Typical blur from camera shake often deviates from the standard uniform convolutional assumption, in part because of problematic rotations which create greater blurring away from some unknown center point. Consequently, successful blind deconvolution for removing shake artifacts requires the estimation of a spatially-varying or non-uniform blur operator. Using ideas from Bayesian inference and convex analysis, this paper derives a non-uniform blind deblurring algorithm with several desirable, yet previously-unexplored attributes. The underlying objective function includes a spatially-adaptive penalty that couples the latent sharp image, non-uniform blur operator, and noise level together. This coupling allows the penalty to automatically adjust its shape based on the estimated degree of local blur and image structure such that regions with large blur or few prominent edges are discounted. Remaining regions with modest blur and revealing edges therefore dominate the overall estimation process without explicitly incorporating structure-selection heuristics. The algorithm can be implemented using an optimization strategy that is virtually parameter free and simpler than existing methods. Detailed theoretical analysis and empirical validation on real images serve to validate the proposed method.

Label propagation is one of the state-of-the-art methods for semi-supervised learning, which estimates labels by propagating label information through a graph. Label propagation assumes that data points (nodes) connected in a graph should have similar labels. Consequently, the label estimation heavily depends on edge weights in a graph which represent similarity of each node pair. We propose a method for a graph to capture the manifold structure of input features using edge weights parameterized by a similarity function. In this approach, edge weights represent both similarity and local reconstruction weight simultaneously, both being reasonable for label propagation. For further justification, we provide analytical considerations including an interpretation as a cross-validation of a propagation model in the feature space, and an error analysis based on a low dimensional manifold model. Experimental results demonstrated the effectiveness of our approach both in synthetic and real datasets.

We provide a detailed study of the estimation of probability distributions---discrete and continuous---in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental tradeoffs between privacy and convergence rate, as well as providing tools to allow movement along the privacy-statistical efficiency continuum. One of the consequences of our results is that Warner's classical work on randomized response is an optimal way to perform survey sampling while maintaining privacy of the respondents.

The idea of computer vision as the Bayesian inverse problem to computer graphics has a long history and an appealing elegance, but it has proved difficult to directly implement. Instead, most vision tasks are approached via complex bottom-up processing pipelines. Here we show that it is possible to write short, simple probabilistic graphics programs that define flexible generative models and to automatically invert them to interpret real-world images. Generative probabilistic graphics programs consist of a stochastic scene generator, a renderer based on graphics software, a stochastic likelihood model linking the renderer's output and the data, and latent variables that adjust the fidelity of the renderer and the tolerance of the likelihood model. Representations and algorithms from computer graphics, originally designed to produce high-quality images, are instead used as the deterministic backbone for highly approximate and stochastic generative models. This formulation combines probabilistic programming, computer graphics, and approximate Bayesian computation, and depends only on general-purpose, automatic inference techniques. We describe two applications: reading sequences of degraded and adversarially obscured alphanumeric characters, and inferring 3D road models from vehicle-mounted camera images. Each of the probabilistic graphics programs we present relies on under 20 lines of probabilistic code, and supports accurate, approximately Bayesian inferences about ambiguous real-world images.

Measuring similarity is crucial to many learning tasks. It is also a richer and broader notion than what most metric learning algorithms can model. For example, similarity can arise from the process of aggregating the decisions of multiple latent components, where each latent component compares data in its own way by focusing on a different subset of features. In this paper, we propose Similarity Component Analysis (SCA), a probabilistic graphical model that discovers those latent components from data. In SCA, a latent component generates a local similarity value, computed with its own metric, independently of other components. The final similarity measure is then obtained by combining the local similarity values with a (noisy-)OR gate. We derive an EM-based algorithm for fitting the model parameters with similarity-annotated data from pairwise comparisons. We validate the SCA model on synthetic datasets where SCA discovers the ground-truth about the latent components. We also apply SCA to a multiway classification task and a link prediction task. For both tasks, SCA attains significantly better prediction accuracies than competing methods. Moreover, we show how SCA can be instrumental in exploratory analysis of data, where we gain insights about the data by examining patterns hidden in its latent components' local similarity values.

EDML is a recently proposed algorithm for learning parameters in Bayesian networks. It was originally derived in terms of approximate inference on a meta-network, which underlies the Bayesian approach to parameter estimation. While this initial derivation helped discover EDML in the first place and provided a concrete context for identifying some of its properties (e.g., in contrast to EM), the formal setting was somewhat tedious in the number of concepts it drew on. In this paper, we propose a greatly simplified perspective on EDML, which casts it as a general approach to continuous optimization. The new perspective has several advantages. First, it makes immediate some results that were non-trivial to prove initially. Second, it facilitates the design of EDML algorithms for new graphical models, leading to a new algorithm for learning parameters in Markov networks. We derive this algorithm in this paper, and show, empirically, that it can sometimes learn better estimates from complete data, several times faster than commonly used optimization methods, such as conjugate gradient and L-BFGS.

Stacked sparse denoising auto-encoders (SSDAs) have recently been shown to be successful at removing noise from corrupted images. However, like most denoising techniques, the SSDA is not robust to variation in noise types beyond what it has seen during training. We present the multi-column stacked sparse denoising autoencoder, a novel technique of combining multiple SSDAs into a multi-column SSDA (MC-SSDA) by combining the outputs of each SSDA. We eliminate the need to determine the type of noise, let alone its statistics, at test time. We show that good denoising performance can be achieved with a single system on a variety of different noise types, including ones not seen in the training set. Additionally, we experimentally demonstrate the efficacy of MC-SSDA denoising by achieving MNIST digit error rates on denoised images at close to that of the uncorrupted images.

We address the problem of estimating the ratio $\frac{q}{p}$ where $p$ is a density function and $q$ is another density, or, more generally an arbitrary function. Knowing or approximating this ratio is needed in various problems of inference and integration, in particular, when one needs to average a function with respect to one probability distribution, given a sample from another. It is often referred as {\it importance sampling} in statistical inference and is also closely related to the problem of {\it covariate shift} in transfer learning as well as to various MCMC methods. Our approach is based on reformulating the problem of estimating the ratio as an inverse problem in terms of an integral operator corresponding to a kernel, and thus reducing it to an integral equation, known as the Fredholm problem of the first kind. This formulation, combined with the techniques of regularization and kernel methods, leads to a principled kernel-based framework for constructing algorithms and for analyzing them theoretically. The resulting family of algorithms (FIRE, for Fredholm Inverse Regularized Estimator) is flexible, simple and easy to implement. We provide detailed theoretical analysis including concentration bounds and convergence rates for the Gaussian kernel for densities defined on $\R^d$ and smooth $d$-dimensional sub-manifolds of the Euclidean space. Model selection for unsupervised or semi-supervised inference is generally a difficult problem. Interestingly, it turns out that in the density ratio estimation setting, when samples from both distributions are available, there are simple completely unsupervised methods for choosing parameters. We call this model selection mechanism CD-CV for Cross-Density Cross-Validation. Finally, we show encouraging experimental results including applications to classification within the covariate shift framework.

Multi-task prediction models are widely being used to couple regressors or classification models by sharing information across related tasks. A common pitfall of these models is that they assume that the output tasks are independent conditioned on the inputs. Here, we propose a multi-task Gaussian process approach to model both the relatedness between regressors as well as the task correlations in the residuals, in order to more accurately identify true sharing between regressors. The resulting Gaussian model has a covariance term that is the sum of Kronecker products, for which efficient parameter inference and out of sample prediction are feasible. On both synthetic examples and applications to phenotype prediction in genetics, we find substantial benefits of modeling structured noise compared to established alternatives.

When a probabilistic model and its prior are given, Bayesian learning offers inference with automatic parameter tuning. However, Bayesian learning is often obstructed by computational difficulty: the rigorous Bayesian learning is intractable in many models, and its variational Bayesian (VB) approximation is prone to suffer from local minima. In this paper, we overcome this difficulty for low-rank subspace clustering (LRSC) by providing an exact global solver and its efficient approximation. LRSC extracts a low-dimensional structure of data by embedding samples into the union of low-dimensional subspaces, and its variational Bayesian variant has shown good performance. We first prove a key property that the VB-LRSC model is highly redundant. Thanks to this property, the optimization problem of VB-LRSC can be separated into small subproblems, each of which has only a small number of unknown variables. Our exact global solver relies on another key property that the stationary condition of each subproblem is written as a set of polynomial equations, which is solvable with the homotopy method. For further computational efficiency, we also propose an efficient approximate variant, of which the stationary condition can be written as a polynomial equation with a single variable. Experimental results show the usefulness of our approach.