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>In applications such as recommendation systems and revenue management, it is important to<lb>predict preferences on items that have not been seen by a user or predict outcomes of comparisons<lb>among those that have never been compared. A popular discrete choice model of multinomial<lb>logit model captures the structure of the hidden preferences with a low-rank matrix. In order to<lb>predict the preferences, we want to learn the underlying model from noisy observations of the<lb>low-rank matrix, collected as revealed preferences in various forms of ordinal data. A natural<lb>approach to learn such a model is to solve a convex relaxation of nuclear norm minimization.<lb>We present the convex relaxation approach in two contexts of interest: collaborative ranking<lb>and bundled choice modeling. In both cases, we show that the convex relaxation is minimax<lb>optimal. We prove an upper bound on the resulting error with finite samples, and provide a<lb>matching information-theoretic lower bound.

Collaboratively Learning Preferences from Ordinal Data