This is a tutorial and survey paper on factor analysis, probabilistic Principal Component Analysis (PCA), variational inference, and Variational Autoencoder (VAE). These methods, which are tightly related, are dimensionality reduction and generative models. They asssume that every data point is generated from or caused by a low-dimensional latent factor. By learning the parameters of distribution of latent space, the corresponding low-dimensional factors are found for the sake of dimensionality reduction. For their stochastic and generative behaviour, these models can also be used for generation of new data points in the data space. In this paper, we first start with variational inference where we derive the Evidence Lower Bound (ELBO) and Expectation Maximization (EM) for learning the parameters. Then, we introduce factor analysis, derive its joint and marginal distributions, and work out its EM steps. Probabilistic PCA is then explained, as a special case of factor analysis, and its closed-form solutions are derived. Finally, VAE is explained where the encoder, decoder and sampling from the latent space are introduced. Training VAE using both EM and backpropagation are explained.

We proposed a novel multilayer correlated topic model (MCTM) to analyze how the main ideas inherit and vary between a document and its different segments, which helps understand an article's structure. The variational expectation-maximization (EM) algorithm was derived to estimate the posterior and parameters in MCTM. We introduced two potential applications of MCTM, including the paragraph-level document analysis and market basket data analysis. The effectiveness of MCTM in understanding the document structure has been verified by the great predictive performance on held-out documents and intuitive visualization. We also showed that MCTM could successfully capture customers' popular shopping patterns in the market basket analysis.

Before diving into this topic, lets first start with some definitions. "Rescaling" a vector means to add or subtract a constant and then multiply or divide by a constant, as you would do to change the units of measurement of the data, for example, to convert a temperature from Celsius to Fahrenheit. "Normalizing" a vector most often means dividing by a norm of the vector. It also often refers to rescaling by the minimum and range of the vector, to make all the elements lie between 0 and 1 thus bringing all the values of numeric columns in the dataset to a common scale. "Standardizing" a vector most often means subtracting a measure of location and dividing by a measure of scale.

An Introduction to Statistical Learning, with Applications in R (ISLR) can be considered a less advanced treatment of the topics found in another classic of the genre written by some of the same authors, The Elements of Statistical Learning. Another major difference between these 2 titles, beyond the level of depth of the material covered, is that ISLR introduces these topics alongside practical implementations in a programming language, in this case R.

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.

In traditional machine learning techniques, the degree of closeness between true and predicted values generally measures the quality of predictions. However, these learning algorithms do not consider prescription problems where the predicted values will be used as input to decision problems. In this paper, we efficiently leverage feature variables, and we propose a new framework directly integrating predictive tasks under prescriptive tasks in order to prescribe consistent decisions. We train the parameters of predictive algorithm within a prescription problem via bilevel optimization techniques. We present the structure of our method and demonstrate its performance using synthetic data compared to classical methods like point-estimate-based, stochastic optimization and recently developed machine learning based optimization methods. In addition, we control generalization error using different penalty approaches and optimize the integration over validation data set.

Bayesian learning is a powerful learning framework which combines the external information of the data (background information) with the internal information (training data) in a logically consistent way in inference and prediction. By Bayes rule, the external information (prior distribution) and the internal information (training data likelihood) are combined coherently, and the posterior distribution and the posterior predictive (marginal) distribution obtained by Bayes rule summarize the total information needed in the inference and prediction, respectively. In this paper, we study the Bayesian framework of the Tensor Network from two perspective. First, we introduce the prior distribution to the weights in the Tensor Network and predict the labels of the new observations by the posterior predictive (marginal) distribution. Since the intractability of the parameter integral in the normalization constant computation, we approximate the posterior predictive distribution by Laplace approximation and obtain the out-product approximation of the hessian matrix of the posterior distribution of the Tensor Network model. Second, to estimate the parameters of the stationary mode, we propose a stable initialization trick to accelerate the inference process by which the Tensor Network can converge to the stationary path more efficiently and stably with gradient descent method. We verify our work on the MNIST, Phishing Website and Breast Cancer data set. We study the Bayesian properties of the Bayesian Tensor Network by visualizing the parameters of the model and the decision boundaries in the two dimensional synthetic data set. For a application purpose, our work can reduce the overfitting and improve the performance of normal Tensor Network model.

Softmax classifiers with a very large number of classes naturally occur in many applications such as natural language processing and information retrieval. The calculation of full-softmax is very expensive from the computational and energy perspective. There have been a variety of sampling approaches to overcome this challenge, popularly known as negative sampling (NS). Ideally, NS should sample negative classes from a distribution that is dependent on the input data, the current parameters, and the correct positive class. Unfortunately, due to the dynamically updated parameters and data samples, there does not exist any sampling scheme that is truly adaptive and also samples the negative classes in constant time every iteration. Therefore, alternative heuristics like random sampling, static frequency-based sampling, or learning-based biased sampling, which primarily trade either the sampling cost or the adaptivity of samples per iteration, are adopted. In this paper, we show a class of distribution where the sampling scheme is truly adaptive and provably generates negative samples in constant time. Our implementation in C++ on commodity CPU is significantly faster, in terms of wall clock time, compared to the most optimized TensorFlow implementations of standard softmax or other sampling approaches on modern GPUs (V100s).

Influence functions approximate the 'influences' of training data-points for test predictions and have a wide variety of applications. Despite the popularity, their computational cost does not scale well with model and training data size. We present FastIF, a set of simple modifications to influence functions that significantly improves their run-time. We use k-Nearest Neighbors (kNN) to narrow the search space down to a subset of good candidate data points, identify the configurations that best balance the speed-quality trade-off in estimating the inverse Hessian-vector product, and introduce a fast parallel variant. Our proposed method achieves about 80x speedup while being highly correlated with the original influence values. With the availability of the fast influence functions, we demonstrate their usefulness in four applications. First, we examine whether influential data-points can 'explain' test time behavior using the framework of simulatability. Second, we visualize the influence interactions between training and test data-points. Third, we show that we can correct model errors by additional fine-tuning on certain influential data-points, improving the accuracy of a trained MNLI model by 2.6% on the HANS challenge set using a small number of gradient updates. Finally, we experiment with a data-augmentation setup where we use influence functions to search for new data-points unseen during training to improve model performance. Overall, our fast influence functions can be efficiently applied to large models and datasets, and our experiments demonstrate the potential of influence functions in model interpretation and correcting model errors. Code is available at https://github.com/salesforce/fast-influence-functions

In this paper, we investigate performing joint dimensionality reduction and classification using a novel histogram neural network. Motivated by a popular dimensionality reduction approach, t-Distributed Stochastic Neighbor Embedding (t-SNE), our proposed method incorporates a classification loss computed on samples in a low-dimensional embedding space. We compare the learned sample embeddings against coordinates found by t-SNE in terms of classification accuracy and qualitative assessment. We also explore use of various divergence measures in the t-SNE objective. The proposed method has several advantages such as readily embedding out-of-sample points and reducing feature dimensionality while retaining class discriminability. Our results show that the proposed approach maintains and/or improves classification performance and reveals characteristics of features produced by neural networks that may be helpful for other applications.