Han Liu

Yuxin Zhang, Mingbao Lin, Xunchao Li et al.

Moire patterns appear frequently when taking photos of digital screens, drastically degrading the image quality. Despite the advance of CNNs in image demoireing, existing networks are with heavy design, causing redundant computation burden for mobile devices. In this paper, we launch the first study on accelerating demoireing networks and propose a dynamic demoireing acceleration method (DDA) towards a real-time deployment on mobile devices. Our stimulus stems from a simple-yet-universal fact that moire patterns often unbalancedly distribute across an image. Consequently, excessive computation is wasted upon non-moire areas. Therefore, we reallocate computation costs in proportion to the complexity of image patches. In order to achieve this aim, we measure the complexity of an image patch by designing a novel moire prior that considers both colorfulness and frequency information of moire patterns. Then, we restore image patches with higher-complexity using larger networks and the ones with lower-complexity are assigned with smaller networks to relieve the computation burden. At last, we train all networks in a parameter-shared supernet paradigm to avoid additional parameter burden. Extensive experiments on several benchmarks demonstrate the efficacy of our proposed DDA. In addition, the acceleration evaluated on the VIVO X80 Pro smartphone equipped with a chip of Snapdragon 8 Gen 1 shows that our method can drastically reduce the inference time, leading to a real-time image demoireing on mobile devices. Source codes and models are released at https://github.com/zyxxmu/DDA

Hao Li, Han Liu, Dewei Hu et al.

Interactive segmentation model leverages prompts from users to produce robust segmentation. This advancement is facilitated by prompt engineering, where interactive prompts serve as strong priors during test-time. However, this is an inherently subjective and hard-to-reproduce process. The variability in user expertise and inherently ambiguous boundaries in medical images can lead to inconsistent prompt selections, potentially affecting segmentation accuracy. This issue has not yet been extensively explored for medical imaging. In this paper, we assess the test-time variability for interactive medical image segmentation with diverse point prompts. For a given target region, the point is classified into three sub-regions: boundary, margin, and center. Our goal is to identify a straightforward and efficient approach for optimal prompt selection during test-time based on three considerations: (1) benefits of additional prompts, (2) effects of prompt placement, and (3) strategies for optimal prompt selection. We conduct extensive experiments on the public Medical Segmentation Decathlon dataset for challenging colon tumor segmentation task. We suggest an optimal strategy for prompt selection during test-time, supported by comprehensive results. The code is publicly available at https://github.com/MedICL-VU/variability

Ho Hin Lee, Yucheng Tang, Han Liu et al.

Recent studies have demonstrated the superior performance of introducing ``scan-wise" contrast labels into contrastive learning for multi-organ segmentation on multi-phase computed tomography (CT). However, such scan-wise labels are limited: (1) a coarse classification, which could not capture the fine-grained ``organ-wise" contrast variations across all organs; (2) the label (i.e., contrast phase) is typically manually provided, which is error-prone and may introduce manual biases of defining phases. In this paper, we propose a novel data-driven contrastive loss function that adapts the similar/dissimilar contrast relationship between samples in each minibatch at organ-level. Specifically, as variable levels of contrast exist between organs, we hypothesis that the contrast differences in the organ-level can bring additional context for defining representations in the latent space. An organ-wise contrast correlation matrix is computed with mean organ intensities under one-hot attention maps. The goal of adapting the organ-driven correlation matrix is to model variable levels of feature separability at different phases. We evaluate our proposed approach on multi-organ segmentation with both non-contrast CT (NCCT) datasets and the MICCAI 2015 BTCV Challenge contrast-enhance CT (CECT) datasets. Compared to the state-of-the-art approaches, our proposed contrastive loss yields a substantial and significant improvement of 1.41% (from 0.923 to 0.936, p-value$<$0.01) and 2.02% (from 0.891 to 0.910, p-value$<$0.01) on mean Dice scores across all organs with respect to NCCT and CECT cohorts. We further assess the trained model performance with the MICCAI 2021 FLARE Challenge CECT datasets and achieve a substantial improvement of mean Dice score from 0.927 to 0.934 (p-value$<$0.01). The code is available at: https://github.com/MASILab/DCC_CL

Chaoqi Wang, Yibo Jiang, Chenghao Yang et al.

The increasing capabilities of large language models (LLMs) raise opportunities for artificial general intelligence but concurrently amplify safety concerns, such as potential misuse of AI systems, necessitating effective AI alignment. Reinforcement Learning from Human Feedback (RLHF) has emerged as a promising pathway towards AI alignment but brings forth challenges due to its complexity and dependence on a separate reward model. Direct Preference Optimization (DPO) has been proposed as an alternative, and it remains equivalent to RLHF under the reverse KL regularization constraint. This paper presents $f$-DPO, a generalized approach to DPO by incorporating diverse divergence constraints. We show that under certain $f$-divergences, including Jensen-Shannon divergence, forward KL divergences and $α$-divergences, the complex relationship between the reward and optimal policy can also be simplified by addressing the Karush-Kuhn-Tucker conditions. This eliminates the need for estimating the normalizing constant in the Bradley-Terry model and enables a tractable mapping between the reward function and the optimal policy. Our approach optimizes LLMs to align with human preferences in a more efficient and supervised manner under a broad set of divergence constraints. Empirically, adopting these divergences ensures a balance between alignment performance and generation diversity. Importantly, $f$-DPO outperforms PPO-based methods in divergence efficiency, and divergence constraints directly influence expected calibration error (ECE).

Lucas W. Remedios, Han Liu, Samuel W. Remedios et al.

Multimodal fusion promises better pancreas segmentation. However, where to perform fusion in models is still an open question. It is unclear if there is a best location to fuse information when analyzing pairs of imperfectly aligned images. Two main alignment challenges in this pancreas segmentation study are 1) the pancreas is deformable and 2) breathing deforms the abdomen. Even after image registration, relevant deformations are often not corrected. We examine how early through late fusion impacts pancreas segmentation. We used 353 pairs of T2-weighted (T2w) and T1-weighted (T1w) abdominal MR images from 163 subjects with accompanying pancreas labels. We used image registration (deeds) to align the image pairs. We trained a collection of basic UNets with different fusion points, spanning from early to late, to assess how early through late fusion influenced segmentation performance on imperfectly aligned images. We assessed generalization of fusion points on nnUNet. The single-modality T2w baseline using a basic UNet model had a Dice score of 0.73, while the same baseline on the nnUNet model achieved 0.80. For the basic UNet, the best fusion approach occurred in the middle of the encoder (early/mid fusion), which led to a statistically significant improvement of 0.0125 on Dice score compared to the baseline. For the nnUNet, the best fusion approach was naïve image concatenation before the model (early fusion), which resulted in a statistically significant Dice score increase of 0.0021 compared to baseline. Fusion in specific blocks can improve performance, but the best blocks for fusion are model specific, and the gains are small. In imperfectly registered datasets, fusion is a nuanced problem, with the art of design remaining vital for uncovering potential insights. Future innovation is needed to better address fusion in cases of imperfect alignment of abdominal image pairs.

Hao Li, Han Liu, Dewei Hu et al.

In this paper, we present PRISM, a Promptable and Robust Interactive Segmentation Model, aiming for precise segmentation of 3D medical images. PRISM accepts various visual inputs, including points, boxes, and scribbles as sparse prompts, as well as masks as dense prompts. Specifically, PRISM is designed with four principles to achieve robustness: (1) Iterative learning. The model produces segmentations by using visual prompts from previous iterations to achieve progressive improvement. (2) Confidence learning. PRISM employs multiple segmentation heads per input image, each generating a continuous map and a confidence score to optimize predictions. (3) Corrective learning. Following each segmentation iteration, PRISM employs a shallow corrective refinement network to reassign mislabeled voxels. (4) Hybrid design. PRISM integrates hybrid encoders to better capture both the local and global information. Comprehensive validation of PRISM is conducted using four public datasets for tumor segmentation in the colon, pancreas, liver, and kidney, highlighting challenges caused by anatomical variations and ambiguous boundaries in accurate tumor identification. Compared to state-of-the-art methods, both with and without prompt engineering, PRISM significantly improves performance, achieving results that are close to human levels. The code is publicly available at https://github.com/MedICL-VU/PRISM.

Tim Tsz-Kit Lau, Han Liu

In many applications in statistics and machine learning, the availability of data samples from multiple possibly heterogeneous sources has become increasingly prevalent. On the other hand, in distributionally robust optimization, we seek data-driven decisions which perform well under the most adverse distribution from a nominal distribution constructed from data samples within a certain discrepancy of probability distributions. However, it remains unclear how to achieve such distributional robustness in model learning and estimation when data samples from multiple sources are available. In this work, we propose constructing the nominal distribution in optimal transport-based distributionally robust optimization problems through the notion of Wasserstein barycenter as an aggregation of data samples from multiple sources. Under specific choices of the loss function, the proposed formulation admits a tractable reformulation as a finite convex program, with powerful finite-sample and asymptotic guarantees. As an illustrative example, we demonstrate with the problem of distributionally robust sparse inverse covariance matrix estimation for zero-mean Gaussian random vectors that our proposed scheme outperforms other widely used estimators in both the low- and high-dimensional regimes.

Kexin Zhang, Baoyu Jing, K. Selçuk Candan et al.

Cross-domain time series imputation is an underexplored data-centric research task that presents significant challenges, particularly when the target domain suffers from high missing rates and domain shifts in temporal dynamics. Existing time series imputation approaches primarily focus on the single-domain setting, which cannot effectively adapt to a new domain with domain shifts. Meanwhile, conventional domain adaptation techniques struggle with data incompleteness, as they typically assume the data from both source and target domains are fully observed to enable adaptation. For the problem of cross-domain time series imputation, missing values introduce high uncertainty that hinders distribution alignment, making existing adaptation strategies ineffective. Specifically, our proposed solution tackles this problem from three perspectives: (i) Data: We introduce a frequency-based time series interpolation strategy that integrates shared spectral components from both domains while retaining domain-specific temporal structures, constructing informative priors for imputation. (ii) Model: We design a diffusion-based imputation model that effectively learns domain-shared representations and captures domain-specific temporal dependencies with dedicated denoising networks. (iii) Algorithm: We further propose a cross-domain consistency alignment strategy that selectively regularizes output-level domain discrepancies, enabling effective knowledge transfer while preserving domain-specific characteristics. Extensive experiments on three real-world datasets demonstrate the superiority of our proposed approach. Our code implementation is available here.

Riqiang Gao, Mamadou Diallo, Han Liu et al.

Radiotherapy (RT) planning is complex, subjective, and time-intensive. Advances with artificial intelligence (AI) promise to improve its precision and efficiency, but progress is often limited by the scarcity of large, standardized datasets. To address this, we introduce the Automated Iterative RT Planning (AIRTP) system, a scalable solution for generating high-quality treatment plans. This scalable solution is designed to generate substantial volumes of consistently high-quality treatment plans, overcoming a key obstacle in the advancement of AI-driven RT planning. Our AIRTP pipeline adheres to clinical guidelines and automates essential steps, including organ-at-risk (OAR) contouring, helper structure creation, beam setup, optimization, and plan quality improvement, using AI integrated with RT planning software like Varian Eclipse. Furthermore, a novel approach for determining optimization parameters to reproduce 3D dose distributions, i.e. a method to convert dose predictions to deliverable treatment plans constrained by machine limitations is proposed. A comparative analysis of plan quality reveals that our automated pipeline produces treatment plans of quality comparable to those generated manually, which traditionally require several hours of labor per plan. Committed to public research, the first data release of our AIRTP pipeline includes nine cohorts covering head-and-neck and lung cancer sites to support an AAPM 2025 challenge. To our best knowledge, this dataset features more than 10 times number of plans compared to the largest existing well-curated public dataset. Repo: https://github.com/RiqiangGao/GDP-HMM_AAPMChallenge.

Han Liu, Zhizhong Han, Yu-Shen Liu et al.

Low-rank metric learning aims to learn better discrimination of data subject to low-rank constraints. It keeps the intrinsic low-rank structure of datasets and reduces the time cost and memory usage in metric learning. However, it is still a challenge for current methods to handle datasets with both high dimensions and large numbers of samples. To address this issue, we present a novel fast low-rank metric learning (FLRML) method.FLRML casts the low-rank metric learning problem into an unconstrained optimization on the Stiefel manifold, which can be efficiently solved by searching along the descent curves of the manifold.FLRML significantly reduces the complexity and memory usage in optimization, which makes the method scalable to both high dimensions and large numbers of samples.Furthermore, we introduce a mini-batch version of FLRML to make the method scalable to larger datasets which are hard to be loaded and decomposed in limited memory. The outperforming experimental results show that our method is with high accuracy and much faster than the state-of-the-art methods under several benchmarks with large numbers of high-dimensional data. Code has been made available at https://github.com/highan911/FLRML

Quanquan Gu, Zhaoran Wang, Han Liu

In this paper, we study the estimation of the $k$-dimensional sparse principal subspace of covariance matrix $Σ$ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-$k$, and attains a $\sqrt{s/n}$ statistical rate of convergence with $s$ being the subspace sparsity level and $n$ the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets.

Zhenyu Pan, Haozheng Luo, Manling Li et al.

We present a Chain-of-Action (CoA) framework for multimodal and retrieval-augmented Question-Answering (QA). Compared to the literature, CoA overcomes two major challenges of current QA applications: (i) unfaithful hallucination that is inconsistent with real-time or domain facts and (ii) weak reasoning performance over compositional information. Our key contribution is a novel reasoning-retrieval mechanism that decomposes a complex question into a reasoning chain via systematic prompting and pre-designed actions. Methodologically, we propose three types of domain-adaptable `Plug-and-Play' actions for retrieving real-time information from heterogeneous sources. We also propose a multi-reference faith score (MRFS) to verify and resolve conflicts in the answers. Empirically, we exploit both public benchmarks and a Web3 case study to demonstrate the capability of CoA over other methods.

Linyao Yang, Hongyang Chen, Xiao Wang et al.

Entity alignment, which is a prerequisite for creating a more comprehensive Knowledge Graph (KG), involves pinpointing equivalent entities across disparate KGs. Contemporary methods for entity alignment have predominantly utilized knowledge embedding models to procure entity embeddings that encapsulate various similarities-structural, relational, and attributive. These embeddings are then integrated through attention-based information fusion mechanisms. Despite this progress, effectively harnessing multifaceted information remains challenging due to inherent heterogeneity. Moreover, while Large Language Models (LLMs) have exhibited exceptional performance across diverse downstream tasks by implicitly capturing entity semantics, this implicit knowledge has yet to be exploited for entity alignment. In this study, we propose a Large Language Model-enhanced Entity Alignment framework (LLMEA), integrating structural knowledge from KGs with semantic knowledge from LLMs to enhance entity alignment. Specifically, LLMEA identifies candidate alignments for a given entity by considering both embedding similarities between entities across KGs and edit distances to a virtual equivalent entity. It then engages an LLM iteratively, posing multiple multi-choice questions to draw upon the LLM's inference capability. The final prediction of the equivalent entity is derived from the LLM's output. Experiments conducted on three public datasets reveal that LLMEA surpasses leading baseline models. Additional ablation studies underscore the efficacy of our proposed framework.

Zihao Li, Yuan Cao, Cheng Gao et al.

Transformers have achieved great success in recent years. Interestingly, transformers have shown particularly strong in-context learning capability -- even without fine-tuning, they are still able to solve unseen tasks well purely based on task-specific prompts. In this paper, we study the capability of one-layer transformers in learning one of the most classical nonparametric estimators, the one-nearest neighbor prediction rule. Under a theoretical framework where the prompt contains a sequence of labeled training data and unlabeled test data, we show that, although the loss function is nonconvex when trained with gradient descent, a single softmax attention layer can successfully learn to behave like a one-nearest neighbor classifier. Our result gives a concrete example of how transformers can be trained to implement nonparametric machine learning algorithms, and sheds light on the role of softmax attention in transformer models.

Zhi Zhang, Chris Chow, Yasi Zhang et al.

Lifelong reinforcement learning (RL) has been developed as a paradigm for extending single-task RL to more realistic, dynamic settings. In lifelong RL, the "life" of an RL agent is modeled as a stream of tasks drawn from a task distribution. We propose EPIC (Empirical PAC-Bayes that Improves Continuously), a novel algorithm designed for lifelong RL using PAC-Bayes theory. EPIC learns a shared policy distribution, referred to as the world policy, which enables rapid adaptation to new tasks while retaining valuable knowledge from previous experiences. Our theoretical analysis establishes a relationship between the algorithm's generalization performance and the number of prior tasks preserved in memory. We also derive the sample complexity of EPIC in terms of RL regret. Extensive experiments on a variety of environments demonstrate that EPIC significantly outperforms existing methods in lifelong RL, offering both theoretical guarantees and practical efficacy through the use of the world policy.

Cheng Gao, Yuan Cao, Zihao Li et al.

Despite the widespread success of Transformers across various domains, their optimization guarantees in large-scale model settings are not well-understood. This paper rigorously analyzes the convergence properties of gradient flow in training Transformers with weight decay regularization. First, we construct the mean-field limit of large-scale Transformers, showing that as the model width and depth go to infinity, gradient flow converges to the Wasserstein gradient flow, which is represented by a partial differential equation. Then, we demonstrate that the gradient flow reaches a global minimum consistent with the PDE solution when the weight decay regularization parameter is sufficiently small. Our analysis is based on a series of novel mean-field techniques that adapt to Transformers. Compared with existing tools for deep networks (Lu et al., 2020) that demand homogeneity and global Lipschitz smoothness, we utilize a refined analysis assuming only $\textit{partial homogeneity}$ and $\textit{local Lipschitz smoothness}$. These new techniques may be of independent interest.

Rihui Xin, Han Liu, Zecheng Wang et al.

Large Language Models have achieved remarkable success in natural language processing tasks, with Reinforcement Learning playing a key role in adapting them to specific applications. However, obtaining ground truth answers for training LLMs in mathematical problem-solving is often challenging, costly, and sometimes unfeasible. This research delves into the utilization of format and length as surrogate signals to train LLMs for mathematical problem-solving, bypassing the need for traditional ground truth answers. Our study shows that a reward function centered on format correctness alone can yield performance improvements comparable to the standard GRPO algorithm in early phases. Recognizing the limitations of format-only rewards in the later phases, we incorporate length-based rewards. The resulting GRPO approach, leveraging format-length surrogate signals, not only matches but surpasses the performance of the standard GRPO algorithm relying on ground truth answers in certain scenarios, achieving 40.0% accuracy on AIME2024 with a 7B base model. Through systematic exploration and experimentation, this research not only offers a practical solution for training LLMs to solve mathematical problems and reducing the dependence on extensive ground truth data collection, but also reveals the essence of why our label-free approach succeeds: the powerful base model is like an excellent student who has already mastered mathematical and logical reasoning skills, but performs poorly on the test paper, it simply needs to develop good answering habits to achieve outstanding results in exams, to unlock the capabilities it already possesses.

Tim Tsz-Kit Lau, Weijian Li, Chenwei Xu et al.

Modern deep neural networks often require distributed training with many workers due to their large size. As the number of workers increases, communication overheads become the main bottleneck in data-parallel minibatch stochastic gradient methods with per-iteration gradient synchronization. Local gradient methods like Local SGD reduce communication by only synchronizing model parameters and/or gradients after several local steps. Despite an understanding of their convergence and the importance of batch sizes for training efficiency and generalization, optimal batch sizes for local gradient methods are difficult to determine. We introduce adaptive batch size strategies for local gradient methods that increase batch sizes adaptively to reduce minibatch gradient variance. We provide convergence guarantees under homogeneous data conditions and support our claims with image classification and language modeling experiments, demonstrating the effectiveness of our strategies for both training efficiency and generalization.

Xin Yu, Qi Yang, Han Liu et al.

2D single-slice abdominal computed tomography (CT) enables the assessment of body habitus and organ health with low radiation exposure. However, single-slice data necessitates the use of 2D networks for segmentation, but these networks often struggle to capture contextual information effectively. Consequently, even when trained on identical datasets, 3D networks typically achieve superior segmentation results. In this work, we propose a novel 3D-to-2D distillation framework, leveraging pre-trained 3D models to enhance 2D single-slice segmentation. Specifically, we extract the prediction distribution centroid from the 3D representations, to guide the 2D student by learning intra- and inter-class correlation. Unlike traditional knowledge distillation methods that require the same data input, our approach employs unpaired 3D CT scans with any contrast to guide the 2D student model. Experiments conducted on 707 subjects from the single-slice Baltimore Longitudinal Study of Aging (BLSA) dataset demonstrate that state-of-the-art 2D multi-organ segmentation methods can benefit from the 3D teacher model, achieving enhanced performance in single-slice multi-organ segmentation. Notably, our approach demonstrates considerable efficacy in low-data regimes, outperforming the model trained with all available training subjects even when utilizing only 200 training subjects. Thus, this work underscores the potential to alleviate manual annotation burdens.

Xingguo Li, Tuo Zhao, Xiaoming Yuan et al.

This paper describes an R package named flare, which implements a family of new high dimensional regression methods (LAD Lasso, SQRT Lasso, $\ell_q$ Lasso, and Dantzig selector) and their extensions to sparse precision matrix estimation (TIGER and CLIME). These methods exploit different nonsmooth loss functions to gain modeling flexibility, estimation robustness, and tuning insensitiveness. The developed solver is based on the alternating direction method of multipliers (ADMM). The package flare is coded in double precision C, and called from R by a user-friendly interface. The memory usage is optimized by using the sparse matrix output. The experiments show that flare is efficient and can scale up to large problems.

Harsh Shrivastava, Xinshi Chen, Binghong Chen et al.

Recovering sparse conditional independence graphs from data is a fundamental problem in machine learning with wide applications. A popular formulation of the problem is an $\ell_1$ regularized maximum likelihood estimation. Many convex optimization algorithms have been designed to solve this formulation to recover the graph structure. Recently, there is a surge of interest to learn algorithms directly based on data, and in this case, learn to map empirical covariance to the sparse precision matrix. However, it is a challenging task in this case, since the symmetric positive definiteness (SPD) and sparsity of the matrix are not easy to enforce in learned algorithms, and a direct mapping from data to precision matrix may contain many parameters. We propose a deep learning architecture, GLAD, which uses an Alternating Minimization (AM) algorithm as our model inductive bias, and learns the model parameters via supervised learning. We show that GLAD learns a very compact and effective model for recovering sparse graphs from data.

Binghong Chen, Bo Dai, Qinjie Lin et al.

We propose a meta path planning algorithm named \emph{Neural Exploration-Exploitation Trees~(NEXT)} for learning from prior experience for solving new path planning problems in high dimensional continuous state and action spaces. Compared to more classical sampling-based methods like RRT, our approach achieves much better sample efficiency in high-dimensions and can benefit from prior experience of planning in similar environments. More specifically, NEXT exploits a novel neural architecture which can learn promising search directions from problem structures. The learned prior is then integrated into a UCB-type algorithm to achieve an online balance between \emph{exploration} and \emph{exploitation} when solving a new problem. We conduct thorough experiments to show that NEXT accomplishes new planning problems with more compact search trees and significantly outperforms state-of-the-art methods on several benchmarks.

Han Liu, Dan Zeng, Qi Tian

Cloud detection plays a very important role in the process of remote sensing images. This paper designs a super-pixel level cloud detection method based on convolutional neural network (CNN) and deep forest. Firstly, remote sensing images are segmented into super-pixels through the combination of SLIC and SEEDS. Structured forests is carried out to compute edge probability of each pixel, based on which super-pixels are segmented more precisely. Segmented super-pixels compose a super-pixel level remote sensing database. Though cloud detection is essentially a binary classification problem, our database is labeled into four categories: thick cloud, cirrus cloud, building and other culture, to improve the generalization ability of our proposed models. Secondly, super-pixel level database is used to train our cloud detection models based on CNN and deep forest. Considering super-pixel level remote sensing images contain less semantic information compared with general object classification database, we propose a Hierarchical Fusion CNN (HFCNN). It takes full advantage of low-level features like color and texture information and is more applicable to cloud detection task. In test phase, every super-pixel in remote sensing images is classified by our proposed models and then combined to recover final binary mask by our proposed distance metric, which is used to determine ambiguous super-pixels. Experimental results show that, compared with conventional methods, HFCNN can achieve better precision and recall.

Kean Ming Tan, Zhaoran Wang, Tong Zhang et al.

Sliced inverse regression is a popular tool for sufficient dimension reduction, which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the covariates. The estimated linear combinations include all covariates, making results difficult to interpret and perhaps unnecessarily variable, particularly when the number of covariates is large. In this paper, we propose a convex formulation for fitting sparse sliced inverse regression in high dimensions. Our proposal estimates the subspace of the linear combinations of the covariates directly and performs variable selection simultaneously. We solve the resulting convex optimization problem via the linearized alternating direction methods of multiplier algorithm, and establish an upper bound on the subspace distance between the estimated and the true subspaces. Through numerical studies, we show that our proposal is able to identify the correct covariates in the high-dimensional setting.

Chris Junchi Li, Zhaoran Wang, Han Liu

Solving statistical learning problems often involves nonconvex optimization. Despite the empirical success of nonconvex statistical optimization methods, their global dynamics, especially convergence to the desirable local minima, remain less well understood in theory. In this paper, we propose a new analytic paradigm based on diffusion processes to characterize the global dynamics of nonconvex statistical optimization. As a concrete example, we study stochastic gradient descent (SGD) for the tensor decomposition formulation of independent component analysis. In particular, we cast different phases of SGD into diffusion processes, i.e., solutions to stochastic differential equations. Initialized from an unstable equilibrium, the global dynamics of SGD transit over three consecutive phases: (i) an unstable Ornstein-Uhlenbeck process slowly departing from the initialization, (ii) the solution to an ordinary differential equation, which quickly evolves towards the desirable local minimum, and (iii) a stable Ornstein-Uhlenbeck process oscillating around the desirable local minimum. Our proof techniques are based upon Stroock and Varadhan's weak convergence of Markov chains to diffusion processes, which are of independent interest.

Carson Eisenach, Han Liu

Motivated by the task of clustering either $d$ variables or $d$ points into $K$ groups, we investigate efficient algorithms to solve the Peng-Wei (P-W) $K$-means semi-definite programming (SDP) relaxation. The P-W SDP has been shown in the literature to have good statistical properties in a variety of settings, but remains intractable to solve in practice. To this end we propose FORCE, a new algorithm to solve this SDP relaxation. Compared to the naive interior point method, our method reduces the computational complexity of solving the SDP from $\tilde{O}(d^7\logε^{-1})$ to $\tilde{O}(d^{6}K^{-2}ε^{-1})$ arithmetic operations for an $ε$-optimal solution. Our method combines a primal first-order method with a dual optimality certificate search, which when successful, allows for early termination of the primal method. We show for certain variable clustering problems that, with high probability, FORCE is guaranteed to find the optimal solution to the SDP relaxation and provide a certificate of exact optimality. As verified by our numerical experiments, this allows FORCE to solve the P-W SDP with dimensions in the hundreds in only tens of seconds. For a variation of the P-W SDP where $K$ is not known a priori a slight modification of FORCE reduces the computational complexity of solving this problem as well: from $\tilde{O}(d^7\logε^{-1})$ using a standard SDP solver to $\tilde{O}(d^{4}ε^{-1})$.

Daniel R. Jiang, Emmanuel Ekwedike, Han Liu

Inspired by recent successes of Monte-Carlo tree search (MCTS) in a number of artificial intelligence (AI) application domains, we propose a model-based reinforcement learning (RL) technique that iteratively applies MCTS on batches of small, finite-horizon versions of the original infinite-horizon Markov decision process. The terminal condition of the finite-horizon problems, or the leaf-node evaluator of the decision tree generated by MCTS, is specified using a combination of an estimated value function and an estimated policy function. The recommendations generated by the MCTS procedure are then provided as feedback in order to refine, through classification and regression, the leaf-node evaluator for the next iteration. We provide the first sample complexity bounds for a tree search-based RL algorithm. In addition, we show that a deep neural network implementation of the technique can create a competitive AI agent for the popular multi-player online battle arena (MOBA) game King of Glory.

Xingguo Li, Junwei Lu, Raman Arora et al.

We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary points and the null space of Hessian matrices \xl{of the objective function} via the lens of invariant groups\removed{for associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks}. As a major motivating example, we apply the proposed general theory to characterize the global \xl{landscape} of the \xl{nonconvex optimization in} low-rank matrix factorization problem. In particular, we illustrate how the rotational symmetry group gives rise to infinitely many nonisolated strict saddle points and equivalent global minima of the objective function. By explicitly identifying all stationary points, we divide the entire parameter space into three regions: ($\cR_1$) the region containing the neighborhoods of all strict saddle points, where the objective has negative curvatures; ($\cR_2$) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and ($\cR_3$) the complement of the above regions, where the gradient has sufficiently large magnitudes. We further extend our result to the matrix sensing problem. Such global landscape implies strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.

Ethan X. Fang, Han Liu, Kim-Chuan Toh et al.

This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-uniform and unknown sampling distributions. This method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear-norm penalized approach, and makes low-rank matrix recovery feasible in more practical settings. Theoretically, we prove that the proposed estimator achieves fast rates of convergence under different settings. Computationally, we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator, which bridges a gap between theory and practice of machine learning methods with max-norm regularization. Further, we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets.

James W. Hennessey, Han Liu, Holger Winnemöller et al.

Accurately drawing 3D objects is difficult for untrained individuals, as it requires an understanding of perspective and its effects on geometry and proportions. Step-by-step tutorials break the complex task of sketching an entire object down into easy-to-follow steps that even a novice can follow. However, creating such tutorials requires expert knowledge and is a time-consuming task. As a result, the availability of tutorials for a given object or viewpoint is limited. How2Sketch addresses this problem by automatically generating easy-to-follow tutorials for arbitrary 3D objects. Given a segmented 3D model and a camera viewpoint,it computes a sequence of steps for constructing a drawing scaffold comprised of geometric primitives, which helps the user draw the final contours in correct perspective and proportion. To make the drawing scaffold easy to construct, the algorithm solves for an ordering among the scaffolding primitives and explicitly makes small geometric modifications to the size and location of the object parts to simplify relative positioning. Technically, we formulate this scaffold construction as a single selection problem that simultaneously solves for the ordering and geometric changes of the primitives. We demonstrate our algorithm for generating tutorials on a variety of man-made objects and evaluate how easily the tutorials can be followed with a user study.

Xingguo Li, Haoming Jiang, Jarvis Haupt et al.

Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these "sacrifices" do not always require more computational efforts. To shed light on such a "free-lunch" phenomenon, we study the square-root-Lasso (SQRT-Lasso) type regression problem. Specifically, we show that the nonsmooth loss functions of SQRT-Lasso type regression ease tuning effort and gain adaptivity to inhomogeneous noise, but is not necessarily more challenging than Lasso in computation. We can directly apply proximal algorithms (e.g. proximal gradient descent, proximal Newton, and proximal Quasi-Newton algorithms) without worrying the nonsmoothness of the loss function. Theoretically, we prove that the proximal algorithms combined with the pathwise optimization scheme enjoy fast convergence guarantees with high probability. Numerical results are provided to support our theory.

Xingguo Li, Raman Arora, Han Liu et al.

We propose a stochastic variance reduced optimization algorithm for solving sparse learning problems with cardinality constraints. Sufficient conditions are provided, under which the proposed algorithm enjoys strong linear convergence guarantees and optimal estimation accuracy in high dimensions. We further extend the proposed algorithm to an asynchronous parallel variant with a near linear speedup. Numerical experiments demonstrate the efficiency of our algorithm in terms of both parameter estimation and computational performance.

Kean Ming Tan, Zhaoran Wang, Han Liu et al.

Sparse generalized eigenvalue problem (GEP) plays a pivotal role in a large family of high-dimensional statistical models, including sparse Fisher's discriminant analysis, canonical correlation analysis, and sufficient dimension reduction. Sparse GEP involves solving a non-convex optimization problem. Most existing methods and theory in the context of specific statistical models that are special cases of the sparse GEP require restrictive structural assumptions on the input matrices. In this paper, we propose a two-stage computational framework to solve the sparse GEP. At the first stage, we solve a convex relaxation of the sparse GEP. Taking the solution as an initial value, we then exploit a nonconvex optimization perspective and propose the truncated Rayleigh flow method (Rifle) to estimate the leading generalized eigenvector. We show that Rifle converges linearly to a solution with the optimal statistical rate of convergence for many statistical models. Theoretically, our method significantly improves upon the existing literature by eliminating structural assumptions on the input matrices for both stages. To achieve this, our analysis involves two key ingredients: (i) a new analysis of the gradient based method on nonconvex objective functions, and (ii) a fine-grained characterization of the evolution of sparsity patterns along the solution path. Thorough numerical studies are provided to validate the theoretical results.

Chris Junchi Li, Mengdi Wang, Han Liu et al.

Principal component analysis (PCA) has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the principal component by processing streaming data are of tremendous practical and theoretical interests. Despite its rich applications, theoretical convergence analysis remains largely open. In this paper, we cast online PCA into a stochastic nonconvex optimization problem, and we analyze the online PCA algorithm as a stochastic approximation iteration. The stochastic approximation iteration processes data points incrementally and maintains a running estimate of the principal component. We prove for the first time a nearly optimal finite-sample error bound for the online PCA algorithm. Under the subgaussian assumption, we show that the finite-sample error bound closely matches the minimax information lower bound.

Yan Li, Han Liu, Warren Powell

We propose a sequential learning policy for noisy discrete global optimization and ranking and selection (R\&S) problems with high dimensional sparse belief functions, where there are hundreds or even thousands of features, but only a small portion of these features contain explanatory power. We aim to identify the sparsity pattern and select the best alternative before the finite budget is exhausted. We derive a knowledge gradient policy for sparse linear models (KGSpLin) with group Lasso penalty. This policy is a unique and novel hybrid of Bayesian R\&S with frequentist learning. Particularly, our method naturally combines B-spline basis expansion and generalizes to the nonparametric additive model (KGSpAM) and functional ANOVA model. Theoretically, we provide the estimation error bounds of the posterior mean estimate and the functional estimate. Controlled experiments show that the algorithm efficiently learns the correct set of nonzero parameters even when the model is imbedded with hundreds of dummy parameters. Also it outperforms the knowledge gradient for a linear model.

Zhaoran Wang, Quanquan Gu, Han Liu

Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we consider two problems: Mean estimation for sparse principal submatrix and edge probability estimation for stochastic block model. We exploit the sum-of-squares relaxation hierarchy to sharply characterize the limits of a broad class of convex relaxations. Our result shows statistical optimality needs to be compromised for achieving computational tractability using convex relaxations. Compared with existing results on computational lower bounds for statistical problems, which consider general polynomial-time algorithms and rely on computational hardness hypotheses on problems like planted clique detection, our theory focuses on a broad class of convex relaxations and does not rely on unproven hypotheses.

Tuo Zhao, Han Liu, Tong Zhang

The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it strong rule for coordinate preselection}. Such a complex algorithmic structure grants superior empirical performance, but also poses significant challenge to theoretical analysis. To tackle this long lasting problem, we develop a new theory showing that these three features play pivotal roles in guaranteeing the outstanding statistical and computational performance of the pathwise coordinate optimization framework. Particularly, we analyze the existing pathwise coordinate optimization algorithms and provide new theoretical insights into them. The obtained insights further motivate the development of several modifications to improve the pathwise coordinate optimization framework, which guarantees linear convergence to a unique sparse local optimum with optimal statistical properties in parameter estimation and support recovery. This is the first result on the computational and statistical guarantees of the pathwise coordinate optimization framework in high dimensions. Thorough numerical experiments are provided to support our theory.

Mengdi Wang, Ethan X. Fang, Han Liu

Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., problems of the form $\min_x \mathbf{E}_v [f_v\big(\mathbf{E}_w [g_w(x)]\big)]$. In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of $f_v,g_{w}$ and use an auxiliary variable to track the unknown quantity $\mathbf{E}_w[g_w(x)]$. We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieve a convergence rate of $O(k^{-1/4})$ in the general case and $O(k^{-2/3})$ in the strongly convex case, after taking $k$ samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of $O(k^{-2/7})$ in the general case and $O(k^{-4/5})$ in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc.

Zhaoran Wang, Huanran Lu, Han Liu

Sparse principal component analysis (PCA) involves nonconvex optimization for which the global solution is hard to obtain. To address this issue, one popular approach is convex relaxation. However, such an approach may produce suboptimal estimators due to the relaxation effect. To optimally estimate sparse principal subspaces, we propose a two-stage computational framework named "tighten after relax": Within the 'relax' stage, we approximately solve a convex relaxation of sparse PCA with early stopping to obtain a desired initial estimator; For the 'tighten' stage, we propose a novel algorithm called sparse orthogonal iteration pursuit (SOAP), which iteratively refines the initial estimator by directly solving the underlying nonconvex problem. A key concept of this two-stage framework is the basin of attraction. It represents a local region within which the `tighten' stage has desired computational and statistical guarantees. We prove that, the initial estimator obtained from the 'relax' stage falls into such a region, and hence SOAP geometrically converges to a principal subspace estimator which is minimax-optimal within a certain model class. Unlike most existing sparse PCA estimators, our approach applies to the non-spiked covariance models, and adapts to non-Gaussianity as well as dependent data settings. Moreover, through analyzing the computational complexity of the two stages, we illustrate an interesting phenomenon that larger sample size can reduce the total iteration complexity. Our framework motivates a general paradigm for solving many complex statistical problems which involve nonconvex optimization with provable guarantees.

Robert Vanderbei, Han Liu, Lie Wang et al.

In this paper we present two new approaches to efficiently solve large-scale compressed sensing problems. These two ideas are independent of each other and can therefore be used either separately or together. We consider all possibilities. For the first approach, we note that the zero vector can be taken as the initial basic (infeasible) solution for the linear programming problem and therefore, if the true signal is very sparse, some variants of the simplex method can be expected to take only a small number of pivots to arrive at a solution. We implemented one such variant and demonstrate a dramatic improvement in computation time on very sparse signals. The second approach requires a redesigned sensing mechanism in which the vector signal is stacked into a matrix. This allows us to exploit the Kronecker compressed sensing (KCS) mechanism. We show that the Kronecker sensing requires stronger conditions for perfect recovery compared to the original vector problem. However, the Kronecker sensing, modeled correctly, is a much sparser linear optimization problem. Hence, algorithms that benefit from sparse problem representation, such as interior-point methods, can solve the Kronecker sensing problems much faster than the corresponding vector problem. In our numerical studies, we demonstrate a ten-fold improvement in the computation time.

Fang Han, Huanran Lu, Han Liu

The vector autoregressive (VAR) model is a powerful tool in modeling complex time series and has been exploited in many fields. However, fitting high dimensional VAR model poses some unique challenges: On one hand, the dimensionality, caused by modeling a large number of time series and higher order autoregressive processes, is usually much higher than the time series length; On the other hand, the temporal dependence structure in the VAR model gives rise to extra theoretical challenges. In high dimensions, one popular approach is to assume the transition matrix is sparse and fit the VAR model using the "least squares" method with a lasso-type penalty. In this manuscript, we propose an alternative way in estimating the VAR model. The main idea is, via exploiting the temporal dependence structure, to formulate the estimating problem into a linear program. There is instant advantage for the proposed approach over the lasso-type estimators: The estimation equation can be decomposed into multiple sub-equations and accordingly can be efficiently solved in a parallel fashion. In addition, our method brings new theoretical insights into the VAR model analysis. So far the theoretical results developed in high dimensions (e.g., Song and Bickel (2011) and Kock and Callot (2012)) mainly pose assumptions on the design matrix of the formulated regression problems. Such conditions are indirect about the transition matrices and not transparent. In contrast, our results show that the operator norm of the transition matrices plays an important role in estimation accuracy. We provide explicit rates of convergence for both estimation and prediction. In addition, we provide thorough experiments on both synthetic and real-world equity data to show that there are empirical advantages of our method over the lasso-type estimators in both parameter estimation and forecasting.

Zhaoran Wang, Fang Han, Han Liu

We study sparse principal component analysis for high dimensional vector autoregressive time series under a doubly asymptotic framework, which allows the dimension $d$ to scale with the series length $T$. We treat the transition matrix of time series as a nuisance parameter and directly apply sparse principal component analysis on multivariate time series as if the data are independent. We provide explicit non-asymptotic rates of convergence for leading eigenvector estimation and extend this result to principal subspace estimation. Our analysis illustrates that the spectral norm of the transition matrix plays an essential role in determining the final rates. We also characterize sufficient conditions under which sparse principal component analysis attains the optimal parametric rate. Our theoretical results are backed up by thorough numerical studies.

Han Liu, Lie Wang, Tuo Zhao

We propose a calibrated multivariate regression method named CMR for fitting high dimensional multivariate regression models. Compared with existing methods, CMR calibrates regularization for each regression task with respect to its noise level so that it simultaneously attains improved finite-sample performance and tuning insensitiveness. Theoretically, we provide sufficient conditions under which CMR achieves the optimal rate of convergence in parameter estimation. Computationally, we propose an efficient smoothed proximal gradient algorithm with a worst-case numerical rate of convergence $\cO(1/ε)$, where $ε$ is a pre-specified accuracy of the objective function value. We conduct thorough numerical simulations to illustrate that CMR consistently outperforms other high dimensional multivariate regression methods. We also apply CMR to solve a brain activity prediction problem and find that it is as competitive as a handcrafted model created by human experts. The R package \texttt{camel} implementing the proposed method is available on the Comprehensive R Archive Network \url{http://cran.r-project.org/web/packages/camel/}.