In this paper we introduce and analyze the learning scenario of \emph{coupled nonlinear dimensionality reduction}, which combines two major steps of machine learning pipeline: projection onto a manifold and subsequent supervised learning. First, we present new generalization bounds for this scenario and, second, we introduce an algorithm that follows from these bounds. The generalization error bound is based on a careful analysis of the empirical Rademacher complexity of the relevant hypothesis set. In particular, we show an upper bound on the Rademacher complexity that is in $\widetilde O(\sqrt{\Lambda_{(r)}/m})$, where $m$ is the sample size and $\Lambda_{(r)}$ the upper bound on the Ky-Fan $r$-norm of the associated kernel matrix. We give both upper and lower bound guarantees in terms of that Ky-Fan $r$-norm, which strongly justifies the definition of our hypothesis set. To the best of our knowledge, these are the first learning guarantees for the problem of coupled dimensionality reduction. Our analysis and learning guarantees further apply to several special cases, such as that of using a fixed kernel with supervised dimensionality reduction or that of unsupervised learning of a kernel for dimensionality reduction followed by a supervised learning algorithm. Based on theoretical analysis, we suggest a structural risk minimization algorithm consisting of the coupled fitting of a low dimensional manifold and a separation function on that manifold.

With rapidly increasing data, clustering algorithms are important tools for data analytics in modern research. They have been successfully applied to a wide range of domains; for instance, bioinformatics, speech recognition, and financial analysis. Formally speaking, given a set of data instances, a clustering algorithm is expected to divide the set of data instances into the subsets which maximize the intra-subset similarity and inter-subset dissimilarity, where a similarity measure is defined beforehand. In this work, the state-of-the-arts clustering algorithms are reviewed from design concept to methodology; Different clustering paradigms are discussed. Advanced clustering algorithms are also discussed. After that, the existing clustering evaluation metrics are reviewed. A summary with future insights is provided at the end.

What is the difference of a prediction that is made with a causal model and a non-causal model? Suppose we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables. Here, we propose to exploit this invariance of a prediction under a causal model for causal inference: given different experimental settings (for example various interventions) we collect all models that do show invariance in their predictive accuracy across settings and interventions. The causal model will be a member of this set of models with high probability. This approach yields valid confidence intervals for the causal relationships in quite general scenarios. We examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable. We further investigate robustness properties of our approach under model misspecification and discuss possible extensions. The empirical properties are studied for various data sets, including large-scale gene perturbation experiments.

This paper addresses the problem of active learning of a multi-output Gaussian process (MOGP) model representing multiple types of coexisting correlated environmental phenomena. In contrast to existing works, our active learning problem involves selecting not just the most informative sampling locations to be observed but also the types of measurements at each selected location for minimizing the predictive uncertainty (i.e., posterior joint entropy) of a target phenomenon of interest given a sampling budget. Unfortunately, such an entropy criterion scales poorly in the numbers of candidate sampling locations and selected observations when optimized. To resolve this issue, we first exploit a structure common to sparse MOGP models for deriving a novel active learning criterion. Then, we exploit a relaxed form of submodularity property of our new criterion for devising a polynomial-time approximation algorithm that guarantees a constant-factor approximation of that achieved by the optimal set of selected observations. Empirical evaluation on real-world datasets shows that our proposed approach outperforms existing algorithms for active learning of MOGP and single-output GP models.

We combine supervised learning with unsupervised learning in deep neural networks. The proposed model is trained to simultaneously minimize the sum of supervised and unsupervised cost functions by backpropagation, avoiding the need for layer-wise pre-training. Our work builds on the Ladder network proposed by Valpola (2015), which we extend by combining the model with supervision. We show that the resulting model reaches state-of-the-art performance in semi-supervised MNIST and CIFAR-10 classification, in addition to permutation-invariant MNIST classification with all labels.

This is a lecture note for the course DS-GA 3001 at the Center for Data Science , New York University in Fall, 2015. As the name of the course suggests, this lecture note introduces readers to a neural network based approach to natural language understanding/processing. In order to make it as self-contained as possible, I spend much time on describing basics of machine learning and neural networks, only after which how they are used for natural languages is introduced. On the language front, I almost solely focus on language modelling and machine translation, two of which I personally find most fascinating and most fundamental to natural language understanding.

We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities where classical HMC is not an option due to intractable gradients, KMC adaptively learns the target's gradient structure by fitting an exponential family model in a Reproducing Kernel Hilbert Space. Computational costs are reduced by two novel efficient approximations to this gradient. While being asymptotically exact, KMC mimics HMC in terms of sampling efficiency, and offers substantial mixing improvements over state-of-the-art gradient free samplers. We support our claims with experimental studies on both toy and real-world applications, including Approximate Bayesian Computation and exact-approximate MCMC.

We derive a statistical model for estimation of a dendrogram from single linkage hierarchical clustering (SLHC) that takes account of uncertainty through noise or corruption in the measurements of separation of data. Our focus is on just the estimation of the hierarchy of partitions afforded by the dendrogram, rather than the heights in the latter. The concept of estimating this "dendrogram structure'' is introduced, and an approximate maximum likelihood estimator (MLE) for the dendrogram structure is described. These ideas are illustrated by a simple Monte Carlo simulation that, at least for small data sets, suggests the method outperforms SLHC in the presence of noise.

To identify the location of objects of a particular class, a passive computer vision system generally processes all the regions in an image to finally output few regions. However, we can use structure in the scene to search for objects without processing the entire image. We propose a search technique that sequentially processes image regions such that the regions that are more likely to correspond to the query class object are explored earlier. We frame the problem as a Markov decision process and use an imitation learning algorithm to learn a search strategy. Since structure in the scene is essential for search, we work with indoor scene images as they contain both unary scene context information and object-object context in the scene. We perform experiments on the NYU-depth v2 dataset and show that the unary scene context features alone can achieve a significantly high average precision while processing only 20-25\% of the regions for classes like bed and sofa. By considering object-object context along with the scene context features, the performance is further improved for classes like counter, lamp, pillow and sofa.

Gaussian process is a theoretically appealing model for nonparametric analysis, but its computational cumbersomeness hinders its use in large scale and the existing reduced-rank solutions are usually heuristic. In this work, we propose a novel construction of Gaussian process as a projection from fixed discrete frequencies to any continuous location. This leads to a valid stochastic process that has a theoretic support with the reduced rank in the spectral density, as well as a high-speed computing algorithm. Our method provides accurate estimates for the covariance parameters and concise form of predictive distribution for spatial prediction. For non-stationary data, we adopt the mixture framework with a customized spectral dependency structure. This enables clustering based on local stationarity, while maintains the joint Gaussianness. Our work is directly applicable in solving some of the challenges in the spatial data, such as large scale computation, anisotropic covariance, spatio-temporal modeling, etc. We illustrate the uses of the model via simulations and an application on a massive dataset.