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.

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.
Heavy-Tailed Analogues of the Covariance Matrix for ICA

<lb>We consider a situation in which we see samples Xn ∈ R drawn i.i.d. from some<lb>distribution with mean zero and unknown covariance A. We wish to compute the<lb>top eigenvector of A in an incremental fashion with an algorithm that maintains<lb>an estimate of the top eigenvector in O(d) space, and incrementally adjusts the<lb>estimate with each new data point that arrives. Two classical such schemes are<lb>due to Krasulina (1969) and Oja (1983). We give finite-sample convergence rates<lb>for both.
The Fast Convergence of Incremental PCA

In a controlled experiment of sequence-tosequence approaches for the task of sentence correction, we find that characterbased models are generally more effective than word-based models and models that encode subword information via convolutions, and that modeling the output data as a series of diffs improves effectiveness over standard approaches. Our strongest sequence-to-sequence model improves over our strongest phrase-based statistical machine translation model, with access to the same data, by 6 M2 (0.5 GLEU) points. Additionally, in the data environment of the standard CoNLL-2014 setup, we demonstrate that modeling (and tuning against) diffs yields similar or better M2 scores with simpler models and/or significantly less data than previous sequence-to-sequence approaches.

Adapting Sequence Models for Sentence Correction