In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Input: Consider Input: Independent Component Analysis (ICA) is the problem of learning a square matrix A, given samples of X = AS, where S is a random vector with independent coordinates. Most existing algorithms are provably efficient only when each Si has finite and moderately valued fourth moment. However, there are practical applications where this assumption need not be true, such as speech and finance. Algorithms have been proposed for heavy-tailed ICA, but they are not practical, using random walks and the full power of the ellipsoid algorithm multiple times. The main contributions of this paper are: (1) A practical algorithm for heavy-tailed ICA that we call HTICA. We provide theoretical guarantees and show that it outperforms other algorithms in some heavy-tailed regimes, both on real and synthetic data. Like the current state-of-the-art, the new algorithm is based on the centroid body (a first moment analogue of the covariance matrix). Unlike the state-of-the-art, our algorithm is practically efficient. To achieve this, we use explicit analytic representations of the centroid body, which bypasses the use of the ellipsoid method and random walks. (2) We study how heavy tails affect different ICA algorithms, including HTICA. Somewhat surprisingly, we show that some algorithms that use the covariance matrix or higher moments can successfully solve a range of ICA instances with infinite second moment. We study this theoretically and experimentally, with both synthetic and real-world heavy-tailed data.

Output: Heavy-Tailed Analogues of the Covariance Matrix for ICA

Input: Consider Input: In classical machine learning, regression is treated as a black box process of identifying a suitable function from a hypothesis set without attempting to gain insight into the mechanism connecting inputs and outputs. In the natural sciences, however, finding an interpretable function for a phenomenon is the prime goal as it allows to understand and generalize results. This paper proposes a novel type of function learning network, called equation learner (EQL), that can learn analytical expressions and is able to extrapolate to unseen domains. It is implemented as an end-to-end differentiable feed-forward network and allows for efficient gradient based training. Due to sparsity regularization concise interpretable expressions can be obtained. Often the true underlying source expression is identified. INTRODUCTION The quality of a model is typically measured by its ability to generalize from a training set to previously unseen data from the same distribution. In regression tasks generalization essentially boils down to interpolation if the training data is sufficiently dense. As long as models are selected correctly, i. e. in a way to not overfit the data, the regression problem is well understood and can at least conceptually be considered solved. However, when working with data from real-world devices, e. g. controlling a robotic arm, interpolation might not be sufficient. It could happen that future data lies outside of the training domain, e. g. when the arm is temporarily operated outside of its specifications. For the sake of robustness and safety it is desirable in such a case to have a regression model that continues to make good predictions, or at least does not fail catastrophically. This setting, which we call extrapolation generalization, is the topic of the present paper. We are particularly interested in regression tasks for systems that can be described by real-valued analytic expression, e. g. mechanical systems such as a pendulum or a robotic arm. These are typically governed by a highly nonlinear function but it is nevertheless possible, in principle, to infer their behavior on an extrapolation domain from their behavior elsewhere. We make two main contributions: 1) a new type of network that can learn analytical expressions and is able to extrapolate to unseen domains and 2) a model selection strategy tailored to the extrapolation setting. The following section describes the setting of regression and extrapolation. Afterwards we introduce our method and discuss the architecture, its training, and its relation to prior art. We present our results in the Section Experimental evaluation and close with conclusions. REGRESSION AND EXTRAPOLATION We consider a multivariate regression problem with a training set {(x1, y1), . . . , (xN , yN )} with x R, y R. Because our main interest lies on extrapolation in the context of learning the dynamics of physical systems we assume the data originates from an unknown analytical function (or system of functions), : R R with additive zero-mean noise, , i. e. y = (x) + and E = 0. The function may, for instance, reflect a system of ordinary differential equations that govern the movements of a robot arm or the like. The general task is to learn a function : R R that approximates the true functional relation as well as possible in the squared loss sense, i. e. achieves minimal expected error E(x) (x)2. In practice, we only have particular examples of the function values available and measure the quality of predicting in terms of the empirical error on

Output: EXTRAPOLATION AND LEARNING EQUATIONS

Input: Consider Input: Pre-trained word embeddings learned from unlabeled text have become a standard component of neural network architectures for NLP tasks. However, in most cases, the recurrent network that operates on word-level representations to produce context sensitive representations is trained on relatively little labeled data. In this paper, we demonstrate a general semi-supervised approach for adding pretrained context embeddings from bidirectional language models to NLP systems and apply it to sequence labeling tasks. We evaluate our model on two standard datasets for named entity recognition (NER) and chunking, and in both cases achieve state of the art results, surpassing previous systems that use other forms of transfer or joint learning with additional labeled data and task specific gazetteers.
Output: Semi-supervised sequence tagging with bidirectional language models