We propose the convex factorization machine (CFM), which is a convex variant of the widely used Factorization Machines (FMs). Specifically, we employ a linear+quadratic model and regularize the linear term with the $\ell_2$-regularizer and the quadratic term with the trace norm regularizer. Then, we formulate the CFM optimization as a semidefinite programming problem and propose an efficient optimization procedure with Hazan's algorithm. A key advantage of CFM over existing FMs is that it can find a globally optimal solution, while FMs may get a poor locally optimal solution since the objective function of FMs is non-convex. In addition, the proposed algorithm is simple yet effective and can be implemented easily. Finally, CFM is a general factorization method and can also be used for other factorization problems including including multi-view matrix factorization and tensor completion problems. Through synthetic and movielens datasets, we first show that the proposed CFM achieves results competitive to FMs. Furthermore, in a toxicogenomics prediction task, we show that CFM outperforms a state-of-the-art tensor factorization method.

In this paper, we provide two main contributions in PAC-Bayesian theory for domain adaptation where the objective is to learn, from a source distribution, a well-performing majority vote on a different target distribution. On the one hand, we propose an improvement of the previous approach proposed by Germain et al. (2013), that relies on a novel distribution pseudodistance based on a disagreement averaging, allowing us to derive a new tighter PAC-Bayesian domain adaptation bound for the stochastic Gibbs classifier. We specialize it to linear classifiers, and design a learning algorithm which shows interesting results on a synthetic problem and on a popular sentiment annotation task. On the other hand, we generalize these results to multisource domain adaptation allowing us to take into account different source domains. This study opens the door to tackle domain adaptation tasks by making use of all the PAC-Bayesian tools.

We propose a procedure for supervised classification that is based on potential functions. The potential of a class is defined as a kernel density estimate multiplied by the class's prior probability. The method transforms the data to a potential-potential (pot-pot) plot, where each data point is mapped to a vector of potentials. Separation of the classes, as well as classification of new data points, is performed on this plot. For this, either the $\alpha$-procedure ($\alpha$-P) or $k$-nearest neighbors ($k$-NN) are employed. For data that are generated from continuous distributions, these classifiers prove to be strongly Bayes-consistent. The potentials depend on the kernel and its bandwidth used in the density estimate. We investigate several variants of bandwidth selection, including joint and separate pre-scaling and a bandwidth regression approach. The new method is applied to benchmark data from the literature, including simulated data sets as well as 50 sets of real data. It compares favorably to known classification methods such as LDA, QDA, max kernel density estimates, $k$-NN, and $DD$-plot classification using depth functions.

The Orthogonal Least Squares (OLS) algorithm sequentially selects columns of the coefficient matrix to greedily find an approximate sparse solution to an underdetermined system of linear equations. Previous work on the analysis of OLS has been limited; in particular, there exist no guarantees on the performance of OLS for sparse linear regression from random measurements. In this paper, the problem of inferring a sparse vector from random linear combinations of its components using OLS is studied. For the noiseless scenario, it is shown that when the entries of a coefficient matrix are samples from a Gaussian or a Bernoulli distribution, OLS with high probability recovers a $k$-sparse $m$-dimensional sparse vector using ${\cal O}\left(k\log m\right)$ measurements. Similar result is established for the bounded-noise scenario where an additional condition on the smallest nonzero element of the unknown vector is required. Moreover, generalizations that reduce computational complexity of OLS and thus extend its practical feasibility are proposed. The generalized OLS algorithm is empirically shown to outperform broadly used existing algorithms in terms of accuracy, running time, or both.

These two types of patterns are complementary. In this paper, we propose a generative topic embedding model to combine the two types of patterns. In our model, topics are represented by embedding vectors, and are shared across documents. The probability of each word is influenced by both its local context and its topic. A variational inference method yields the topic embeddings as well as the topic mixing proportions for each document. Jointly they represent the document in a low-dimensional continuous space. In two document classification tasks, our method performs better than eight existing methods, with fewer features. In addition, we illustrate with an example that our method can generate coherent topics even based on only one document.

Quantitative analysis of soccer players' passing ability focuses on descriptive statistics without considering the players' real contribution to the passing and ball possession strategy of their team. Which player is able to help the build-up of an attack, or to maintain the possession of the ball? We introduce a novel methodology called QPass to answer questions like these quantitatively. Based on the analysis of an entire season, we rank the players based on the intrinsic value of their passes using QPass. We derive an album of pass trajectories for different gaming styles. Our methodology reveals a quite counterintuitive paradigm: losing the ball possession could lead to better chances to win a game.

We study the problem of learning a tree graphical model from samples such that low-order marginals are accurate. We define a distance ("small set TV" or ssTV) between distributions $P$ and $Q$ by taking the maximum, over all subsets $\mathcal{S}$ of a given size, of the total variation between the marginals of P and Q on $\mathcal{S}$. Approximating a distribution to within small ssTV allows making predictions based on partial observations. Focusing on pairwise marginals and tree-structured Ising models on $p$ nodes with maximum edge strength $\beta$, we prove that $\max\{e^{2\beta}\log p, \eta^{-2}\log(p/\eta)\} $ i.i.d. samples suffices to get a distribution (from the same class) with ssTV at most $\eta$ from the one generating the samples.

Recently, a general method for analyzing the statistical accuracy of the EM algorithm has been developed and applied to some simple latent variable models [Balakrishnan et al. 2016]. In that method, the basin of attraction for valid initialization is required to be a ball around the truth. Using Stein's Lemma, we extend these results in the case of estimating the centers of a two-component Gaussian mixture in $d$ dimensions. In particular, we significantly expand the basin of attraction to be the intersection of a half space and a ball around the origin. If the signal-to-noise ratio is at least a constant multiple of $ \sqrt{d\log d} $, we show that a random initialization strategy is feasible.

Stochastic gradient Markov chain Monte Carlo (SG-MCMC) methods are Bayesian analogs to popular stochastic optimization methods; however, this connection is not well studied. We explore this relationship by applying simulated annealing to an SGMCMC algorithm. Furthermore, we extend recent SG-MCMC methods with two key components: i) adaptive preconditioners (as in ADAgrad or RMSprop), and ii) adaptive element-wise momentum weights. The zero-temperature limit gives a novel stochastic optimization method with adaptive element-wise momentum weights, while conventional optimization methods only have a shared, static momentum weight. Under certain assumptions, our theoretical analysis suggests the proposed simulated annealing approach converges close to the global optima. Experiments on several deep neural network models show state-of-the-art results compared to related stochastic optimization algorithms.

In this paper, we investigate a divide and conquer approach to Kernel Ridge Regression (KRR). Given n samples, the division step involves separating the points based on some underlying disjoint partition of the input space (possibly via clustering), and then computing a KRR estimate for each partition. The conquering step is simple: for each partition, we only consider its own local estimate for prediction. We establish conditions under which we can give generalization bounds for this estimator, as well as achieve optimal minimax rates. We also show that the approximation error component of the generalization error is lesser than when a single KRR estimate is fit on the data: thus providing both statistical and computational advantages over a single KRR estimate over the entire data (or an averaging over random partitions as in other recent work, [30]). Lastly, we provide experimental validation for our proposed estimator and our assumptions.