Given one feature of a novel animal, humans readily make inferences about other features of the animal. For example, winged creatures often fly, and creatures that eat fish often live in the water. We explore the knowledge that supports these inferences and compare two approaches. The first approach proposes that humans rely on abstract representations of dependency relationships between features, and is formalized here as a graphical model. The second approach proposes that humans rely on specific knowledge of previously encountered animals, and is formalized here as a family of exemplar models. We evaluate these models using a task where participants reason about chimeras, or animals with pairs of features that have not previously been observed to co-occur. The results support the hypothesis that humans rely on explicit representations of relationships between features.

Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a non-linear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. Although the resulting optimization problems are non-convex and non-smooth, we provide an efficient first-order scheme which scales to large graphs. Moreover, our approach comes with the quality guarantee that given any partition as initialization the algorithm either outputs a better partition or it stops immediately.

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy. Furthermore, VT converges faster and results in sparser (simpler) models, thus realizing an automatic Occam's razor for HMM learning. For more general scenario VT can be worse compared to ML but still capable of correctly recovering most of the parameters.

Predicting the nodes of a given graph is a fascinating theoretical problem with applications in several domains. Since graph sparsification via spanning trees retains enough information while making the task much easier, trees are an important special case of this problem. Although it is known how to predict the nodes of an unweighted tree in a nearly optimal way, in the weighted case a fully satisfactory algorithm is not available yet. We fill this hole and introduce an efficient node predictor, Shazoo, which is nearly optimal on any weighted tree. Moreover, we show that Shazoo can be viewed as a common nontrivial generalization of both previous approaches for unweighted trees and weighted lines. Experiments on real-world datasets confirm that Shazoo performs well in that it fully exploits the structure of the input tree, and gets very close to (and sometimes better than) less scalable energy minimization methods.

We prove a new oracle inequality for support vector machines with Gaussian RBF kernels solving the regularized least squares regression problem. To this end, we apply the modulus of smoothness. With the help of the new oracle inequality we then derive learning rates that can also be achieved by a simple data-dependent parameter selection method. Finally, it turns out that our learning rates are asymptotically optimal for regression functions satisfying certain standard smoothness conditions.

The goal of this paper is to investigate the advantages and disadvantages of learning in Banach spaces over Hilbert spaces. While many works have been carried out in generalizing Hilbert methods to Banach spaces, in this paper, we consider the simple problem of learning a Parzen window classifier in a reproducing kernel Banach space (RKBS)---which is closely related to the notion of embedding probability measures into an RKBS---in order to carefully understand its pros and cons over the Hilbert space classifier. We show that while this generalization yields richer distance measures on probabilities compared to its Hilbert space counterpart, it however suffers from serious computational drawback limiting its practical applicability, which therefore demonstrates the need for developing efficient learning algorithms in Banach spaces.

In this paper we describe a maximum likelihood likelihood approach for dictionary learning in the multiplicative exponential noise model. This model is prevalent in audio signal processing where it underlies a generative composite model of the power spectrogram. Maximum joint likelihood estimation of the dictionary and expansion coefficients leads to a nonnegative matrix factorization problem where the Itakura-Saito divergence is used. The optimality of this approach is in question because the number of parameters (which include the expansion coefficients) grows with the number of observations. In this paper we describe a variational procedure for optimization of the marginal likelihood, i.e., the likelihood of the dictionary where the activation coefficients have been integrated out (given a specific prior). We compare the output of both maximum joint likelihood estimation (i.e., standard Itakura-Saito NMF) and maximum marginal likelihood estimation (MMLE) on real and synthetical datasets. The MMLE approach is shown to embed automatic model order selection, akin to automatic relevance determination.

We derive an instantaneous (per-round) data-dependent regret bound for stochastic multiarmed bandits with side information (also known as contextual bandits). The scaling of our regret bound with the number of states (contexts) $N$ goes as $\sqrt{N I_{\rho_t}(S;A)}$, where $I_{\rho_t}(S;A)$ is the mutual information between states and actions (the side information) used by the algorithm at round $t$. If the algorithm uses all the side information, the regret bound scales as $\sqrt{N \ln K}$, where $K$ is the number of actions (arms). However, if the side information $I_{\rho_t}(S;A)$ is not fully used, the regret bound is significantly tighter. In the extreme case, when $I_{\rho_t}(S;A) = 0$, the dependence on the number of states reduces from linear to logarithmic. Our analysis allows to provide the algorithm large amount of side information, let the algorithm to decide which side information is relevant for the task, and penalize the algorithm only for the side information that it is using de facto. We also present an algorithm for multiarmed bandits with side information with computational complexity that is a linear in the number of actions.

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the second term. We show that the basic proximal-gradient method, the basic proximal-gradient method with a strong convexity assumption, and the accelerated proximal-gradient method achieve the same convergence rates as in the error-free case, provided the errors decrease at an appropriate rate. Our experimental results on a structured sparsity problem indicate that sequences of errors with these appealing theoretical properties can lead to practical performance improvements.

We consider the problem of recovering a matrix $\mathbf{M}$ that is the sum of a low-rank matrix $\mathbf{L}$ and a sparse matrix $\mathbf{S}$ from a small set of linear measurements of the form $\mathbf{y} = \mathcal{A}(\mathbf{M}) = \mathcal{A}({\bf L}+{\bf S})$. This model subsumes three important classes of signal recovery problems: compressive sensing, affine rank minimization, and robust principal component analysis. We propose a natural optimization problem for signal recovery under this model and develop a new greedy algorithm called SpaRCS to solve it. SpaRCS inherits a number of desirable properties from the state-of-the-art CoSaMP and ADMiRA algorithms, including exponential convergence and efficient implementation. Simulation results with video compressive sensing, hyperspectral imaging, and robust matrix completion data sets demonstrate both the accuracy and efficacy of the algorithm.