Submodular and supermodular functions have found wide applicability in machine learning, capturing notions such as diversity and regularity, respectively. These notions have deep consequences for optimization, and the problem of (approximately) optimizing submodular functions has received much attention. However, beyond optimization, these notions allow specifying expressive probabilistic models that can be used to quantify predictive uncertainty via marginal inference. Prominent, well-studied special cases include Ising models and determinantal point processes, but the general class of log-submodular and log-supermodular models is much richer and little studied. In this paper, we investigate the use of Markov chain Monte Carlo sampling to perform approximate inference in general log-submodular and log-supermodular models. In particular, we consider a simple Gibbs sampling procedure, and establish two sufficient conditions, the first guaranteeing polynomial-time, and the second fast (O(nlogn)) mixing. We also evaluate the efficiency of the Gibbs sampler on three examples of such models, and compare against a recently proposed variational approach.

The robust principal component analysis (RPCA) problem seeks to separate low-rank trends from sparse outlierswithin a data matrix, that is, to approximate a $n\times d$ matrix $D$ as the sum of a low-rank matrix $L$ and a sparse matrix $S$.We examine the robust principal component analysis (RPCA) problem under data compression, wherethe data $Y$ is approximately given by $(L + S)\cdot C$, that is, a low-rank $+$ sparse data matrix that has been compressed to size $n\times m$ (with $m$ substantially smaller than the original dimension $d$) via multiplication witha compression matrix $C$. We give a convex program for recovering the sparse component $S$ along with the compressed low-rank component $L\cdot C$, along with upper bounds on the error of this reconstructionthat scales naturally with the compression dimension $m$ and coincides with existing results for the uncompressedsetting $m=d$. Our results can also handle error introduced through additive noise or through missing data.The scaling of dimension, compression, and signal complexity in our theoretical results is verified empirically through simulations, and we also apply our method to a data set measuring chlorine concentration acrossa network of sensors, to test its performance in practice.

Modeling the distribution of natural images is challenging, partly because of strong statistical dependencies which can extend over hundreds of pixels. Recurrent neural networks have been successful in capturing long-range dependencies in a number of problems but only recently have found their way into generative image models. We here introduce a recurrent image model based on multi-dimensional long short-term memory units which are particularly suited for image modeling due to their spatial structure. Our model scales to images of arbitrary size and its likelihood is computationally tractable. We find that it outperforms the state of the art in quantitative comparisons on several image datasets and produces promising results when used for texture synthesis and inpainting.

Perception is often described as a predictive process based on an optimal inference with respect to a generative model. We study here the principled construction of a generative model specifically crafted to probe motion perception. In that context, we first provide an axiomatic, biologically-driven derivation of the model. This model synthesizes random dynamic textures which are defined by stationary Gaussian distributions obtained by the random aggregation of warped patterns. Importantly, we show that this model can equivalently be described as a stochastic partial differential equation. Using this characterization of motion in images, it allows us to recast motion-energy models into a principled Bayesian inference framework. Finally, we apply these textures in order to psychophysically probe speed perception in humans. In this framework, while the likelihood is derived from the generative model, the prior is estimated from the observed results and accounts for the perceptual bias in a principled fashion.

In personalized recommendation systems, it is important to predict preferences of a user on items that have not been seen by that user yet. Similarly, in revenue management, it is important to predict outcomes of comparisons among those items that have never been compared so far. The MultiNomial Logit model, a popular discrete choice model, captures the structure of the hidden preferences with a low-rank matrix. In order to predict the preferences, we want to learn the underlying model from noisy observations of the low-rank matrix, collected as revealed preferences in various forms of ordinal data. A natural approach to learn such a model is to solve a convex relaxation of nuclear norm minimization. We present the convex relaxation approach in two contexts of interest: collaborative ranking and bundled choice modeling. In both cases, we show that the convex relaxation is minimax optimal. We prove an upper bound on the resulting error with finite samples, and provide a matching information-theoretic lower bound.

In many applications, the data is of rich structure that can be represented by a hypergraph, where the data items are represented by vertices and the associations among items are represented by hyperedges. Equivalently, we are given an input bipartite graph with two types of vertices: items, and associations (which we refer to as topics). We consider the problem of partitioning the set of items into a given number of parts such that the maximum number of topics covered by a part of the partition is minimized. This is a natural clustering problem, with various applications, e.g. partitioning of a set of information objects such as documents, images, and videos, and load balancing in the context of computation platforms.In this paper, we focus on the streaming computation model for this problem, in which items arrive online one at a time and each item must be assigned irrevocably to a part of the partition at its arrival time. Motivated by scalability requirements, we focus on the class of streaming computation algorithms with memory limited to be at most linear in the number of the parts of the partition. We show that a greedy assignment strategy is able to recover a hidden co-clustering of items under a natural set of recovery conditions. We also report results of an extensive empirical evaluation, which demonstrate that this greedy strategy yields superior performance when compared with alternative approaches.

We present a method for learning Bayesian networks from data sets containing thousands of variables without the need for structure constraints. Our approach is made of two parts. The first is a novel algorithm that effectively explores the space of possible parent sets of a node. It guides the exploration towards the most promising parent sets on the basis of an approximated score function that is computed in constant time. The second part is an improvement of an existing ordering-based algorithm for structure optimization. The new algorithm provably achieves a higher score compared to its original formulation. Our novel approach consistently outperforms the state of the art on very large data sets.

We propose rectified factor networks (RFNs) to efficiently construct very sparse, non-linear, high-dimensional representations of the input. RFN models identify rare and small events, have a low interference between code units, have a small reconstruction error, and explain the data covariance structure. RFN learning is a generalized alternating minimization algorithm derived from the posterior regularization method which enforces non-negative and normalized posterior means. We proof convergence and correctness of the RFN learning algorithm.On benchmarks, RFNs are compared to other unsupervised methods like autoencoders, RBMs, factor analysis, ICA, and PCA. In contrast to previous sparse coding methods, RFNs yield sparser codes, capture the data's covariance structure more precisely, and have a significantly smaller reconstruction error. We test RFNs as pretraining technique of deep networks on different vision datasets, where RFNs were superior to RBMs and autoencoders. On gene expression data from two pharmaceutical drug discovery studies, RFNs detected small and rare gene modules that revealed highly relevant new biological insights which were so far missed by other unsupervised methods.RFN package for GPU/CPU is available at http://www.bioinf.jku.at/software/rfn.

Compact explicit feature maps provide a practical framework to scale kernel methods to large-scale learning, but deriving such maps for many types of kernels remains a challenging open problem. Among the commonly used kernels for nonlinear classification are polynomial kernels, for which low approximation error has thus far necessitated explicit feature maps of large dimensionality, especially for higher-order polynomials. Meanwhile, because polynomial kernels are unbounded, they are frequently applied to data that has been normalized to unit l2 norm. The question we address in this work is: if we know a priori that data is so normalized, can we devise a more compact map? We show that a putative affirmative answer to this question based on Random Fourier Features is impossible in this setting, and introduce a new approximation paradigm, Spherical Random Fourier (SRF) features, which circumvents these issues and delivers a compact approximation to polynomial kernels for data on the unit sphere. Compared to prior work, SRF features are less rank-deficient, more compact, and achieve better kernel approximation, especially for higher-order polynomials. The resulting predictions have lower variance and typically yield better classification accuracy.

Recently, strong results have been demonstrated by Deep Recurrent Neural Networks on natural language transduction problems. In this paper we explore the representational power of these models using synthetic grammars designed to exhibit phenomena similar to those found in real transduction problems such as machine translation. These experiments lead us to propose new memory-based recurrent networks that implement continuously differentiable analogues of traditional data structures such as Stacks, Queues, and DeQues. We show that these architectures exhibit superior generalisation performance to Deep RNNs and are often able to learn the underlying generating algorithms in our transduction experiments.