Rank aggregation systems collect ordinal preferences from individuals to produce a global ranking that represents the social preference. Rank-breaking is a common practice to reduce the computational complexity of learning the global ranking. The individual preferences are broken into pairwise comparisons and applied to efficient algorithms tailored for independent paired comparisons. However, due to the ignored dependencies in the data, naive rank-breaking approaches can result in inconsistent estimates. The key idea to produce accurate and consistent estimates is to treat the pairwise comparisons unequally, depending on the topology of the collected data. In this paper, we provide the optimal rank-breaking estimator, which not only achieves consistency but also achieves the best error bound. This allows us to characterize the fundamental tradeoff between accuracy and complexity. Further, the analysis identifies how the accuracy depends on the spectral gap of a corresponding comparison graph.

Commonly used in computer vision and other applications, robust PCA represents an algorithmic attempt to reduce the sensitivity of classical PCA to outliers. The basic idea is to learn a decomposition of some data matrix of interest into low rank and sparse components, the latter representing unwanted outliers. Although the resulting optimization problem is typically NP-hard, convex relaxations provide a computationally-expedient alternative with theoretical support. However, in practical regimes performance guarantees break down and a variety of non-convex alternatives, including Bayesian-inspired models, have been proposed to boost estimation quality. Unfortunately though, without additional a priori knowledge none of these methods can significantly expand the critical operational range such that exact principal subspace recovery is possible. Into this mix we propose a novel pseudo-Bayesian algorithm that explicitly compensates for design weaknesses in many existing non-convex approaches leading to state-of-the-art performance with a sound analytical foundation. Surprisingly, our algorithm can even outperform convex matrix completion despite the fact that the latter is provided with perfect knowledge of which entries are not corrupted.

While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool for reasoning about computer programs, we consider this problem from the perspective of integrating probability and logic. Recent work on assigning probabilities to mathematical statements has used the concept of coherent distributions, which satisfy logical constraints such as the probability of a sentence and its negation summing to one. Although there are algorithms which converge to a coherent probability distribution in the limit, this yields only weak guarantees about finite approximations of these distributions. In our setting, this is a significant limitation: Coherent distributions assign probability one to all statements provable in a specific logical theory, such as Peano Arithmetic, which can prove what the output of any terminating computation is; thus, a coherent distribution must assign probability one to the output of any terminating computation. To model uncertainty about computations, we propose to work with approximations to coherent distributions. We introduce inductive coherence, a strengthening of coherence that provides appropriate constraints on finite approximations, and propose an algorithm which satisfies this criterion.

How you see the world may be very different than how someone else sees it. And recognizing that has long been thought to be a uniquely human ability. But when it comes to understanding others' perspectives, humans might not be alone. "Reading others' mind is not our special skill," says Fumihiro Kano, a comparative psychologist at Kyoto University in Japan. Nonhuman apes can do it, too, according to Dr. Kano's research, published Thursday in the journal Science, a finding that could further blur the line between the cognitive capacities of humans and nonhuman apes.

A strange new video will show you that apes might be a lot more perceptive than you thought. The video, part of a new study, shows chimpanzees, bonobos, and orangutans are able to understand people's beliefs, desires and intentions – a phenomenon that was previously considered to be unique to humans. The findings, which challenge our understanding of how intelligent apes are, were presented in a hilarious video. The video, part of a new study, shows chimpanzees, bonobos, and orangutans are able to understand people's beliefs, desires and intentions – a phenomenon that was previously considered to be unique to humans. The apes watched as a human witnessed an object being hidden in one location.

The sunny beaches of summer are already losing their popularity, as increasingly chilly waters drive vacationers to less frigid pursuits. Cold water has long challenged humans, who have no natural defenses against a wintry marine environment. Beavers and sea otters, on the other hand, thrive in cold water, despite lacking the insulating blubber that protects other marine mammals. The secret is in their fur, where warm air is trapped among the hairs in their thick pelts, keeping them warm even as they dodge ice floes. Researchers at Massachusetts Institute of Technology in Cambridge, inspired by this evolutionary strategy, have created synthetic pelts modeled after the mechanism through which beavers warm themselves.

Almost every machine learning algorithm comes with a large number of settings that we, the machine learning researchers and practitioners, need to specify. These tuning knobs, the so-called hyperparameters, help us control the behavior of machine learning algorithms when optimizing for performance, finding the right balance between bias and variance. Hyperparameter tuning for performance optimization is an art in itself, and there are no hard-and-fast rules that guarantee best performance on a given dataset. In Part I and Part II, we saw different holdout and bootstrap techniques for estimating the generalization performance of a model. We learned about the bias-variance trade-off, and we computed the uncertainty of our estimates. In this third part, we will focus on different methods of cross-validation for model evaluation and model selection. We will use these cross-validation techniques to rank models from several hyperparameter configurations and estimate how well they generalize to independent datasets. Previously, we used the holdout method or different flavors of bootstrapping to estimate the generalization performance of our predictive models.

Nasa has announced that it has found evidence of flowing water on Mars. Scientists have long speculated that Recurring Slope Lineae -- or dark patches -- on Mars were made up of briny water but the new findings prove that those patches are caused by liquid water, which it has established by finding hydrated salts. Several hundred camped outside the London store in Covent Garden. The 6s will have new features like a vastly improved camera and a pressure-sensitive "3D Touch" display

Nasa has announced that it has found evidence of flowing water on Mars. Scientists have long speculated that Recurring Slope Lineae -- or dark patches -- on Mars were made up of briny water but the new findings prove that those patches are caused by liquid water, which it has established by finding hydrated salts. Several hundred camped outside the London store in Covent Garden. The 6s will have new features like a vastly improved camera and a pressure-sensitive "3D Touch" display

A team including CNRS astrophysicists have calculated the size and surface properties of the planet dubbed Proxima b, and concluded it may be an'ocean planet' similar to Earth. Scientists announced Proxima b's discovery in August, and said it may be the first exoplanet--planet outside our Solar System--to one day be visited by robots from Earth. A team including CNRS astrophysicists have calculated the size and surface properties of the planet dubbed Proxima b, and concluded it may be an'ocean planet' similar to Earth. It is estimated to have a mass about 1.3 times that of Earth, and orbits about 7.5 million kilometres (4.6 million miles) from its star--about a tenth the distance of innermost planet Mercury from the Sun. 'Contrary to what one might expect, such proximity does not necessarily mean that Proxima b's surface is too hot' for water to exist in liquid form, said a CNRS statement.