Lican Kang

Han Yuan, Mingxuan Liu, Lican Kang et al.

Nowadays, the interpretation of why a machine learning (ML) model makes certain inferences is as crucial as the accuracy of such inferences. Some ML models like the decision tree possess inherent interpretability that can be directly comprehended by humans. Others like artificial neural networks (ANN), however, rely on external methods to uncover the deduction mechanism. SHapley Additive exPlanations (SHAP) is one of such external methods, which requires a background dataset when interpreting ANNs. Generally, a background dataset consists of instances randomly sampled from the training dataset. However, the sampling size and its effect on SHAP remain to be unexplored. In our empirical study on the MIMIC-III dataset, we show that the two core explanations - SHAP values and variable rankings fluctuate when using different background datasets acquired from random sampling, indicating that users cannot unquestioningly trust the one-shot interpretation from SHAP. Luckily, such fluctuation decreases with the increase of the background dataset size. Also, we notice an U-shape in the stability assessment of SHAP variable rankings, demonstrating that SHAP is more reliable in ranking the most and least important variables compared to moderately important ones. Overall, our results suggest that users should take into account how background data affects SHAP results, with improved SHAP stability as the background sample size increases.

Jinyuan Chang, Yuling Jiao, Lican Kang et al.

In this work, we propose a novel deep bootstrap framework for nonparametric regression based on conditional diffusion models. Specifically, we construct a conditional diffusion model to learn the distribution of the response variable given the covariates. This model is then used to generate bootstrap samples by pairing the original covariates with newly synthesized responses. We reformulate nonparametric regression as conditional sample mean estimation, which is implemented directly via the learned conditional diffusion model. Unlike traditional bootstrap methods that decouple the estimation of the conditional distribution, sampling, and nonparametric regression, our approach integrates these components into a unified generative framework. With the expressive capacity of diffusion models, our method facilitates both efficient sampling from high-dimensional or multimodal distributions and accurate nonparametric estimation. We establish rigorous theoretical guarantees for the proposed method. In particular, we derive optimal end-to-end convergence rates in the Wasserstein distance between the learned and target conditional distributions. Building on this foundation, we further establish the convergence guarantees of the resulting bootstrap procedure. Numerical studies demonstrate the effectiveness and scalability of our approach for complex regression tasks.

Yuling Jiao, Lican Kang, Huazhen Lin et al.

This paper aims to conduct a comprehensive theoretical analysis of current diffusion models. We introduce a novel generative learning methodology utilizing the Schr{ö}dinger bridge diffusion model in latent space as the framework for theoretical exploration in this domain. Our approach commences with the pre-training of an encoder-decoder architecture using data originating from a distribution that may diverge from the target distribution, thus facilitating the accommodation of a large sample size through the utilization of pre-existing large-scale models. Subsequently, we develop a diffusion model within the latent space utilizing the Schr{ö}dinger bridge framework. Our theoretical analysis encompasses the establishment of end-to-end error analysis for learning distributions via the latent Schr{ö}dinger bridge diffusion model. Specifically, we control the second-order Wasserstein distance between the generated distribution and the target distribution. Furthermore, our obtained convergence rates effectively mitigate the curse of dimensionality, offering robust theoretical support for prevailing diffusion models.

Yuling Jiao, Lican Kang, Jin Liu et al.

Deep nonparametric regression, characterized by the utilization of deep neural networks to learn target functions, has emerged as a focus of research attention in recent years. Despite considerable progress in understanding convergence rates, the absence of asymptotic properties hinders rigorous statistical inference. To address this gap, we propose a novel framework that transforms the deep estimation paradigm into a platform conducive to conditional mean estimation, leveraging the conditional diffusion model. Theoretically, we develop an end-to-end convergence rate for the conditional diffusion model and establish the asymptotic normality of the generated samples. Consequently, we are equipped to construct confidence regions, facilitating robust statistical inference. Furthermore, through numerical experiments, we empirically validate the efficacy of our proposed methodology.

Han Yuan, Lican Kang, Yong Li

While deep learning has exhibited remarkable predictive capabilities in various medical image tasks, its inherent black-box nature has hindered its widespread implementation in real-world healthcare settings. Our objective is to unveil the decision-making processes of deep learning models in the context of glaucoma classification by employing several Class Activation Map (CAM) techniques to generate model focus regions and comparing them with clinical domain knowledge of the anatomical area (optic cup, optic disk, and blood vessels). Four deep neural networks, including VGG-11, ResNet-18, DeiT-Tiny, and Swin Transformer-Tiny, were developed using binary diagnostic labels of glaucoma and five CAM methods (Grad-CAM, XGrad-CAM, Score-CAM, Eigen-CAM, and Layer-CAM) were employed to highlight the model focus area. We applied the paired-sample t-test to compare the percentage of anatomies in the model focus area to the proportion of anatomies in the entire image. After that, Pearson's and Spearman's correlation tests were implemented to examine the relationship between model predictive ability and the percentage of anatomical structures in the model focus area. On five public glaucoma datasets, all deep learning models consistently displayed statistically significantly higher percentages of anatomical structures in the focus area than the proportions of anatomical structures in the entire image. Also, we validated the positive relationship between the percentage of anatomical structures in the focus area and model predictive performance. Our study provides evidence of the convergence of decision logic between deep neural networks and human clinicians through rigorous statistical tests. We anticipate that it can help alleviate clinicians' concerns regarding the trustworthiness of deep learning in healthcare. For reproducibility, the code and dataset have been released at GitHub.

Peili Li, Yuling Jiao, Xiliang Lu et al.

In this work, we consider the algorithm to the (nonlinear) regression problems with $\ell_0$ penalty. The existing algorithms for $\ell_0$ based optimization problem are often carried out with a fixed step size, and the selection of an appropriate step size depends on the restricted strong convexity and smoothness for the loss function, hence it is difficult to compute in practical calculation. In sprite of the ideas of support detection and root finding \cite{HJK2020}, we proposes a novel and efficient data-driven line search rule to adaptively determine the appropriate step size. We prove the $\ell_2$ error bound to the proposed algorithm without much restrictions for the cost functional. A large number of numerical comparisons with state-of-the-art algorithms in linear and logistic regression problems show the stability, effectiveness and superiority of the proposed algorithms.

Yuling Jiao, Lican Kang, Yanyan Liu et al.

Schrödinger-Föllmer sampler (SFS) is a novel and efficient approach for sampling from possibly unnormalized distributions without ergodicity. SFS is based on the Euler-Maruyama discretization of Schrödinger-Föllmer diffusion process $$\mathrm{d} X_{t}=-\nabla U\left(X_t, t\right) \mathrm{d} t+\mathrm{d} B_{t}, \quad t \in[0,1],\quad X_0=0$$ on the unit interval, which transports the degenerate distribution at time zero to the target distribution at time one. In \cite{sfs21}, the consistency of SFS is established under a restricted assumption that %the drift term $b(x,t)$ the potential $U(x,t)$ is uniformly (on $t$) strongly %concave convex (on $x$). In this paper we provide a nonasymptotic error bound of SFS in Wasserstein distance under some smooth and bounded conditions on the density ratio of the target distribution over the standard normal distribution, but without requiring the strongly convexity of the potential.

Jian Huang, Yuling Jiao, Lican Kang et al.

Screening and working set techniques are important approaches to reducing the size of an optimization problem. They have been widely used in accelerating first-order methods for solving large-scale sparse learning problems. In this paper, we develop a new screening method called Newton screening (NS) which is a generalized Newton method with a built-in screening mechanism. We derive an equivalent KKT system for the Lasso and utilize a generalized Newton method to solve the KKT equations. Based on this KKT system, a built-in working set with a relatively small size is first determined using the sum of primal and dual variables generated from the previous iteration, then the primal variable is updated by solving a least-squares problem on the working set and the dual variable updated based on a closed-form expression. Moreover, we consider a sequential version of Newton screening (SNS) with a warm-start strategy. We show that NS possesses an optimal convergence property in the sense that it achieves one-step local convergence. Under certain regularity conditions on the feature matrix, we show that SNS hits a solution with the same signs as the underlying true target and achieves a sharp estimation error bound with high probability. Simulation studies and real data analysis support our theoretical results and demonstrate that SNS is faster and more accurate than several state-of-the-art methods in our comparative studies.

Jian Huang, Yuling Jiao, Lican Kang et al.

Feature selection is important for modeling high-dimensional data, where the number of variables can be much larger than the sample size. In this paper, we develop a support detection and root finding procedure to learn the high dimensional sparse generalized linear models and denote this method by GSDAR. Based on the KKT condition for $\ell_0$-penalized maximum likelihood estimations, GSDAR generates a sequence of estimators iteratively. Under some restricted invertibility conditions on the maximum likelihood function and sparsity assumption on the target coefficients, the errors of the proposed estimate decays exponentially to the optimal order. Moreover, the oracle estimator can be recovered if the target signal is stronger than the detectable level. We conduct simulations and real data analysis to illustrate the advantages of our proposed method over several existing methods, including Lasso and MCP.