Definition: 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: <lb>Nonlinear component analysis such as kernel Principle Component Analysis (KPCA) and kernel<lb>Canonical Correlation Analysis (KCCA) are widely used in machine learning, statistics and data analysis,<lb>but they can not scale up to big datasets. Recent attempts have employed random feature approximations<lb>to convert the problem to the primal form for linear computational complexity. However, to obtain high<lb>quality solutions, the number of random features should be the same order of magnitude as the number<lb>of data points, making such approach not directly applicable to the regime with millions of data points.<lb>We propose a simple, computationally efficient, and memory friendly algorithm based on the “doubly<lb>stochastic gradients” to scale up a range of kernel nonlinear component analysis, such as kernel PCA, CCA<lb>and SVD. Despite the non-convex nature of these problems, our method enjoys theoretical guarantees<lb>that it converges at the rate Õ(1/t) to the global optimum, even for the top k eigen subspace. Unlike<lb>many alternatives, our algorithm does not require explicit orthogonalization, which is infeasible on big<lb>datasets. We demonstrate the effectiveness and scalability of our algorithm on large scale synthetic and<lb>real world datasets.

Scale Up Nonlinear Component Analysis with Doubly Stochastic Gradients