Learning Mixtures of Gaussians in High Dimensions

Efficiently learning mixture of Gaussians is a fundamental problem in statistics and learning theory. Given samples coming from a random one out of k Gaussian distributions in Rn, the learning problem asks to estimate the means and the covariance matrices of these Gaussians. This learning problem arises in many areas ranging from the natural sciences to the social sciences, and has also found many machine learning applications. Unfortunately, learning mixture of Gaussians is an information theoretically hard problem: in order to learn the parameters up to a reasonable accuracy, the number of samples required is exponential in the number of Gaussian components in the worst case. In this work, we show that provided we are in high enough dimensions, the class of Gaussian mixtures is learnable in its most general form under a smoothed analysis framework, where the parameters are randomly perturbed from an adversarial starting point. In particular, given samples from a mixture of Gaussians with randomly perturbed parameters, when n > {\Omega}(k^2), we give an algorithm that learns the parameters with polynomial running time and using polynomial number of samples. The central algorithmic ideas consist of new ways to decompose the moment tensor of the Gaussian mixture by exploiting its structural properties. The symmetries of this tensor are derived from the combinatorial structure of higher order moments of Gaussian distributions (sometimes referred to as Isserlis' theorem or Wick's theorem). We also develop new tools for bounding smallest singular values of structured random matrices, which could be useful in other smoothed analysis settings

Brief Studies

In spite of many fine Bible translations the pastor will constantly turn to his Greek New Testament, for no translation can reproduce fully the many fine shades of meaning in the original. A careful study of the original will frequently bring the exact meaning of a text or a word into sharper focus, change wholly or in part the meaning conveyed in the translation, or may even correct a misleading and inadequate translation. A few samples chosen at random will illustrate this

Momentum and Mass Fluxes in a Gas Confined between Periodically Structured Surfaces at Different Temperatures

It is well known that in a gas-filled duct or channel along which a temperature gradient is applied, a thermal creep flow is created. Here we show that a mass and momentum flux can also be induced in a gas confined between two parallel structured surfaces at different temperatures, i.e. \textit{orthogonal} to the temperature gradient. We use both analytical and numerical methods to compute the resulting fluxes. The momentum flux assumes its maximum value in the free-molecular flow regime, the (normalized) mass flux in the transition flow regime. The discovered phenomena could find applications in novel methods for energy-conversion and thermal pumping of gases.Comment: 6 pages, 5 figures, updated fig.5, updated text for the numerical metho

Biased Information Search in Homogeneous Groups: Confidence as a Moderator for the Effect of Anticipated Task Requirements

When searching for information, groups that are homogeneous regarding their members’ prediscussion decision preferences show a strong bias for information that supports rather than conflicts with the prevailing opinion (confirmation bias). The present research examined whether homogeneous groups blindly search for information confirming their beliefs irrespective of the anticipated task or whether they are sensitive to the usefulness of new information for this forthcoming task. Results of three experiments show that task sensitivity depends on the groups’ confidence in the correctness of their decision: Moderately confident groups displayed a strong confirmation bias when they anticipated having to give reasons for their decision but showed a balanced information search or even a disconfirmation bias (i.e., predominately seeking conflicting information) when they anticipated having to refute unterarguments. In contrast, highly confident groups demonstrated a strong confirmation bias independent of the anticipated task requirements

Language Barriers in Health Care Settings: An Annotated Bibliography of Research Literature

Provides an overview of resources related to the prevalence, role, and effects of language barriers and access in health care

Paradoxes in Fair Computer-Aided Decision Making

Computer-aided decision making--where a human decision-maker is aided by a computational classifier in making a decision--is becoming increasingly prevalent. For instance, judges in at least nine states make use of algorithmic tools meant to determine "recidivism risk scores" for criminal defendants in sentencing, parole, or bail decisions. A subject of much recent debate is whether such algorithmic tools are "fair" in the sense that they do not discriminate against certain groups (e.g., races) of people. Our main result shows that for "non-trivial" computer-aided decision making, either the classifier must be discriminatory, or a rational decision-maker using the output of the classifier is forced to be discriminatory. We further provide a complete characterization of situations where fair computer-aided decision making is possible