Jian Huang

Siming Zheng, Yuanyuan Lin, Jian Huang

We propose a mutual information-based sufficient representation learning (MSRL) approach, which uses the variational formulation of the mutual information and leverages the approximation power of deep neural networks. MSRL learns a sufficient representation with the maximum mutual information with the response and a user-selected distribution. It can easily handle multi-dimensional continuous or categorical response variables. MSRL is shown to be consistent in the sense that the conditional probability density function of the response variable given the learned representation converges to the conditional probability density function of the response variable given the predictor. Non-asymptotic error bounds for MSRL are also established under suitable conditions. To establish the error bounds, we derive a generalized Dudley's inequality for an order-two U-process indexed by deep neural networks, which may be of independent interest. We discuss how to determine the intrinsic dimension of the underlying data distribution. Moreover, we evaluate the performance of MSRL via extensive numerical experiments and real data analysis and demonstrate that MSRL outperforms some existing nonlinear sufficient dimension reduction methods.

Yuan Gao, Jian Huang, Yuling Jiao et al.

Continuous normalizing flows (CNFs) are a generative method for learning probability distributions, which is based on ordinary differential equations. This method has shown remarkable empirical success across various applications, including large-scale image synthesis, protein structure prediction, and molecule generation. In this work, we study the theoretical properties of CNFs with linear interpolation in learning probability distributions from a finite random sample, using a flow matching objective function. We establish non-asymptotic error bounds for the distribution estimator based on CNFs, in terms of the Wasserstein-2 distance. The key assumption in our analysis is that the target distribution satisfies one of the following three conditions: it either has a bounded support, is strongly log-concave, or is a finite or infinite mixture of Gaussian distributions. We present a convergence analysis framework that encompasses the error due to velocity estimation, the discretization error, and the early stopping error. A key step in our analysis involves establishing the regularity properties of the velocity field and its estimator for CNFs constructed with linear interpolation. This necessitates the development of uniform error bounds with Lipschitz regularity control of deep ReLU networks that approximate the Lipschitz function class, which could be of independent interest. Our nonparametric convergence analysis offers theoretical guarantees for using CNFs to learn probability distributions from a finite random sample.

Ding Huang, Jian Huang, Ting Li et al.

We propose a conditional stochastic interpolation (CSI) method for learning conditional distributions. CSI is based on estimating probability flow equations or stochastic differential equations that transport a reference distribution to the target conditional distribution. This is achieved by first learning the conditional drift and score functions based on CSI, which are then used to construct a deterministic process governed by an ordinary differential equation or a diffusion process for conditional sampling. In our proposed approach, we incorporate an adaptive diffusion term to address the instability issues arising in the diffusion process. We derive explicit expressions of the conditional drift and score functions in terms of conditional expectations, which naturally lead to an nonparametric regression approach to estimating these functions. Furthermore, we establish nonasymptotic error bounds for learning the target conditional distribution. We illustrate the application of CSI on image generation using a benchmark image dataset.

Ding Huang, Ting Li, Jian Huang

We propose a Bayesian framework for fine-tuning large diffusion models with a novel network structure called Bayesian Power Steering (BPS). We clarify the meaning behind adaptation from a \textit{large probability space} to a \textit{small probability space} and explore the task of fine-tuning pre-trained models using learnable modules from a Bayesian perspective. BPS extracts task-specific knowledge from a pre-trained model's learned prior distribution. It efficiently leverages large diffusion models, differentially intervening different hidden features with a head-heavy and foot-light configuration. Experiments highlight the superiority of BPS over contemporary methods across a range of tasks even with limited amount of data. Notably, BPS attains an FID score of 10.49 under the sketch condition on the COCO17 dataset.

Shiao Liu, Xingyu Zhou, Yuling Jiao et al.

Conditional distribution is a fundamental quantity for describing the relationship between a response and a predictor. We propose a Wasserstein generative approach to learning a conditional distribution. The proposed approach uses a conditional generator to transform a known distribution to the target conditional distribution. The conditional generator is estimated by matching a joint distribution involving the conditional generator and the target joint distribution, using the Wasserstein distance as the discrepancy measure for these joint distributions. We establish non-asymptotic error bound of the conditional sampling distribution generated by the proposed method and show that it is able to mitigate the curse of dimensionality, assuming that the data distribution is supported on a lower-dimensional set. We conduct numerical experiments to validate proposed method and illustrate its applications to conditional sample generation, nonparametric conditional density estimation, prediction uncertainty quantification, bivariate response data, image reconstruction and image generation.

Changming Zhao, Dongrui Wu, Jian Huang et al.

Bootstrap aggregating (Bagging) and boosting are two popular ensemble learning approaches, which combine multiple base learners to generate a composite model for more accurate and more reliable performance. They have been widely used in biology, engineering, healthcare, etc. This paper proposes BoostForest, which is an ensemble learning approach using BoostTree as base learners and can be used for both classification and regression. BoostTree constructs a tree model by gradient boosting. It increases the randomness (diversity) by drawing the cut-points randomly at node splitting. BoostForest further increases the randomness by bootstrapping the training data in constructing different BoostTrees. BoostForest generally outperformed four classical ensemble learning approaches (Random Forest, Extra-Trees, XGBoost and LightGBM) on 35 classification and regression datasets. Remarkably, BoostForest tunes its parameters by simply sampling them randomly from a parameter pool, which can be easily specified, and its ensemble learning framework can also be used to combine many other base learners.