Sparse pseudo-point approximations for Gaussian process (GP) models provide a suite of methods that support deployment of GPs in the large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which both the posterior distribution over function values and the hyperparameter estimates are updated in an online fashion. The small number of existing approaches either use suboptimal hand-crafted heuristics for hyperparameter learning, or suffer from catastrophic forgetting or slow updating when new data arrive. This paper develops a new principled framework for deploying Gaussian process probabilistic models in the streaming setting, providing methods for learning hyperparameters and optimising pseudo-input locations. The proposed framework is assessed using synthetic and real-world datasets.

Learning an encoding of feature vectors in terms of an over-complete dictionary or a information geometric (Fisher vectors) construct is wide-spread in statistical signal processing and computer vision. In content based information retrieval using deep-learning classifiers, such encodings are learnt on the flattened last layer, without adherence to the multi-linear structure of the underlying feature tensor. We illustrate a variety of feature encodings incl. sparse dictionary coding and Fisher vectors along with proposing that a structured tensor factorization scheme enables us to perform retrieval that can be at par, in terms of average precision, with Fisher vector encoded image signatures. In short, we illustrate how structural constraints increase retrieval fidelity.

Topological data analysis (TDA) has emerged as one of the most promising techniques to reconstruct the unknown shapes of high-dimensional spaces from observed data samples. TDA, thus, yields key shape descriptors in the form of persistent topological features that can be used for any supervised or unsupervised learning task, including multi-way classification. Sparse sampling, on the other hand, provides a highly efficient technique to reconstruct signals in the spatial-temporal domain from just a few carefully-chosen samples. Here, we present a new method, referred to as the Sparse-TDA algorithm, that combines favorable aspects of the two techniques. This combination is realized by selecting an optimal set of sparse pixel samples from the persistent features generated by a vector-based TDA algorithm. These sparse samples are selected from a low-rank matrix representation of persistent features using QR pivoting. We show that the Sparse-TDA method demonstrates promising performance on three benchmark problems related to human posture recognition and image texture classification.

Variational inference (VI) is widely used as an efficient alternative to Markov chain Monte Carlo. It posits a family of approximating distributions $q$ and finds the closest member to the exact posterior $p$. Closeness is usually measured via a divergence $D(q || p)$ from $q$ to $p$. While successful, this approach also has problems. Notably, it typically leads to underestimation of the posterior variance. In this paper we propose CHIVI, a black-box variational inference algorithm that minimizes $D_{\chi}(p || q)$, the $\chi$-divergence from $p$ to $q$. CHIVI minimizes an upper bound of the model evidence, which we term the $\chi$ upper bound (CUBO). Minimizing the CUBO leads to improved posterior uncertainty, and it can also be used with the classical VI lower bound (ELBO) to provide a sandwich estimate of the model evidence. We study CHIVI on three models: probit regression, Gaussian process classification, and a Cox process model of basketball plays. When compared to expectation propagation and classical VI, CHIVI produces better error rates and more accurate estimates of posterior variance.

The objective of the paper is to study accuracy of multi-class classification in high-dimensional setting, where the number of classes is also large ("large $L$, large $p$, small $n$" model). While this problem arises in many practical applications and many techniques have been recently developed for its solution, to the best of our knowledge nobody provided a rigorous theoretical analysis of this important setup. The purpose of the present paper is to fill in this gap. We consider one of the most common settings, classification of high-dimensional normal vectors where, unlike standard assumptions, the number of classes could be large. We derive non-asymptotic conditions on effects of significant features, and the low and the upper bounds for distances between classes required for successful feature selection and classification with a given accuracy. Furthermore, we study an asymptotic setup where the number of classes is growing with the dimension of feature space and while the number of samples per class is possibly limited. We discover an interesting and, at first glance, somewhat counter-intuitive phenomenon that a large number of classes may be a "blessing" rather than a "curse" since, in certain settings, the precision of classification can improve as the number of classes grows. This is due to more accurate feature selection since even weaker significant features, which are not sufficiently strong to be manifested in a coarse classification, can nevertheless have a strong impact when the number of classes is large. We supplement our theoretical investigation by a simulation study and a real data example where we again observe the above phenomenon.

There is no need to get confused with multiple linear regression, generalized linear model or general linear methods. The general linear model or multivariate regression model is a statistical linear model and is written as Y XB U. Usually, a linear model includes a number of different statistical models such as ANOVA, ANCOVA, MANOVA, MANCOVA, ordinary linear regression, t-test and F-test. The GLM is a generalization of multiple linear regression models to the case of more than one dependent variable. So if Y, B, and U represent column vectors, the matrix equation above will portray a multiple linear regression. Which are the key assumptions made in a multiple linear regression analysis?

This article targets whom have a basic understanding for TensorFlow Core API. Learning TensorFlow Core API, which is the lowest level API in TensorFlow, is a very good step for starting learning TensorFlow because it let you understand the kernel of the library. Here is a very simple example of TensorFlow Core API in which we create and train a linear regression model. The loss returned is 53.76. Existence of error, specially for large error, means that the parameters used must be updated.

A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces $\mathscr{U}=\underset{i=1}{\overset{M}\bigcup}S_i$. The similarity matrices thus constructed give the exact clustering in the noise-free case. A simple adaptation of the technique also allows clustering of noisy data. Two known methods for subspace clustering can be derived from the CUR technique. Experiments on synthetic and real data are presented to test the method.

While linear mixed model (LMM) has shown a competitive performance in correcting spurious associations raised by population stratification, family structures, and cryptic relatedness, more challenges are still to be addressed regarding the complex structure of genotypic and phenotypic data. For example, geneticists have discovered that some clusters of phenotypes are more co-expressed than others. Hence, a joint analysis that can utilize such relatedness information in a heterogeneous data set is crucial for genetic modeling. We proposed the sparse graph-structured linear mixed model (sGLMM) that can incorporate the relatedness information from traits in a dataset with confounding correction. Our method is capable of uncovering the genetic associations of a large number of phenotypes together while considering the relatedness of these phenotypes. Through extensive simulation experiments, we show that the proposed model outperforms other existing approaches and can model correlation from both population structure and shared signals. Further, we validate the effectiveness of sGLMM in the real-world genomic dataset on two different species from plants and humans. In Arabidopsis thaliana data, sGLMM behaves better than all other baseline models for 63.4% traits. We also discuss the potential causal genetic variation of Human Alzheimer's disease discovered by our model and justify some of the most important genetic loci.

Analyzing the temporal behavior of nodes in time-varying graphs is useful for many applications such as targeted advertising, community evolution and outlier detection. In this paper, we present a novel approach, STWalk, for learning trajectory representations of nodes in temporal graphs. The proposed framework makes use of structural properties of graphs at current and previous time-steps to learn effective node trajectory representations. STWalk performs random walks on a graph at a given time step (called space-walk) as well as on graphs from past time-steps (called time-walk) to capture the spatio-temporal behavior of nodes. We propose two variants of STWalk to learn trajectory representations. In one algorithm, we perform space-walk and time-walk as part of a single step. In the other variant, we perform space-walk and time-walk separately and combine the learned representations to get the final trajectory embedding. Extensive experiments on three real-world temporal graph datasets validate the effectiveness of the learned representations when compared to three baseline methods. We also show the goodness of the learned trajectory embeddings for change point detection, as well as demonstrate that arithmetic operations on these trajectory representations yield interesting and interpretable results.