Linear autoregressive models serve as basic representations of discrete time stochastic processes. Different attempts have been made to provide non-linear versions of the basic autoregressive process, including different versions based on kernel methods. Motivated by the powerful framework of Hilbert space embeddings of distributions, in this paper we apply this methodology for the kernel embedding of an autoregressive process of order $p$. By doing so, we provide a non-linear version of an autoregressive process, that shows increased performance over the linear model in highly complex time series. We use the method proposed for one-step ahead forecasting of different time-series, and compare its performance against other non-linear methods.

Pathological conditions of the retina examined through regular screening [1], [2] can heavily assist prevention of visual blindness. Fundus imaging is the most widely used modality for early screening and detection of such blindness causing diseases like diabetic retinopathy, glucoma, agerelated macular degeneration [3], hypertension and stroke induced changes [4]. Imaging of fundus has largely improved with progress from the film based photography camera to use of electronic imaging sensors; as well as red free imaging, stereo photography, hyperspectral imaging, angiography, etc. [5], thereby reducing inter-and intra-observer reporting variability. Retinal image analysis has also significantly contributed to this technological development [5], [6]. Since fundus imaging is predominantly used for first level of abnormality screening, research focus includes: (i) detection and segmentation of retinal structures (vessels, fovea, optic disc), (ii) segmentation of abnormalities, and (iii) quality quantification of images acquired to assess reporting fitness [5]. Related Work: The process of clinical reporting of retinal abnormalities is systematic and lesions are reported with respect to their location from vessels or optic disc. Computer assisted diagnosis systems are accordingly being developed to improve the clinical workflow [5].

Sparse coding (Sc) has been studied very well as a powerful data representation method. It attempts to represent the feature vector of a data sample by reconstructing it as the sparse linear combination of some basic elements, and a $L_2$ norm distance function is usually used as the loss function for the reconstruction error. In this paper, we investigate using Sc as the representation method within multi-instance learning framework, where a sample is given as a bag of instances, and further represented as a histogram of the quantized instances. We argue that for the data type of histogram, using $L_2$ norm distance is not suitable, and propose to use the earth mover's distance (EMD) instead of $L_2$ norm distance as a measure of the reconstruction error. By minimizing the EMD between the histogram of a sample and the its reconstruction from some basic histograms, a novel sparse coding method is developed, which is refereed as SC-EMD. We evaluate its performances as a histogram representation method in tow multi-instance learning problems --- abnormal image detection in wireless capsule endoscopy videos, and protein binding site retrieval. The encouraging results demonstrate the advantages of the new method over the traditional method using $L_2$ norm distance.

Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing and evaluating linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design for graph signals, lead to algorithms for reconstructing missing information from graph signals via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of a graph signal in the vertex and graph spectral domains, our approach is to generalize the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform whose atoms are jointly localized in the vertex and graph spectral domains. One challenge we highlight is that due to the inhomogeneity of the underlying graph data domain, the local structure in a single small region of the graph can drastically affect the uncertainty bounds for signals concentrated in different regions of the graph, limiting the information provided by global uncertainty principles. Accordingly, we suggest a new way to incorporate a notion of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the center vertex of the given atom. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals.

We consider a composite convex minimization problem associated with regularized empirical risk minimization, which often arises in machine learning. We propose two new stochastic gradient methods that are based on stochastic dual averaging method with variance reduction. Our methods generate a sparser solution than the existing methods because we do not need to take the average of the history of the solutions. This is favorable in terms of both interpretability and generalization. Moreover, our methods have theoretical support for both a strongly and a non-strongly convex regularizer and achieve the best known convergence rates among existing nonaccelerated stochastic gradient methods.

Multiple Kernel Learning, or MKL, extends (kernelized) SVM by attempting to learn not only a classifier/regressor but also the best kernel for the training task, usually from a combination of existing kernel functions. Most MKL methods seek the combined kernel that performs best over every training example, sacrificing performance in some areas to seek a global optimum. Localized kernel learning (LKL) overcomes this limitation by allowing the training algorithm to match a component kernel to the examples that can exploit it best. Several approaches to the localized kernel learning problem have been explored in the last several years. We unify many of these approaches under one simple system and design a new algorithm with improved performance. We also develop enhanced versions of existing algorithms, with an eye on scalability and performance.

We introduce a deep multitask architecture to integrate multityped representations of multimodal objects. This multitype exposition is less abstract than the multimodal characterization, but more machine-friendly, and thus is more precise to model. For example, an image can be described by multiple visual views, which can be in the forms of bag-of-words (counts) or color/texture histograms (real-valued). At the same time, the image may have several social tags, which are best described using a sparse binary vector. Our deep model takes as input multiple type-specific features, narrows the cross-modality semantic gaps, learns cross-type correlation, and produces a high-level homogeneous representation. At the same time, the model supports heterogeneously typed tasks. We demonstrate the capacity of the model on two applications: social image retrieval and multiple concept prediction. The deep architecture produces more compact representation, naturally integrates multiviews and multimodalities, exploits better side information, and most importantly, performs competitively against baselines.

We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\tilde{r}/\varepsilon^2)$ rows. Here $\tilde{r}$ is the maximum stable rank, i.e. squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [MZ11, KVZ14], and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future. Our main theorem, via connections with spectral error matrix multiplication shown in prior work, implies quantitative improvements for approximate least squares regression and low rank approximation. Our main result has also already been applied to improve dimensionality reduction guarantees for $k$-means clustering [CEMMP14], and implies new results for nonparametric regression [YPW15]. We also separately point out that the proof of the "BSS" deterministic row-sampling result of [BSS12] can be modified to show that for any matrices $A, B$ of stable rank at most $\tilde{r}$, one can achieve the spectral norm guarantee for approximate matrix multiplication of $A^T B$ by deterministically sampling $O(\tilde{r}/\varepsilon^2)$ rows that can be found in polynomial time. The original result of [BSS12] was for rank instead of stable rank. Our observation leads to a stronger version of a main theorem of [KMST10].

Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient variational inference algorithm, freeing the scientist to refine and explore many models. ADVI supports a broad class of models-no conjugacy assumptions are required. We study ADVI across ten different models and apply it to a dataset with millions of observations. ADVI is integrated into Stan, a probabilistic programming system; it is available for immediate use.

Developing efficient and scalable algorithms for Latent Dirichlet Allocation (LDA) is of wide interest for many applications. Previous work has developed an O(1) Metropolis-Hastings sampling method for each token. However, the performance is far from being optimal due to random accesses to the parameter matrices and frequent cache misses. In this paper, we first carefully analyze the memory access efficiency of existing algorithms for LDA by the scope of random access, which is the size of the memory region in which random accesses fall, within a short period of time. We then develop WarpLDA, an LDA sampler which achieves both the best O(1) time complexity per token and the best O(K) scope of random access. Our empirical results in a wide range of testing conditions demonstrate that WarpLDA is consistently 5-15x faster than the state-of-the-art Metropolis-Hastings based LightLDA, and is comparable or faster than the sparsity aware F+LDA. With WarpLDA, users can learn up to one million topics from hundreds of millions of documents in a few hours, at an unprecedentedly throughput of 11G tokens per second.