The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using $\Theta(n^2)$ variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans (2010), we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to $\Theta(n \log n)$ in theory and $\Theta(n \log^2 n)$ in practice. We modify the recent convex formulation of the 2-SUM problem introduced by Fogel et al. (2013) to use this polytope, and demonstrate how we can attain results of similar quality in significantly less computational time for large $n$. To our knowledge, this is the first usage of Goemans' compact formulation of the permutahedron in a convex optimization problem. We also introduce a simpler regularization scheme for this convex formulation of the 2-SUM problem that yields good empirical results.

We propose a class of closed-form estimators for sparsity-structured graphical models, expressed as exponential family distributions, under high-dimensional settings. Our approach builds on observing the precise manner in which the classical graphical model MLE ``breaks down'' under high-dimensional settings. Our estimator uses a carefully constructed, well-defined and closed-form backward map, and then performs thresholding operations to ensure the desired sparsity structure. We provide a rigorous statistical analysis that shows that surprisingly our simple class of estimators recovers the same asymptotic convergence rates as those of the $\ell_1$-regularized MLEs that are much more difficult to compute. We corroborate this statistical performance, as well as significant computational advantages via simulations of both discrete and Gaussian graphical models.

Patch-based methods have been widely used for noise reduction in recent years. In this paper, we propose a general statistical aggregation method which combines image patches denoised with several commonly-used algorithms. We show that weakly denoised versions of the input image obtained with standard methods, can serve to compute an efficient patch-based aggregated estimator. In our approach, we evaluate the Stein's Unbiased Risk Estimator (SURE) of each denoised candidate imagepatch and use this information to compute the exponential weighted aggregation (EWA) estimator. The aggregation method is flexible enough to combine anystandard denoising algorithm and has an interpretation with Gibbs distribution. Thedenoising algorithm (PEWA) is based on a MCMC sampling and is able to produce results that are comparable to the current state-of-the-art.

Deep learning has been successfully applied to multimodal representation learning problems, with a common strategy to learning joint representations that are shared across multiple modalities on top of layers of modality-specific networks. Nonetheless, there still remains a question how to learn a good association between data modalities; in particular, a good generative model of multimodal data should be able to reason about missing data modality given the rest of data modalities. In this paper, we propose a novel multimodal representation learning framework that explicitly aims this goal. Rather than learning with maximum likelihood, we train the model to minimize the variation of information. We provide a theoretical insight why the proposed learning objective is sufficient to estimate the data-generating joint distribution of multimodal data. We apply our method to restricted Boltzmann machines and introduce learning methods based on contrastive divergence and multi-prediction training. In addition, we extend to deep networks with recurrent encoding structure to finetune the whole network. In experiments, we demonstrate the state-of-the-art visual recognition performance on MIR-Flickr database and PASCAL VOC 2007 database with and without text features.

We present a theoretical analysis of F-measures for binary, multiclass and multilabel classification. These performance measures are non-linear, but in many scenarios they are pseudo-linear functions of the per-class false negative/false positive rate. Based on this observation, we present a general reduction of F-measure maximization to cost-sensitive classification with unknown costs. We then propose an algorithm with provable guarantees to obtain an approximately optimal classifier for the F-measure by solving a series of cost-sensitive classification problems. The strength of our analysis is to be valid on any dataset and any class of classifiers, extending the existing theoretical results on F-measures, which are asymptotic in nature. We present numerical experiments to illustrate the relative importance of cost asymmetry and thresholding when learning linear classifiers on various F-measure optimization tasks.

Stochastic variational inference (SVI) uses stochastic optimization to scale up Bayesian computation to massive data. We present an alternative perspective on SVI as approximate parallel coordinate ascent. SVI trades-off bias and variance to step close to the unknown true coordinate optimum given by batch variational Bayes (VB). We define a model to automate this process. The model infers the location of the next VB optimum from a sequence of noisy realizations. As a consequence of this construction, we update the variational parameters using Bayes rule, rather than a hand-crafted optimization schedule. When our model is a Kalman filter this procedure can recover the original SVI algorithm and SVI with adaptive steps. We may also encode additional assumptions in the model, such as heavy-tailed noise. By doing so, our algorithm outperforms the original SVI schedule and a state-of-the-art adaptive SVI algorithm in two diverse domains.

We propose a general and versatile framework that significantly speeds-up graphical model optimization while maintaining an excellent solution accuracy. The proposed approach, refereed as Inference by Learning or IbyL, relies on a multi-scale pruning scheme that progressively reduces the solution space by use of a coarse-to-fine cascade of learnt classifiers. We thoroughly experiment with classic computer vision related MRF problems, where our novel framework constantly yields a significant time speed-up (with respect to the most efficient inference methods) and obtains a more accurate solution than directly optimizing the MRF. We make our code available on-line.

Recursive neural networks comprise a class of architecture that can operate on structured input. They have been previously successfully applied to model compositionality in natural language using parse-tree-based structural representations. Even though these architectures are deep in structure, they lack the capacity for hierarchical representation that exists in conventional deep feed-forward networks as well as in recently investigated deep recurrent neural networks. In this work we introduce a new architecture --- a deep recursive neural network (deep RNN) --- constructed by stacking multiple recursive layers. We evaluate the proposed model on the task of fine-grained sentiment classification. Our results show that deep RNNs outperform associated shallow counterparts that employ the same number of parameters. Furthermore, our approach outperforms previous baselines on the sentiment analysis task, including a multiplicative RNN variant as well as the recently introduced paragraph vectors, achieving new state-of-the-art results. We provide exploratory analyses of the effect of multiple layers and show that they capture different aspects of compositionality in language.

The success of machine learning, particularly in supervised settings, has led to numerous attempts to apply it in adversarial settings such as spam and malware detection. The core challenge in this class of applications is that adversaries are not static data generators, but make a deliberate effort to evade the classifiers deployed to detect them. We investigate both the problem of modeling the objectives of such adversaries, as well as the algorithmic problem of accounting for rational, objective-driven adversaries. In particular, we demonstrate severe shortcomings of feature reduction in adversarial settings using several natural adversarial objective functions, an observation that is particularly pronounced when the adversary is able to substitute across similar features (for example, replace words with synonyms or replace letters in words). We offer a simple heuristic method for making learning more robust to feature cross-substitution attacks. We then present a more general approach based on mixed-integer linear programming with constraint generation, which implicitly trades off overfitting and feature selection in an adversarial setting using a sparse regularizer along with an evasion model. Our approach is the first method for combining an adversarial classification algorithm with a very general class of models of adversarial classifier evasion. We show that our algorithmic approach significantly outperforms state-of-the-art alternatives.

Self-paced learning (SPL) is a recently proposed learning regime inspired by the learning process of humans and animals that gradually incorporates easy to more complex samples into training. Existing methods are limited in that they ignore an important aspect in learning: diversity. To incorporate this information, we propose an approach called self-paced learning with diversity (SPLD) which formalizes the preference for both easy and diverse samples into a general regularizer. This regularization term is independent of the learning objective, and thus can be easily generalized into various learning tasks. Albeit non-convex, the optimization of the variables included in this SPLD regularization term for sample selection can be globally solved in linearithmic time. We demonstrate that our method significantly outperforms the conventional SPL on three real-world datasets. Specifically, SPLD achieves the best MAP so far reported in literature on the Hollywood2 and Olympic Sports datasets.