We consider an online model for recommendation systems, with each user being recommended an item at each time-step and providing ’like’ or ’dislike’ feedback. A latent variable model specifies the user preferences: both users and items are clustered into types. The model captures structure in both the item and user spaces, as used by item-item and user-user collaborative filtering algorithms. We study the situation in which the type preference matrix has i.i.d. entries. Our main contribution is an algorithm that simultaneously uses both item and user structures, proved to be near-optimal via corresponding information-theoretic lower bounds. In particular, our analysis highlights the sub-optimality of using only one of item or user structure (as is done in most collaborative filtering algorithms).

For different parameterizations (mappings from parameters to predictors), we study the regularization cost in predictor space induced by $l_2$ regularization on the parameters (weights). We focus on linear neural networks as parameterizations of linear predictors. We identify the representation cost of certain sparse linear ConvNets and residual networks. In order to get a better understanding of how the architecture and parameterization affect the representation cost, we also study the reverse problem, identifying which regularizers on linear predictors (e.g., $l_p$ norms, group norms, the $k$-support-norm, elastic net) can be the representation cost induced by simple $l_2$ regularization, and designing the parameterizations that do so.

Graphical models are often used to facilitate efficient computation of posteriors in order to make predictions. With this objective in mind, we consider the problem of estimating the parameters of a graphical model with known structure from samples such that posteriors computed using the model are accurate. Focusing on tree-structured binary Markov random fields, our main result is a sharp characterization of the dependence on number of samples needed for all pairwise marginals (and hence posteriors of one variable given another) to be accurate: n = Θ(η-2log p) samples are necessary and sufficient to estimate model parameters such that all marginals of arbitrary order k are accurate to within √kη. The result implies that prediction error is bounded uniformly, with no dependence on the strength of interactions. We will also show that these guarantees are achievable using moment matching techniques.

The possibility of latency arbitrage in financial markets has led to the deployment of high-speed communication links between distant financial centers. These links are noisy and so there is a need for coding. In this paper, we develop a game-theoretic model of trading behavior where two traders compete to capture latency arbitrage opportunities using binary signalling. Different coding schemes are strategies that trade off between reliability and latency. When one trader has a better channel, the second trader should not compete. With statistically identical channels, we find there are two different regimes of channel noise for which: there is a unique Nash equilibrium yielding ties; and there are two Nash equilibria with different winners.

Color image demosaicking is a key process in the digital imaging pipeline. In this paper, we study a well-known and influential demosaicking algorithm based upon alternating projections (AP), proposed by Gunturk, Altunbasak and Mersereau in 2002. Since its publication, the AP algorithm has been widely cited and compared against in a series of more recent papers in the demosaicking literature. Despite good performances, a limitation of the AP algorithm is its high computational complexity. We provide three main contributions in this paper. First, we present a rigorous analysis of the convergence property of the AP demosaicking algorithm, showing that it is a contraction mapping, with a unique fixed point. Second, we show that this fixed point is in fact the solution to a constrained quadratic minimization problem, thus, establishing the optimality of the AP algorithm. Finally, using the tool of polyphase representation, we show how to obtain the results of the AP algorithm in a single step, implemented as linear filtering in the polyphase domain. Replacing the original iterative procedure by the proposed one-step solution leads to substantial computational savings, by about an order of magnitude in our experiments.