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.

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 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.

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.

So I understand that variable selection is a part of model selection. But what exactly does model selection consist of? I ask this because I am reading an article Burnham & Anderson: AIC vs BIC where they talk about AIC and BIC in model selection. Reading this article I realize I have been thinking of'model selection' as'variable selection' (ref. An excerpt from the article where they talk about 12 models with increasing degrees of "generality" and these models show "tapering effects" (Figure 1) when KL-Information is plotted against the 12 models: DIFFERENT PHILOSOPHIES AND TARGET MODELS ... Despite that the target of BIC is a more general model than the target model for AIC, the model most often selected here by BIC will be less general than Model 7 unless n is very large.

The question of which model type to apply to a Machine Learning task can be a daunting one given the immense number of algorithms available in the literature. It can be difficult to compare the relative merits of two methods, as one can outperform the other in a certain class of problems while consistently coming in behind for another class. In this post, the last one of our series of posts about Logistic Regression, we'll explore the differences between Decision Trees and Logistic Regression for classification problems, and try to highlight scenarios where one might be recommended over the other. Logistic Regression and trees differ in the way that they generate decision boundaries i.e. the lines that are drawn to separate different classes. To illustrate this difference, let's look at the results of the two model types on the following 2-class problem: Decision Trees bisect the space into smaller and smaller regions, whereas Logistic Regression fits a single line to divide the space exactly into two.