Jun Lu

Jun Lu

Clustering has become a core technology in machine learning, largely due to its application in the field of unsupervised learning, clustering, classification, and density estimation. A frequentist approach exists to hand clustering based on mixture model which is known as the EM algorithm where the parameters of the mixture model are usually estimated into a maximum likelihood estimation framework. Bayesian approach for finite and infinite Gaussian mixture model generates point estimates for all variables as well as associated uncertainty in the form of the whole estimates' posterior distribution. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in Bayesian inference for finite and infinite Gaussian mixture model in order to seamlessly introduce their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning this field and given the paucity of scope to present this discussion, e.g., the separated analysis of the generation of Dirichlet samples by stick-breaking and Polya's Urn approaches. We refer the reader to literature in the field of the Dirichlet process mixture model for a much detailed introduction to the related fields. Some excellent examples include (Frigyik et al., 2010; Murphy, 2012; Gelman et al., 2014; Hoff, 2009). This survey is primarily a summary of purpose, significance of important background and techniques for Gaussian mixture model, e.g., Dirichlet prior, Chinese restaurant process, and most importantly the origin and complexity of the methods which shed light on their modern applications. The mathematical prerequisite is a first course in probability. Other than this modest background, the development is self-contained, with rigorous proofs provided throughout.

Xiaofeng Liu, Bo Hu, Linghao Jin et al.

In this work, we propose a domain generalization (DG) approach to learn on several labeled source domains and transfer knowledge to a target domain that is inaccessible in training. Considering the inherent conditional and label shifts, we would expect the alignment of $p(x|y)$ and $p(y)$. However, the widely used domain invariant feature learning (IFL) methods relies on aligning the marginal concept shift w.r.t. $p(x)$, which rests on an unrealistic assumption that $p(y)$ is invariant across domains. We thereby propose a novel variational Bayesian inference framework to enforce the conditional distribution alignment w.r.t. $p(x|y)$ via the prior distribution matching in a latent space, which also takes the marginal label shift w.r.t. $p(y)$ into consideration with the posterior alignment. Extensive experiments on various benchmarks demonstrate that our framework is robust to the label shift and the cross-domain accuracy is significantly improved, thereby achieving superior performance over the conventional IFL counterparts.

Jun Lu

This book is meant to provide an introduction to linear models and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to ordinary least squares. In machine learning, the output is usually a nonlinear function of the input. Deep learning even aims to find a nonlinear dependence with many layers, which require a large amount of computation. However, most of these algorithms build upon simple linear models. We then describe linear models from different perspectives and find the properties and theories behind the models. The linear model is the main technique in regression problems, and the primary tool for it is the least squares approximation, which minimizes a sum of squared errors. This is a natural choice when we're interested in finding the regression function which minimizes the corresponding expected squared error. This book is primarily a summary of purpose, significance of important theories behind linear models, e.g., distribution theory and the minimum variance estimator. We first describe ordinary least squares from three different points of view, upon which we disturb the model with random noise and Gaussian noise. Through Gaussian noise, the model gives rise to the likelihood so that we introduce a maximum likelihood estimator. It also develops some distribution theories via this Gaussian disturbance. The distribution theory of least squares will help us answer various questions and introduce related applications. We then prove least squares is the best unbiased linear model in the sense of mean squared error, and most importantly, it actually approaches the theoretical limit. We end up with linear models with the Bayesian approach and beyond.

Xiaofeng Liu, Xiongchang Liu, Bo Hu et al.

Recent advances in unsupervised domain adaptation (UDA) show that transferable prototypical learning presents a powerful means for class conditional alignment, which encourages the closeness of cross-domain class centroids. However, the cross-domain inner-class compactness and the underlying fine-grained subtype structure remained largely underexplored. In this work, we propose to adaptively carry out the fine-grained subtype-aware alignment by explicitly enforcing the class-wise separation and subtype-wise compactness with intermediate pseudo labels. Our key insight is that the unlabeled subtypes of a class can be divergent to one another with different conditional and label shifts, while inheriting the local proximity within a subtype. The cases of with or without the prior information on subtype numbers are investigated to discover the underlying subtype structure in an online fashion. The proposed subtype-aware dynamic UDA achieves promising results on medical diagnosis tasks.

Xiaofeng Liu, Linghao Jin, Xu Han et al.

This paper targets to explore the inter-subject variations eliminated facial expression representation in the compressed video domain. Most of the previous methods process the RGB images of a sequence, while the off-the-shelf and valuable expression-related muscle movement already embedded in the compression format. In the up to two orders of magnitude compressed domain, we can explicitly infer the expression from the residual frames and possible to extract identity factors from the I frame with a pre-trained face recognition network. By enforcing the marginal independent of them, the expression feature is expected to be purer for the expression and be robust to identity shifts. We do not need the identity label or multiple expression samples from the same person for identity elimination. Moreover, when the apex frame is annotated in the dataset, the complementary constraint can be further added to regularize the feature-level game. In testing, only the compressed residual frames are required to achieve expression prediction. Our solution can achieve comparable or better performance than the recent decoded image based methods on the typical FER benchmarks with about 3$\times$ faster inference with compressed data.

Xiaofeng Liu, Bo Hu, Xiongchang Liu et al.

Unsupervised domain adaptation (UDA) aims to transfer the knowledge on a labeled source domain distribution to perform well on an unlabeled target domain. Recently, the deep self-training involves an iterative process of predicting on the target domain and then taking the confident predictions as hard pseudo-labels for retraining. However, the pseudo-labels are usually unreliable, and easily leading to deviated solutions with propagated errors. In this paper, we resort to the energy-based model and constrain the training of the unlabeled target sample with the energy function minimization objective. It can be applied as a simple additional regularization. In this framework, it is possible to gain the benefits of the energy-based model, while retaining strong discriminative performance following a plug-and-play fashion. We deliver extensive experiments on the most popular and large scale UDA benchmarks of image classification as well as semantic segmentation to demonstrate its generality and effectiveness.

Jun Lu, Wei Ma, Boi Faltings

It is often the case that the performance of a neural network can be improved by adding layers. In real-world practices, we always train dozens of neural network architectures in parallel which is a wasteful process. We explored $CompNet$, in which case we morph a well-trained neural network to a deeper one where network function can be preserved and the added layer is compact. The work of the paper makes two contributions: a). The modified network can converge fast and keep the same functionality so that we do not need to train from scratch again; b). The layer size of the added layer in the neural network is controlled by removing the redundant parameters with sparse optimization. This differs from previous network morphism approaches which tend to add more neurons or channels beyond the actual requirements and result in redundance of the model. The method is illustrated using several neural network structures on different data sets including MNIST and CIFAR10.

Jun Lu, Meng Li, David Dunson

Dirichlet process mixture (DPM) models tend to produce many small clusters regardless of whether they are needed to accurately characterize the data - this is particularly true for large data sets. However, interpretability, parsimony, data storage and communication costs all are hampered by having overly many clusters. We propose a powered Chinese restaurant process to limit this kind of problem and penalize over clustering. The method is illustrated using some simulation examples and data with large and small sample size including MNIST and the Old Faithful Geyser data.

Wei Ma, Jun Lu

This article demonstrates that convolutional operation can be converted to matrix multiplication, which has the same calculation way with fully connected layer. The article is helpful for the beginners of the neural network to understand how fully connected layer and the convolutional layer work in the backend. To be concise and to make the article more readable, we only consider the linear case. It can be extended to the non-linear case easily through plugging in a non-linear encapsulation to the values like this $σ(x)$ denoted as $x^{\prime}$.

Jun Lu

In this article we introduce how to put vague hyperprior on Dirichlet distribution, and we update the parameter of it by adaptive rejection sampling (ARS). Finally we analyze this hyperprior in an over-fitted mixture model by some synthetic experiments.

Jun Lu

Machine learning has been used in all kinds of fields. In this article, we introduce how machine learning can be applied into time series problem. Especially, we use the airline ticket prediction problem as our specific problem. Airline companies use many different variables to determine the flight ticket prices: indicator whether the travel is during the holidays, the number of free seats in the plane etc. Some of the variables are observed, but some of them are hidden. Based on the data over a 103 day period, we trained our models, getting the best model - which is AdaBoost-Decision Tree Classification. This algorithm has best performance over the observed 8 routes which has 61.35$\%$ better performance than the random purchase strategy, and relatively small variance over these routes. And we also considered the situation that we cannot get too much historical datas for some routes (for example the route is new and does not have historical data) or we do not want to train historical data to predict to buy or wait quickly, in which problem, we used HMM Sequence Classification based AdaBoost-Decision Tree Classification to perform our prediction on 12 new routes. Finally, we got 31.71$\%$ better performance than the random purchase strategy.