Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in practice. We prove that the commonly used non-convex objective function for matrix completion has no spurious local minima --- all local minima must also be global. Therefore, many popular optimization algorithms such as (stochastic) gradient descent can provably solve matrix completion with \textit{arbitrary} initialization in polynomial time.

We introduce the framework of {\em blind regression} motivated by {\em matrix completion} for recommendation systems: given $m$ users, $n$ movies, and a subset of user-movie ratings, the goal is to predict the unobserved user-movie ratings given the data, i.e., to complete the partially observed matrix. Following the framework of non-parametric statistics, we posit that user $u$ and movie $i$ have features $x_1(u)$ and $x_2(i)$ respectively, and their corresponding rating $y(u,i)$ is a noisy measurement of $f(x_1(u), x_2(i))$ for some unknown function $f$. In contrast with classical regression, the features $x = (x_1(u), x_2(i))$ are not observed, making it challenging to apply standard regression methods to predict the unobserved ratings. Inspired by the classical Taylor's expansion for differentiable functions, we provide a prediction algorithm that is consistent for all Lipschitz functions. In fact, the analysis through our framework naturally leads to a variant of collaborative filtering, shedding insight into the widespread success of collaborative filtering in practice. Assuming each entry is sampled independently with probability at least $\max(m^{-1+\delta},n^{-1/2+\delta})$ with $\delta > 0$, we prove that the expected fraction of our estimates with error greater than $\epsilon$ is less than $\gamma^2 / \epsilon^2$ plus a polynomially decaying term, where $\gamma^2$ is the variance of the additive entry-wise noise term. Experiments with the MovieLens and Netflix datasets suggest that our algorithm provides principled improvements over basic collaborative filtering and is competitive with matrix factorization methods.

Word embeddings are a powerful approach to capturing semantic similarity among terms in a vocabulary. In this paper, we develop exponential family embeddings, which extends the idea of word embeddings to other types of high-dimensional data. As examples, we studied several types of data: neural data with real-valued observations, count data from a market basket analysis, and ratings data from a movie recommendation system. The main idea is that each observation is modeled conditioned on a set of latent embeddings and other observations, called the context, where the way the context is defined depends on the problem. In language the context is the surrounding words; in neuroscience the context is close-by neurons; in market basket data the context is other items in the shopping cart. Each instance of an embedding defines the context, the exponential family of conditional distributions, and how the embedding vectors are shared across data. We infer the embeddings with stochastic gradient descent, with an algorithm that connects closely to generalized linear models. On all three of our applications--neural activity of zebrafish, users' shopping behavior, and movie ratings--we found that exponential family embedding models are more effective than other dimension reduction methods. They better reconstruct held-out data and find interesting qualitative structure.

In this paper, we introduce the public-private framework of data summarization motivated by privacy concerns in personalized recommender systems and online social services. Such systems have usually access to massive data generated by a large pool of users. A major fraction of the data is public and is visible to (and can be used for) all users. However, each user can also contribute some private data that should not be shared with other users to ensure her privacy. The goal is to provide a succinct summary of massive dataset, ideally as small as possible, from which customized summaries can be built for each user, i.e. it can contain elements from the public data (for diversity) and users' private data (for personalization). To formalize the above challenge, we assume that the scoring function according to which a user evaluates the utility of her summary satisfies submodularity, a widely used notion in data summarization applications.

Hybrid methods that utilize both content and rating information are commonly used in many recommender systems. However, most of them use either handcrafted features or the bag-of-words representation as a surrogate for the content information but they are neither effective nor natural enough. To address this problem, we develop a collaborative recurrent autoencoder (CRAE) which is a denoising recurrent autoencoder (DRAE) that models the generation of content sequences in the collaborative filtering (CF) setting.

A reciprocal recommendation problem is one where the goal of learning is not just to predict a user's preference towards a passive item (e.g., a book), but to recommend the targeted user on one side another user from the other side such that a mutual interest between the two exists. The problem thus is sharply different from the more traditional items-to-users recommendation, since a good match requires meeting the preferences of both users. We initiate a rigorous theoretical investigation of the reciprocal recommendation task in a specific framework of sequential learning. We point out general limitations, formulate reasonable assumptions enabling effective learning and, under these assumptions, we design and analyze a computationally efficient algorithm that uncovers mutual likes at a pace comparable to those achieved by a clairvoyant algorithm knowing all user preferences in advance. Finally, we validate our algorithm against synthetic and real-world datasets, showing improved empirical performance over simple baselines.

Implicit feedback is widely used in collaborative filtering methods for recommendation. It is well known that implicit feedback contains a large number of values that are \emph{missing not at random} (MNAR); and the missing data is a mixture of negative and unknown feedback, making it difficult to learn user's negative preferences. Recent studies modeled \emph{exposure}, a latent missingness variable which indicates whether an item is missing to a user, to give each missing entry a confidence of being negative feedback. However, these studies use static models and ignore the information in temporal dependencies among items, which seems to be a essential underlying factor to subsequent missingness. To model and exploit the dynamics of missingness, we propose a latent variable named ``\emph{user intent}'' to govern the temporal changes of item missingness, and a hidden Markov model to represent such a process. The resulting framework captures the dynamic item missingness and incorporate it into matrix factorization (MF) for recommendation. We also explore two types of constraints to achieve a more compact and interpretable representation of \emph{user intents}. Experiments on real-world datasets demonstrate the superiority of our method against state-of-the-art recommender systems.

A common assumption in recommender systems (RS) is the existence of a best fixed recommendation strategy. Such strategy may be simple and work at the item level (e.g., in multi-armed bandit it is assumed one best fixed arm/item exists) or implement more sophisticated RS (e.g., the objective of A/B testing is to find the best fixed RS and execute it thereafter). We argue that this assumption is rarely verified in practice, as the recommendation process itself may impact the user's preferences. For instance, a user may get bored by a strategy, while she may gain interest again, if enough time passed since the last time that strategy was used. In this case, a better approach consists in alternating different solutions at the right frequency to fully exploit their potential. In this paper, we first cast the problem as a Markov decision process, where the rewards are a linear function of the recent history of actions, and we show that a policy considering the long-term influence of the recommendations may outperform both fixed-action and contextual greedy policies. We then introduce an extension of the UCRL algorithm ( L IN UCRL) to effectively balance exploration and exploitation in an unknown environment, and we derive a regret bound that is independent of the number of states.

Conniving couple whose greedy'pervert' plot'drove innocent disabled man to suicide' given stunningly short sentences Obama Center asks for 100 unpaid volunteers despite hiring the former president's'close friend' as CEO on $740K Harry Styles shares gay kiss with SNL star in wild opening monologue as he addresses'queerbaiting' claims Housing nightmare in America's'best state to buy a home' as banks suddenly seize thousands of properties Insufferable blowhard Stephen Colbert is being taken out like the trash... and thank God! What he's done is so diabolical: MAUREEN CALLAHAN JFK Jr's mortifying night of phone sex... day Sarah Jessica Parker ditched her underwear to seduce him in public... and the girlfriend he REALLY wanted to marry: All the women before Carolyn Mass cancellations as Southwest Airlines pulls out of two of America's biggest airports Truth about'super secretive' Michael B. Jordan's love life... and real reason he is perpetually single: Years of private'heartache' and'loneliness' laid bare Beloved young dad and inspiring female'Air Force superstar' among US heroes killed in Iran mission crash as all six are named Insane moment NYC cab plows into pedestrians... and the miracle that saved them from death Caitlin Clark goes viral for bizarre behavior after Team USA's win over Italy: 'What are you doing? We fled Trump to chase the REAL American dream in the most idyllic European hotspot... here's why we're coming back to a red state I looked like a monster after a car accident burned off my face... but a pioneering face transplant gave me my life back. Furious flower farm owner blasts'feral' customers after they trampled tulips to get perfect photos Harry and Meghan hit back at new book claims she was accused by Camilla of'brainwashing' him - dismissing accusations as'deranged conspiracy' Extramarital sex with witches, cursed bloodlines and possessed politicians: DC's chief exorcist reveals the potent stench of evil among America's elite Iran's deadly drone arsenal is a'wake-up call for America': Expert warns US defenses may be unprepared for swarm attacks Iran's foreign minister admits Islamic Republic is receiving military support from Russia and China Hollywood costume designer names VILE A-Listers including pervert James Bond star, slob female sitcom icon... and details the hilarious evil of Shannen Doherty Scientists reveal exactly where you're going WRONG on your dating profile - and the simple changes you can make to bag a date Are you struggling to find love online? Scientists have revealed where you're probably going wrong. A new study has uncovered nine different types of profile photos that singletons tend to use - and what they convey about you to fellow daters.

The company hopes to use its Bee chatbot to connect compatible users without the need for swipes. Bumble is testing an AI dating assistant called Bee that it hopes will get users on dates without them having to swipe through profiles, writes . The company announced the AI assistant during its fourth quarter earnings, and intends to use the AI in a new experience it calls Dates. When a user opts in to Bumble's Dates feature, Bee performs an onboarding chat where it learns about the users' values, relationship goals, communications style, lifestyle and dating intentions, and then attempts to find other users who share some or all of those traits. Once Bee finds someone compatible, both users are notified in the app that they could be a great match, and receive a summary generated by Bee explaining why.