In the superset label learning problem (SLL), each training instance provides a set of candidate labels of which one is the true label of the instance. As in ordinary regression, the candidate label set is a noisy version of the true label. In this work, we solve the problem by maximizing the likelihood of the candidate label sets of training instances. We propose a probabilistic model, the Logistic Stick-Breaking Conditional Multinomial Model (LSB-CMM), to do the job. The LSB-CMM is derived from the logistic stick-breaking process. It first maps data points to mixture components and then assigns to each mixture component a label drawn from a component-specific multinomial distribution.

Sum-product networks are a new deep architecture that can perform fast, exact inference on high-treewidth models. Only generative methods for training SPNs have been proposed to date. In this paper, we present the first discriminative training algorithms for SPNs, combining the high accuracy of the former with the representational power and tractability of the latter. We show that the class of tractable discriminative SPNs is broader than the class of tractable generative ones, and propose an efficient backpropagation-style algorithm for computing the gradient of the conditional log likelihood. Standard gradient descent suffers from the diffusion problem, but networks with many layers can be learned reliably using "hard" gradient descent, where marginal inference is replaced by MPE inference (i.e., inferring the most probable state of the non-evidence variables). The resulting updates have a simple and intuitive form. We test discriminative SPNs on standard image classification tasks. We obtain the best results to date on the CIFAR-10 dataset, using fewer features than prior methods with an SPN architecture that learns local image structure discriminatively. We also report the highest published test accuracy on STL-10 even though we only use the labeled portion of the dataset.

The expected return is a widely used objective in decision making under uncertainty. Many algorithms, such as value iteration, have been proposed to optimize it. In risk-aware settings, however, the expected return is often not an appropriate objective to optimize. We propose a new optimization objective for risk-aware planning and show that it has desirable theoretical properties. We also draw connections to previously proposed objectives for risk-aware planing: minmax, exponential utility, percentile and mean minus variance. Our method applies to an extended class of Markov decision processes: we allow costs to be stochastic as long as they are bounded. Additionally, we present an efficient algorithm for optimizing the proposed objective. Synthetic and real-world experiments illustrate the effectiveness of our method, at scale.

We introduce a class of discrete divergences on sets (equivalently binary vectors) that we call the submodular-Bregman divergences. We consider two kinds, defined either from tight modular upper or tight modular lower bounds of a submodular function. We show that the properties of these divergences are analogous to the (standard continuous) Bregman divergence. We demonstrate how they generalize many useful divergences, including the weighted Hamming distance, squared weighted Hamming, weighted precision, recall, conditional mutual information, and a generalized KL-divergence on sets. We also show that the generalized Bregman divergence on the Lovász extension of a submodular function, which we call the Lovász-Bregman divergence, is a continuous extension of a submodular Bregman divergence. We point out a number of applications, and in particular show that a proximal algorithm defined through the submodular Bregman divergence provides a framework for many mirror-descent style algorithms related to submodular function optimization. We also show that a generalization of the k-means algorithm using the Lovász Bregman divergence is natural in clustering scenarios where ordering is important. A unique property of this algorithm is that computing the mean ordering is extremely efficient unlike other order based distance measures.

Modelling natural images with sparse coding (SC) has faced two main challenges: flexibly representing varying pixel intensities and realistically representing lowlevel image components. This paper proposes a novel multiple-cause generative model of low-level image statistics that generalizes the standard SC model in two crucial points: (1) it uses a spike-and-slab prior distribution for a more realistic representation of component absence/intensity, and (2) the model uses the highly nonlinear combination rule of maximal causes analysis (MCA) instead of a linear combination. The major challenge is parameter optimization because a model with either (1) or (2) results in strongly multimodal posteriors. We show for the first time that a model combining both improvements can be trained efficiently while retaining the rich structure of the posteriors. We design an exact piecewise Gibbs sampling method and combine this with a variational method based on preselection of latent dimensions. This combined training scheme tackles both analytical and computational intractability and enables application of the model to a large number of observed and hidden dimensions.

We investigate a curious relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. Based on our population-level results, we show how the graphical Lasso may be used to recover the edge structure of certain classes of discrete graphical models, and present simulations to verify our theoretical results.

A fundamental problem in the analysis of structured relational data like graphs, networks, databases, and matrices is to extract a summary of the common structure underlying relations between individual entities. Relational data are typically encoded in the form of arrays; invariance to the ordering of rows and columns corresponds to exchangeable arrays. Results in probability theory due to Aldous, Hoover and Kallenberg show that exchangeable arrays can be represented in terms of a random measurable function which constitutes the natural model parameter in a Bayesian model. We obtain a flexible yet simple Bayesian nonparametric model by placing a Gaussian process prior on the parameter function. Efficient inference utilises elliptical slice sampling combined with a random sparse approximation to the Gaussian process. We demonstrate applications of the model to network data and clarify its relation to models in the literature, several of which emerge as special cases.

R is a convex but not necessarily differentiable function whose proximal mapping can be evaluated efficiently. We derive a generalization of Newton-type methods to handle such convex but nonsmooth objective functions. We prove such methods are globally convergent and achieve superlinear rates of convergence in the vicinity of an optimal solution. We also demonstrate the performance of these methods using problems of relevance in machine learning and statistics.

A gaze concurrence is a point in 3D where the gaze directions of two or more people intersect. It is a strong indicator of social saliency because the attention of the participating group is focused on that point. In scenes occupied by large groups of people, multiple concurrences may occur and transition over time. In this paper, we present a method to construct a 3D social saliency field and locate multiple gaze concurrences that occur in a social scene from videos taken by head-mounted cameras. We model the gaze as a cone-shaped distribution emanating from the center of the eyes, capturing the variation of eye-in-head motion. We calibrate the parameters of this distribution by exploiting the fixed relationship between the primary gaze ray and the head-mounted camera pose. The resulting gaze model enables us to build a social saliency field in 3D. We estimate the number and 3D locations of the gaze concurrences via provably convergent modeseeking in the social saliency field. Our algorithm is applied to reconstruct multiple gaze concurrences in several real world scenes and evaluated quantitatively against motion-captured ground truth.

While long term human visual memory can store a remarkable amount of visual information, it tends to degrade over time. Recent works have shown that image memorability is an intrinsic property of an image that can be reliably estimated using state-of-the-art image features and machine learning algorithms. However, the class of features and image information that is forgotten has not been explored yet. In this work, we propose a probabilistic framework that models how and which local regions from an image may be forgotten using a data-driven approach that combines local and global images features. The model automatically discovers memorability maps of individual images without any human annotation. We incorporate multiple image region attributes in our algorithm, leading to improved memorability prediction of images as compared to previous works.