Change detection in multivariate time series has applications in many domains, including health care and network monitoring. A common approach to detect changes is to compare the divergence between the distributions of a reference window and a test window. When the number of dimensions is very large, however, the naive approach has both quality and efficiency issues: to ensure robustness the window size needs to be large, which not only leads to missed alarms but also increases runtime. To this end, we propose LIGHT, a linear-time algorithm for robustly detecting non-linear changes in massively high dimensional time series. Importantly, LIGHT provides high flexibility in choosing the window size, allowing the domain expert to fit the level of details required. To do such, we 1) perform scalable PCA to reduce dimensionality, 2) perform scalable factorization of the joint distribution, and 3) scalably compute divergences between these lower dimensional distributions. Extensive empirical evaluation on both synthetic and real-world data show that LIGHT outperforms state of the art with up to 100% improvement in both quality and efficiency.

This paper studies the problem of estimating the covariance of a collection of vectors using only extremely compressed measurements of each vector. An estimator based on back-projections of these compressive samples is proposed and analyzed. A distribution-free analysis shows that by observing just a single compressive measurement of each vector, one can consistently estimate the covariance matrix, in both infinity and spectral norm, and this same analysis leads to precise rates of convergence in both norms. Via information-theoretic techniques, lower bounds showing that this estimator is minimax-optimal for both infinity and spectral norm estimation problems are established. These results are also specialized to give matching upper and lower bounds for estimating the population covariance of a collection of Gaussian vectors, again in the compressive measurement model. The analysis conducted in this paper shows that the effective sample complexity for this problem is scaled by a factor of $m^2/d^2$ where $m$ is the compression dimension and $d$ is the ambient dimension. Applications to subspace learning (Principal Components Analysis) and learning over distributed sensor networks are also discussed.

Kernel methods obtain superb performance in terms of accuracy for various machine learning tasks since they can effectively extract nonlinear relations. However, their time complexity can be rather large especially for clustering tasks. In this paper we define a general class of kernels that can be easily approximated by randomization. These kernels appear in various applications, in particular, traditional spectral clustering, landmark-based spectral clustering and landmark-based subspace clustering. We show that for $n$ data points from $K$ clusters with $D$ landmarks, the randomization procedure results in an algorithm of complexity $O(KnD)$. Furthermore, we bound the error between the original clustering scheme and its randomization. To illustrate the power of this framework, we propose a new fast landmark subspace (FLS) clustering algorithm. Experiments over synthetic and real datasets demonstrate the superior performance of FLS in accelerating subspace clustering with marginal sacrifice of accuracy.

We aim to analyze the relation between two random vectors that may potentially have both different number of attributes as well as realizations, and which may even not have a joint distribution. This problem arises in many practical domains, including biology and architecture. Existing techniques assume the vectors to have the same domain or to be jointly distributed, and hence are not applicable. To address this, we propose Canonical Divergence Analysis (CDA). We introduce three instantiations, each of which permits practical implementation. Extensive empirical evaluation shows the potential of our method.

Dictionary learning algorithms have been successfully used for both reconstructive and discriminative tasks, where an input signal is represented with a sparse linear combination of dictionary atoms. While these methods are mostly developed for single-modality scenarios, recent studies have demonstrated the advantages of feature-level fusion based on the joint sparse representation of the multimodal inputs. In this paper, we propose a multimodal task-driven dictionary learning algorithm under the joint sparsity constraint (prior) to enforce collaborations among multiple homogeneous/heterogeneous sources of information. In this task-driven formulation, the multimodal dictionaries are learned simultaneously with their corresponding classifiers. The resulting multimodal dictionaries can generate discriminative latent features (sparse codes) from the data that are optimized for a given task such as binary or multiclass classification. Moreover, we present an extension of the proposed formulation using a mixed joint and independent sparsity prior which facilitates more flexible fusion of the modalities at feature level. The efficacy of the proposed algorithms for multimodal classification is illustrated on four different applications -- multimodal face recognition, multi-view face recognition, multi-view action recognition, and multimodal biometric recognition. It is also shown that, compared to the counterpart reconstructive-based dictionary learning algorithms, the task-driven formulations are more computationally efficient in the sense that they can be equipped with more compact dictionaries and still achieve superior performance.

Many practical applications such as gene expression analysis, multi-task learning, image recognition, signal processing, and medical data analysis pursue a sparse solution for the feature selection purpose and particularly favor the nonzeros \emph{evenly} distributed in different groups. The exclusive sparsity norm has been widely used to serve to this purpose. However, it still lacks systematical studies for exclusive sparsity norm optimization. This paper offers two main contributions from the optimization perspective: 1) We provide several efficient algorithms to solve exclusive sparsity norm minimization with either smooth loss or hinge loss (non-smooth loss). All algorithms achieve the optimal convergence rate $O(1/k^2)$ ($k$ is the iteration number). To the best of our knowledge, this is the first time to guarantee such convergence rate for the general exclusive sparsity norm minimization; 2) When the group information is unavailable to define the exclusive sparsity norm, we propose to use the random grouping scheme to construct groups and prove that if the number of groups is appropriately chosen, the nonzeros (true features) would be grouped in the ideal way with high probability. Empirical studies validate the efficiency of proposed algorithms, and the effectiveness of random grouping scheme on the proposed exclusive SVM formulation.

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family of schemes with similar convergence rates. The ODE interpretation also suggests restarting Nesterov's scheme leading to an algorithm, which can be rigorously proven to converge at a linear rate whenever the objective is strongly convex.

Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a $d$-dimensional compact submanifold $M$ in $\mathbb{R}^D$, we establish the spectral convergence rate of the graph Laplacian. It implies the consistency of the spectral clustering algorithm via a standard perturbation argument. A simple numerical study indicates the necessity of a denoising step before applying spectral algorithms.

The modern scale of data has brought new challenges to Bayesian inference. In particular, conventional MCMC algorithms are computationally very expensive for large data sets. A promising approach to solve this problem is embarrassingly parallel MCMC (EP-MCMC), which first partitions the data into multiple subsets and runs independent sampling algorithms on each subset. The subset posterior draws are then aggregated via some combining rules to obtain the final approximation. Existing EP-MCMC algorithms are limited by approximation accuracy and difficulty in resampling. In this article, we propose a new EP-MCMC algorithm PART that solves these problems. The new algorithm applies random partition trees to combine the subset posterior draws, which is distribution-free, easy to resample from and can adapt to multiple scales. We provide theoretical justification and extensive experiments illustrating empirical performance.

We study the problem of reducing test-time acquisition costs in classification systems. Our goal is to learn decision rules that adaptively select sensors for each example as necessary to make a confident prediction. We model our system as a directed acyclic graph (DAG) where internal nodes correspond to sensor subsets and decision functions at each node choose whether to acquire a new sensor or classify using the available measurements. This problem can be naturally posed as an empirical risk minimization over training data. Rather than jointly optimizing such a highly coupled and non-convex problem over all decision nodes, we propose an efficient algorithm motivated by dynamic programming. We learn node policies in the DAG by reducing the global objective to a series of cost sensitive learning problems. Our approach is computationally efficient and has proven guarantees of convergence to the optimal system for a fixed architecture. In addition, we present an extension to map other budgeted learning problems with large number of sensors to our DAG architecture and demonstrate empirical performance exceeding state-of-the-art algorithms for data composed of both few and many sensors.