Long Ma, Yuxi Zhang, Xin Fan, Zhongxuan Luo

Plenty of experimental results of underexposed image correction demonstrate that our proposed method performs favorably against the state-of-the-art methods on both subjective and objective assessments.

Face Detection Single Image Haze Removal

Task-Oriented Convex Bilevel Optimization with Latent Feasibility

no code implementations6 Jul 2019 Risheng Liu, Long Ma, Xiaoming Yuan, Shangzhi Zeng, Jin Zhang

This paper firstly proposes a convex bilevel optimization paradigm to formulate and optimize popular learning and vision problems in real-world scenarios.

Bilevel Optimization

Real-world Underwater Enhancement: Challenges, Benchmarks, and Solutions

1 code implementation15 Jan 2019 Risheng Liu, Xin Fan, Ming Zhu, Minjun Hou, Zhongxuan Luo

Underwater image enhancement is such an important low-level vision task with many applications that numerous algorithms have been proposed in recent years.

Image Enhancement object-detection +1

A Bridging Framework for Model Optimization and Deep Propagation

no code implementations NeurIPS 2018 Risheng Liu, Shichao Cheng, Xiaokun Liu, Long Ma, Xin Fan, Zhongxuan Luo

Different from these existing network based iterations, which often lack theoretical investigations, we provide strict convergence analysis for PODM in the challenging nonconvex and nonsmooth scenarios.

Model Optimization

Exploiting Local Feature Patterns for Unsupervised Domain Adaptation

no code implementations12 Nov 2018 Jun Wen, Risheng Liu, Nenggan Zheng, Qian Zheng, Zhefeng Gong, Junsong Yuan

In this paper, we present a method for learning domain-invariant local feature patterns and jointly aligning holistic and local feature statistics.

Unsupervised Domain Adaptation

A Theoretically Guaranteed Deep Optimization Framework for Robust Compressive Sensing MRI

no code implementations9 Nov 2018 Risheng Liu, Yuxi Zhang, Shichao Cheng, Xin Fan, Zhongxuan Luo

Magnetic Resonance Imaging (MRI) is one of the most dynamic and safe imaging techniques available for clinical applications.

Compressive Sensing

Task Embedded Coordinate Update: A Realizable Framework for Multivariate Non-convex Optimization

no code implementations5 Nov 2018 Yiyang Wang, Risheng Liu, Long Ma, Xiaoliang Song

Integrating both numerical algorithms and advanced techniques together, TECU is proposed in a unified framework for solving a class of non-convex problems.

Learning Converged Propagations with Deep Prior Ensemble for Image Enhancement

1 code implementation9 Oct 2018 Risheng Liu, Long Ma, Yiyang Wang, Lei Zhang

Enhancing visual qualities of images plays very important roles in various vision and learning applications.

Image Enhancement

On the Convergence of Learning-based Iterative Methods for Nonconvex Inverse Problems

no code implementations16 Aug 2018 Risheng Liu, Shichao Cheng, Yi He, Xin Fan, Zhouchen Lin, Zhongxuan Luo

Moreover, there is a lack of rigorous analysis about the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague.


Toward Designing Convergent Deep Operator Splitting Methods for Task-specific Nonconvex Optimization

no code implementations28 Apr 2018 Risheng Liu, Shichao Cheng, Yi He, Xin Fan, Zhongxuan Luo

Operator splitting methods have been successfully used in computational sciences, statistics, learning and vision areas to reduce complex problems into a series of simpler subproblems.


Self-Reinforced Cascaded Regression for Face Alignment

no code implementations23 Nov 2017 Xin Fan, Risheng Liu, Kang Huyan, Yuyao Feng, Zhongxuan Luo

Cascaded regression is prevailing in face alignment thanks to its accuracy and robustness, but typically demands manually annotated examples having low discrepancy between shape-indexed features and shape updates.

Face Alignment Philosophy +1

An Optimization Framework with Flexible Inexact Inner Iterations for Nonconvex and Nonsmooth Programming

no code implementations28 Feb 2017 Yiyang Wang, Risheng Liu, Xiaoliang Song, Zhixun Su

In recent years, numerous vision and learning tasks have been (re)formulated as nonconvex and nonsmooth programmings(NNPs).

Sparse Coding and Counting for Robust Visual Tracking

no code implementations28 May 2016 Risheng Liu, Jing Wang, Yiyang Wang, Zhixun Su, Yu Cai

In this paper, we propose a novel sparse coding and counting method under Bayesian framwork for visual tracking.

Visual Tracking

Adaptive Partial Differential Equation Learning for Visual Saliency Detection

no code implementations CVPR 2014 Risheng Liu, Junjie Cao, Zhouchen Lin, Shiguang Shan

Then by optimizing a discrete submodular function constrained with this LESD and a uniform matroid, the saliency seeds (i. e., boundary conditions) can be learnt for this image, thus achieving an optimal PDE system to model the evolution of visual saliency.

Saliency Detection

Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty for Separable Convex Programs in Machine Learning

no code implementations18 Oct 2013 Zhouchen Lin, Risheng Liu, Huan Li

However, the traditional alternating direction method (ADM) and its linearized version (LADM, obtained by linearizing the quadratic penalty term) are for the two-block case and cannot be naively generalized to solve the multi-block case.

Distributed Computing

Kernel Estimation from Salient Structure for Robust Motion Deblurring

no code implementations5 Dec 2012 Jinshan Pan, Risheng Liu, Zhixun Su, Xianfeng GU

One effective way to eliminate these details is to apply image denoising model based on the Total Variation (TV).

Blind Image Deblurring Image Deblurring +2

Linearized Alternating Direction Method with Adaptive Penalty for Low-Rank Representation

no code implementations NeurIPS 2011 Zhouchen Lin, Risheng Liu, Zhixun Su

It suffers from $O(n^3)$ computation complexity due to the matrix-matrix multiplications and matrix inversions, even if partial SVD is used.

Optimization and Control

Solving Principal Component Pursuit in Linear Time via $l_1$ Filtering

no code implementations26 Aug 2011 Risheng Liu, Zhouchen Lin, Siming Wei, Zhixun Su

In this paper, we propose a novel algorithm, called $l_1$ filtering, for \emph{exactly} solving PCP with an $O(r^2(m+n))$ complexity, where $m\times n$ is the size of data matrix and $r$ is the rank of the matrix to recover, which is supposed to be much smaller than $m$ and $n$.

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