TASK 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.
PROBLEM: The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections the computational bottleneck in many applications replacing it by a linear optimization step. Despite this advantage, the known convergence rates of the FW method fall behind standard first order methods for most settings of interest. It is an active line of research to derive faster linear optimization-based algorithms for various settings of convex optimization. In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of 1 t2 . This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order 1t , and known to be tight. We show that various balls induced by `p norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. We further show how several previous fastrate results for the FW method follow easily from our analysis.

SOLUTION: Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets

PROBLEM: The paper addresses the problem of learning a regression model parameterized by a fixedrank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks.

SOLUTION: Regression on fixed-rank positive semidefinite matrices: a Riemannian approach

PROBLEM: Nowadays, geographic information related to Twitter is crucially important for fine-grained applications. However, the amount of geographic information available on Twitter is low, which makes the pursuit of many applications challenging. Under such circumstances, estimating the location of a tweet is an important goal of the study. Unlike most previous studies that estimate the pre-defined district as the classification task, this study employs a probability distribution to represent richer information of the tweet, not only the location but also its ambiguity. To realize this modeling, we propose the convolutional mixture density network (CMDN), which uses text data to estimate the mixture model parameters. Experimentally obtained results reveal that CMDN achieved the highest prediction performance among the method for predicting the exact coordinates. It also provides a quantitative representation of the location ambiguity for each tweet that properly works for extracting the reliable location estimations.

SOLUTION:
Density Estimation for Geolocation via Convolutional Mixture Density Network