This paper introduces the notion of exact common information, which is the minimum description length of the common randomness needed for the exact distributed generation of two correlated random variables (X, Y ). We introduce the quantity G(X; Y ) = minX→W →Y H(W ) as a natural bound on the exact common information and study its properties and computation. We then introduce the exact common information rate, which is the minimum description rate of the common randomness for the exact generation of a 2-DMS (X, Y ). We give a multiletter characterization for it as the limit G(X; Y ) = limn→∞(1/n)G(X n ; Y n ). While in general G(X; Y ) is greater than or equal to the Wyner common information, we show that they are equal for the Symmetric Binary Erasure Source. We do not know, however, if the exact common information rate has a single letter characterization in general.

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We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let X n be i.i.d. Bernoulli(1/2), and let Y n be the result of passing X n through a memoryless binary symmetric channel with crossover probability α. For any Boolean function b : {0, 1} n → {0, 1}, we conjecture that H(α). While the conjecture remains open, we provide substantial evidence supporting its validity.

In online shopping, users usually express their intent through search queries. However, these queries are often ambiguous. For example, it is more likely (and easier) for users to write a query like "high-end bike" than "21 speed carbon frames jamis or giant road bike". It is challenging to interpret these ambiguous queries and thus search result accuracy suffers. A user oftentimes needs to go through the frustrating process of refining search queries or self-teaching from possibly unstructured information. However, shopping is indeed a structured domain, that is composed of category hierarchy, brands, product lines, features, etc. It would be much better if a shopping site could understand users' intent through this structure, present organized information, and then find the items with the right categories, brands or features.In this paper we study the problem of inferring the latent intent from unstructured queries and mapping them to structured attributes. We present a novel framework that jointly learns this knowledge from user consumption behaviors and product metadata. We present a hybrid Long Shortterm Memory (LSTM) [10] joint model that is accurate and robust, even though user queries are noisy and product catalog is rapidly growing. Our study is conducted on a largescale dataset from Google Shopping, that is composed of millions of items and user queries along with their click responses. Extensive qualitative and quantitative evaluation shows that the proposed model is more accurate, concise, and robust than multiple possible alternatives. In terms of information retrieval (IR) performance, our model is able to improve the quality of current Google Shopping production system, which is a very strong baseline.

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