Compressed sensing is a powerful tool in applications such as magnetic resonance imaging (MRI). It enables accurate recovery of images from highly undersampled measurements by exploiting the sparsity of the images or image patches in a transform domain or dictionary. In this work, we focus on blind compressed sensing (BCS), where the underlying sparse signal model is a priori unknown, and propose a framework to simultaneously reconstruct the underlying image as well as the unknown model from highly undersampled measurements. Specifically, our model is that the patches of the underlying image(s) are approximately sparse in a transform domain. We also extend this model to a union of transforms model that better captures the diversity of features in natural images. The proposed block coordinate descent type algorithms for blind compressed sensing are highly efficient, and are guaranteed to converge to at least the partial global and partial local minimizers of the highly non-convex BCS problems. Our numerical experiments show that the proposed framework usually leads to better quality of image reconstructions in MRI compared to several recent image reconstruction methods. Importantly, the learning of a union of sparsifying transforms leads to better image reconstructions than a single adaptive transform.

Machine learning is now pervasive in every field of inquiry and has lead to breakthroughs in various fields from medical diagnoses to online advertising. Practical machine learning is quite computationally intensive, whether it involves millions of repetitions of simple mathematical methods such as Euclidian Distance or more intricate optimizers or backpropagation algorithms. Such computationally intensive techniques need a fast and expressive language – one that enables scientists to write simple, readable code that performs well. In this post, we introduce a simple machine learning algorithm called K Nearest Neighbors, and demonstrate certain Julia features that allow for its easy and efficient implementation. We will demonstrate that the code we write is inherently generic, and show the use of the same code to run on GPUs via the ArrayFire package.

I doubt that any phenomenon, real or imagined, has inspired more perplexing, convoluted, and ultimately futile philosophical analysis than time travel has. In his classic textbook, An Introduction to Philosophical Analysis, John Hospers tackles the question: "Is it logically possible to go back in time--say, to 3000 B.C., and help the Egyptians build the pyramids? We must be very careful about this one." It's easy to say--we habitually use the same words to talk about time as we do when talking about space--and it's easy to imagine. "In fact, H. G. Wells did imagine it in The Time Machine (1895), and every reader imagines it with him." Hospers was a bit of a kook, actually, who achieved the unusual distinction for a philosopher of having received one electoral vote for President of the United States. But his textbook, first published in 1953, remained standard through four editions and 40 years. His answer to the rhetorical question is an emphatic no. Time travel à la Wells is not just impossible, it is logically impossible. It is a contradiction in terms. In an argument that runs for four dense pages, Hospers proves this by power of reason. "How can we be in the 20th century A.D. and the 30th century B.C. at the same time? Here already is one contradiction … It is not logically possible to be in one century of time and in another century of time at the same time."

While state-of-the-art kernels for graphs with discrete labels scale well to graphs with thousands of nodes, the few existing kernels for graphs with continuous attributes, unfortunately, do not scale well. To overcome this limitation, we present hash graph kernels, a general framework to derive kernels for graphs with continuous attributes from discrete ones. The idea is to iteratively turn continuous attributes into discrete labels using randomized hash functions. We illustrate hash graph kernels for the Weisfeiler-Lehman subtree kernel and for the shortest-path kernel. The resulting novel graph kernels are shown to be, both, able to handle graphs with continuous attributes and scalable to large graphs and data sets. This is supported by our theoretical analysis and demonstrated by an extensive experimental evaluation.

Last time we investigated the k-nearest-neighbors algorithm and the underlying idea that one can learn a classification rule by copying the known classification of nearby data points. This required that we view our data as sitting inside a metric space; that is, we imposed a kind of geometric structure on our data. One glaring problem is that there may be no reasonable way to do this. While we mentioned scaling issues and provided a number of possible metrics in our primer, a more common problem is that the data simply isn't numeric. For instance, a poll of US citizens might ask the respondent to select which of a number of issues he cares most about. There could be 50 choices, and there is no reasonable way to assign these numerical values so that all are equidistant in the resulting metric space. Another issue is that the quality of the data could be bad. For instance, there may be missing values for some attributes (e.g., a respondent may neglect to answer one or more questions).

We study fast learning rates when the losses are not necessarily bounded and may have a distribution with heavy tails. To enable such analyses, we introduce two new conditions: (i) the envelope function $\sup_{f \in \mathcal{F}}|\ell \circ f|$, where $\ell$ is the loss function and $\mathcal{F}$ is the hypothesis class, exists and is $L^r$-integrable, and (ii) $\ell$ satisfies the multi-scale Bernstein's condition on $\mathcal{F}$. Under these assumptions, we prove that learning rate faster than $O(n^{-1/2})$ can be obtained and, depending on $r$ and the multi-scale Bernstein's powers, can be arbitrarily close to $O(n^{-1})$. We then verify these assumptions and derive fast learning rates for the problem of vector quantization by $k$-means clustering with heavy-tailed distributions. The analyses enable us to obtain novel learning rates that extend and complement existing results in the literature from both theoretical and practical viewpoints.

The number of methods available for classification of multi-label data has increased rapidly over recent years, yet relatively few links have been made with the related task of classification of sequential data. If labels indices are considered as time indices, the problems can often be seen as equivalent. In this paper we detect and elaborate on connections between multi-label methods and Markovian models, and study the suitability of multi-label methods for prediction in sequential data. From this study we draw upon the most suitable techniques from the area and develop two novel competitive approaches which can be applied to either kind of data. We carry out an empirical evaluation investigating performance on real-world sequential-prediction tasks: electricity demand, and route prediction. As well as showing that several popular multi-label algorithms are in fact easily applicable to sequencing tasks, our novel approaches, which benefit from a unified view of these areas, prove very competitive against established methods. Keywords: multi-label classification; problem transformation; sequential data; sequence prediction; Markov models 1. Introduction Multi-label classification is the supervised learning problem where an instance is associated with multiple class variables (i.e., labels), rather than with a single class, as in traditional classification problems. See [1] for a review. Corresponding author, jesse.read@polytechnique.edu Preprint submitted to Pattern Recognition September 29, 2016 labels were modelled independently - at the expense of an increased computational cost. The case of binary labels is most common, where a positive class value denotes the relevance of the label (and the negative or null class denotes irrelevance). Typical examples of binary multi-label classification involve categorizing text documents and images, which can be assigned any subset of a particular label set. For example, an image can be associated with both labels beach and sunset. The multi-label classification paradigm has been successfully considered also in many other domains, such as text, video, audio, and bioinformatics - see [1] and references therein for further examples.

Gaussian graphical models are widely utilized to infer and visualize networks of dependencies between continuous variables. However, inferring the graph is difficult when the sample size is small compared to the number of variables. To reduce the number of parameters to estimate in the model, we propose a non-asymptotic model selection procedure supported by strong theoretical guarantees based on an oracle type inequality and a minimax lower bound. The covariance matrix of the model is approximated by a block-diagonal matrix. The structure of this matrix is detected by thresholding the sample covariance matrix, where the threshold is selected using the slope heuristic. Based on the block-diagonal structure of the covariance matrix, the estimation problem is divided into several independent problems: subsequently, the network of dependencies between variables is inferred using the graphical lasso algorithm in each block. The performance of the procedure is illustrated on simulated data. An application to a real gene expression dataset with a limited sample size is also presented: the dimension reduction allows attention to be objectively focused on interactions among smaller subsets of genes, leading to a more parsimonious and interpretable modular network. Contents 1. Introduction 2 2. A method to detect block-diagonal covariance structure 3 3. Theoretical results for non-asymptotic model selection 5 4. Simulation study 8 4.1.

Recent contributions have framed linear system identification as a nonparametric regularized inverse problem. Relying on $\ell_2$-type regularization which accounts for the stability and smoothness of the impulse response to be estimated, these approaches have been shown to be competitive w.r.t classical parametric methods. In this paper, adopting Maximum Entropy arguments, we derive a new $\ell_2$ penalty deriving from a vector-valued kernel; to do so we exploit the structure of the Hankel matrix, thus controlling at the same time complexity, measured by the McMillan degree, stability and smoothness of the identified models. As a special case we recover the nuclear norm penalty on the squared block Hankel matrix. In contrast with previous literature on reweighted nuclear norm penalties, our kernel is described by a small number of hyper-parameters, which are iteratively updated through marginal likelihood maximization; constraining the structure of the kernel acts as a (hyper)regularizer which helps controlling the effective degrees of freedom of our estimator. To optimize the marginal likelihood we adapt a Scaled Gradient Projection (SGP) algorithm which is proved to be significantly computationally cheaper than other first and second order off-the-shelf optimization methods. The paper also contains an extensive comparison with many state-of-the-art methods on several Monte-Carlo studies, which confirms the effectiveness of our procedure.

I'm curious if it's possible to use the reparametrization trick to move the stochasticity outside the model, like this: Now that the stochasticity is outside the model, you can just use vanilla stochastic gradient descent in tensorflow. What is interesting about this approach seems to be the many different gradient estimators available, and i'm wondering if these tricks can bleed back into the variational deep learning literature.