Source coding is the canonical problem of data compression in information theory. In a {\em locally encodable} source coding, each compressed bit depends on only few bits of the input. In this paper, we show that a recently popular model of semisupervised clustering is equivalent to locally encodable source coding. In this model, the task is to perform multiclass labeling of unlabeled elements. At the beginning, we can ask in parallel a set of simple queries to an oracle who provides (possibly erroneous) binary answers to the queries. The queries cannot involve more than two (or a fixed constant number $\Delta$ of) elements. Now the labeling of all the elements (or clustering) must be performed based on the (noisy) query answers. The goal is to recover all the correct labelings while minimizing the number of such queries. The equivalence to locally encodable source codes leads us to find lower bounds on the number of queries required in variety of scenarios. We are also able to show fundamental limitations of pairwise `same cluster' queries - and propose pairwise AND queries, that provably performs better in many situations.

Gradient descent is an algorithm to find the minimum of a function. In machine learning, the difference between the predicted and the actual value is called error. All error, across all datapoints, when added together is known as the cost. Naturally, we would want to minimize the function that represents this cost -- Cost Function. When the cost function cannot be differentiated easily Gradient Descent does the job for us.

Stationarity is a very general, qualitative assumption, that can be assessed on the basis of application specifics. It is thus a rather attractive assumption to base statistical analysis on, especially for problems for which less general qualitative assumptions, such as independence or finite memory, clearly fail. However, it has long been considered too general to allow for statistical inference to be made. One of the reasons for this is that rates of convergence, even of frequencies to the mean, are not available under this assumption alone. Recently, it has been shown that, while some natural and simple problems such as homogeneity, are indeed provably impossible to solve if one only assumes that the data is stationary (or stationary ergodic), many others can be solved using rather simple and intuitive algorithms. The latter problems include clustering and change point estimation. In this volume I summarize these results. The emphasis is on asymptotic consistency, since this the strongest property one can obtain assuming stationarity alone. While for most of the problems for which a solution is found this solution is algorithmically realizable, the main objective in this area of research, the objective which is only partially attained, is to understand what is possible and what is not possible to do for stationary time series. The considered problems include homogeneity testing, clustering with respect to distribution, clustering with respect to independence, change-point estimation, identity testing, and the general question of composite hypotheses testing. For the latter problem, a topological criterion for the existence of a consistent test is presented. In addition, several open questions are discussed.

Density Estimation is one of the central areas of statistics whose purpose is to estimate the probability density function underlying the observed data. It serves as a building block for many tasks in statistical inference, visualization, and machine learning. Density Estimation is widely adopted in the domain of unsupervised learning especially for the application of clustering. As big data become pervasive in almost every area of data sciences, analyzing high-dimensional data that have many features and variables appears to be a major focus in both academia and industry. High-dimensional data pose challenges not only from the theoretical aspects of statistical inference, but also from the algorithmic/computational considerations of machine learning and data analytics. This paper reviews a collection of selected nonparametric density estimation algorithms for high-dimensional data, some of them are recently published and provide interesting mathematical insights. The important application domain of nonparametric density estimation, such as { modal clustering}, are also included in this paper. Several research directions related to density estimation and high-dimensional data analysis are suggested by the authors.

In most recent years, zero-shot recognition (ZSR) has gained increasing attention in machine learning and image processing fields. It aims at recognizing unseen class instances with knowledge transferred from seen classes. This is typically achieved by exploiting a pre-defined semantic feature space (FS), i.e., semantic attributes or word vectors, as a bridge to transfer knowledge between seen and unseen classes. However, due to the absence of unseen classes during training, the conventional ZSR easily suffers from domain shift and hubness problems. In this paper, we propose a novel ZSR learning framework that can handle these two issues well by adaptively adjusting semantic FS. To the best of our knowledge, our work is the first to consider the adaptive adjustment of semantic FS in ZSR. Moreover, our solution can be formulated to a more efficient framework that significantly boosts the training. Extensive experiments show the remarkable performance improvement of our model compared with other existing methods.

We propose a neural network for unsupervised anomaly detection with a novel robust subspace recovery layer (RSR layer). This layer seeks to extract the underlying subspace from a latent representation of the given data and remove outliers that lie away from this subspace. It is used together with an encoder and a decoder. The encoder maps the data into the latent space, from which the RSR layer extracts the subspace. The decoder then smoothly maps back the underlying subspace to a ``manifold" close to the original data. We illustrate algorithmic choices and performance for artificial data with corrupted manifold structure. We also demonstrate competitive precision and recall for image datasets.

Laboratory testing and medication prescription are two of the most important routines in daily clinical practice. Developing an artificial intelligence system that can automatically make lab test imputations and medication recommendations can save cost on potentially redundant lab tests and inform physicians in more effective prescription. We present an intelligent model that can automatically recommend the patients' medications based on their incomplete lab tests, and can even accurately estimate the lab values that have not been taken. We model the complex relations between multiple types of medical entities with their inherent features in a heterogeneous graph. Then we learn a distributed representation for each entity in the graph based on graph convolutional networks to make the representations integrate information from multiple types of entities. Since the entity representations incorporate multiple types of medical information, they can be used for multiple medical tasks. In our experiments, we construct a graph to associate patients, encounters, lab tests and medications, and conduct the two tasks: medication recommendation and lab test imputation. The experimental results demonstrate that our model can outperform the state-of-the-art models in both tasks.

Time series forecasting is one of the challenging problems for humankind. Traditional forecasting methods using mean regression models have severe shortcomings in reflecting real-world fluctuations. While new probabilistic methods rush to rescue, they fight with technical difficulties like quantile crossing or selecting a prior distribution. To meld the different strengths of these fields while avoiding their weaknesses as well as to push the boundary of the state-of-the-art, we introduce ForGAN - one step ahead probabilistic forecasting with generative adversarial networks. ForGAN utilizes the power of the conditional generative adversarial network to learn the data generating distribution and compute probabilistic forecasts from it. We argue how to evaluate ForGAN in opposition to regression methods. To investigate probabilistic forecasting of ForGAN, we create a new dataset and demonstrate our method abilities on it. This dataset will be made publicly available for comparison. Furthermore, we test ForGAN on two publicly available datasets, namely Mackey-Glass dataset and Internet traffic dataset (A5M) where the impressive performance of ForGAN demonstrate its high capability in forecasting future values.

Adaptive importance sampling for stochastic optimization is a promising approach that offers improved convergence through variance reduction. In this work, we propose a new framework for variance reduction that enables the use of mixtures over predefined sampling distributions, which can naturally encode prior knowledge about the data. While these sampling distributions are fixed, the mixture weights are adapted during the optimization process. We propose VRM, a novel and efficient adaptive scheme that asymptotically recovers the best mixture weights in hindsight and can also accommodate sampling distributions over sets of points. We empirically demonstrate the versatility of VRM in a range of applications.

Estimating properties of discrete distributions is a fundamental problem in statistical learning. We design the first unified, linear-time, competitive, property estimator that for a wide class of properties and for all underlying distributions uses just $2n$ samples to achieve the performance attained by the empirical estimator with $n\sqrt{\log n}$ samples. This provides off-the-shelf, distribution-independent, "amplification" of the amount of data available relative to common-practice estimators. We illustrate the estimator's practical advantages by comparing it to existing estimators for a wide variety of properties and distributions. In most cases, its performance with $n$ samples is even as good as that of the empirical estimator with $n\log n$ samples, and for essentially all properties, its performance is comparable to that of the best existing estimator designed specifically for that property.