Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry

Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession

Spectral dimensionality reduction for HMMs

Hidden Markov Models (HMMs) can be accurately approximated using co-occurrence frequencies of pairs and triples of observations by using a fast spectral method in contrast to the usual slow methods like EM or Gibbs sampling. We provide a new spectral method which significantly reduces the number of model parameters that need to be estimated, and generates a sample complexity that does not depend on the size of the observation vocabulary. We present an elementary proof giving bounds on the relative accuracy of probability estimates from our model. (Correlaries show our bounds can be weakened to provide either L1 bounds or KL bounds which provide easier direct comparisons to previous work.) Our theorem uses conditions that are checkable from the data, instead of putting conditions on the unobservable Markov transition matrix

What Twitter Profile and Posted Images Reveal About Depression and Anxiety

Previous work has found strong links between the choice of social media images and users' emotions, demographics and personality traits. In this study, we examine which attributes of profile and posted images are associated with depression and anxiety of Twitter users. We used a sample of 28,749 Facebook users to build a language prediction model of survey-reported depression and anxiety, and validated it on Twitter on a sample of 887 users who had taken anxiety and depression surveys. We then applied it to a different set of 4,132 Twitter users to impute language-based depression and anxiety labels, and extracted interpretable features of posted and profile pictures to uncover the associations with users' depression and anxiety, controlling for demographics. For depression, we find that profile pictures suppress positive emotions rather than display more negative emotions, likely because of social media self-presentation biases. They also tend to show the single face of the user (rather than show her in groups of friends), marking increased focus on the self, emblematic for depression. Posted images are dominated by grayscale and low aesthetic cohesion across a variety of image features. Profile images of anxious users are similarly marked by grayscale and low aesthetic cohesion, but less so than those of depressed users. Finally, we show that image features can be used to predict depression and anxiety, and that multitask learning that includes a joint modeling of demographics improves prediction performance. Overall, we find that the image attributes that mark depression and anxiety offer a rich lens into these conditions largely congruent with the psychological literature, and that images on Twitter allow inferences about the mental health status of users.Comment: ICWSM 201

Structural Logistic Regression for Link Analysis

We present Structural Logistic Regression, an extension of logistic regression to modeling relational data. It is an integrated approach to building regression models from data stored in relational databases in which potential predictors, both boolean and real-valued, are generated by structured search in the space of queries to the database, and then tested with statistical information criteria for inclusion in a logistic regression. Using statistics and relational representation allows modeling in noisy domains with complex structure. Link prediction is a task of high interest with exactly such characteristics. Be it in the domain of scientific citations, social networks or hypertext, the underlying data are extremely noisy and the features useful for prediction are not readily available in a flat file format. We propose the application of Structural Logistic Regression to building link prediction models, and present experimental results for the task of predicting citations made in scientific literature using relational data taken from the CiteSeer search engine. This data includes the citation graph, authorship and publication venues of papers, as well as their word content

A Risk Comparison of Ordinary Least Squares vs Ridge Regression

We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a Principal Component Analysis) and then performs an ordinary (un-regularized) least squares regression in this subspace. This note shows that the risk of this ordinary least squares method is within a constant factor (namely 4) of the risk of ridge regression.Comment: Appearing in JMLR 14, June 201