This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact subset of $\mathbb{R}^d$. On this approximate eigenbasis the eigenvalues of the covariance function can be expressed as simple functions of the spectral density of the Gaussian process, which allows the GP inference to be solved under a computational cost scaling as $\mathcal{O}(nm^2)$ (initial) and $\mathcal{O}(m^3)$ (hyperparameter learning) with $m$ basis functions and $n$ data points. The approach also allows for rigorous error analysis with Hilbert space theory, and we show that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity. The expansion generalizes to Hilbert spaces with an inner product which is defined as an integral over a specified input density. The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data.

The question of how to determine the number of independent latent factors (topics) in mixture models such as Latent Dirichlet Allocation (LDA) is of great practical importance. In most applications, the exact number of topics is unknown, and depends on the application and the size of the data set. Bayesian nonparametric methods can avoid the problem of topic number selection, but they can be impracticably slow for large sample sizes and are subject to local optima. We develop a guaranteed procedure for topic number recovery that does not necessitate learning the model's latent parameters beforehand. Our procedure relies on adapting results from random matrix theory. Performance of our topic number recovery procedure is superior to hLDA, a nonparametric method. We also discuss some implications of our results on the sample complexity and accuracy of popular spectral learning algorithms for LDA. Our results and procedure can be extended to spectral learning algorithms for other exchangeable mixture models as well as Hidden Markov Models.

In this paper, we consider decentralized sequential decision making in distributed online recommender systems, where items are recommended to users based on their search query as well as their specific background including history of bought items, gender and age, all of which comprise the context information of the user. In contrast to centralized recommender systems, in which there is a single centralized seller who has access to the complete inventory of items as well as the complete record of sales and user information, in decentralized recommender systems each seller/learner only has access to the inventory of items and user information for its own products and not the products and user information of other sellers, but can get commission if it sells an item of another seller. Therefore the sellers must distributedly find out for an incoming user which items to recommend (from the set of own items or items of another seller), in order to maximize the revenue from own sales and commissions. We formulate this problem as a cooperative contextual bandit problem, analytically bound the performance of the sellers compared to the best recommendation strategy given the complete realization of user arrivals and the inventory of items, as well as the context-dependent purchase probabilities of each item, and verify our results via numerical examples on a distributed data set adapted based on Amazon data. We evaluate the dependence of the performance of a seller on the inventory of items the seller has, the number of connections it has with the other sellers, and the commissions which the seller gets by selling items of other sellers to its users.

We revisit the classical decision-theoretic problem of weighted expert voting from a statistical learning perspective. In particular, we examine the consistency (both asymptotic and finitary) of the optimal Nitzan-Paroush weighted majority and related rules. In the case of known expert competence levels, we give sharp error estimates for the optimal rule. When the competence levels are unknown, they must be empirically estimated. We provide frequentist and Bayesian analyses for this situation. Some of our proof techniques are non-standard and may be of independent interest. The bounds we derive are nearly optimal, and several challenging open problems are posed. Experimental results are provided to illustrate the theory.

In this paper we analyze the asymptotic properties of l1 penalized maximum likelihood estimation of signals with piece-wise constant mean values and/or variances. The focus is on segmentation of a non-stationary time series with respect to changes in these model parameters. This change point detection and estimation problem is also referred to as total variation denoising or l1 -mean filtering and has many important applications in most fields of science and engineering. We establish the (approximate) sparse consistency properties, including rate of convergence, of the so-called fused lasso signal approximator (FLSA). We show that this only holds if the sign of the corresponding consecutive changes are all different, and that this estimator is otherwise incapable of correctly detecting the underlying sparsity pattern. The key idea is to notice that the optimality conditions for this problem can be analyzed using techniques related to brownian bridge theory.

This survey presents in some detail the main advances that have been recently taking place in Computational Linguistics towards the unification of the two prominent semantic paradigms: the compositional formal semantics view and the distributional models of meaning based on vector spaces. After an introduction to these two approaches, I review the most important models that aim to provide compositionality in distributional semantics. Then I proceed and present in more detail a particular framework by Coecke, Sadrzadeh and Clark (2010) based on the abstract mathematical setting of category theory, as a more complete example capable to demonstrate the diversity of techniques and scientific disciplines that this kind of research can draw from. This paper concludes with a discussion about important open issues that need to be addressed by the researchers in the future.

The accurate detection of small deviations in given density matrices is important for quantum information processing. Here we propose a new method based on the concept of data mining. We demonstrate that the proposed method can more accurately detect small erroneous deviations in reconstructed density matrices, which contain intrinsic fluctuations due to the limited number of samples, than a naive method of checking the trace distance from the average of the given density matrices. This method has the potential to be a key tool in broad areas of physics where the detection of small deviations of quantum states reconstructed using a limited number of samples are essential.

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.

Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of structural network properties. The present paper advocates two graph embedding approaches with centrality considerations to comply with node hierarchy. The problem is formulated first as one of constrained multi-dimensional scaling (MDS), and it is solved via block coordinate descent iterations with successive approximations and guaranteed convergence to a KKT point. In addition, a regularization term enforcing graph smoothness is incorporated with the goal of reducing edge crossings. A second approach leverages the locally-linear embedding (LLE) algorithm which assumes that the graph encodes data sampled from a low-dimensional manifold. Closed-form solutions to the resulting centrality-constrained optimization problems are determined yielding meaningful embeddings. Experimental results demonstrate the efficacy of both approaches, especially for visualizing large networks on the order of thousands of nodes.

This paper deals with chain graphs under the Andersson-Madigan-Perlman (AMP) interpretation. In particular, we present a constraint based algorithm for learning an AMP chain graph a given probability distribution is faithful to. Moreover, we show that the extension of Meek's conjecture to AMP chain graphs does not hold, which compromises the development of efficient and correct score+search learning algorithms under assumptions weaker than faithfulness. We also introduce a new family of graphical models that consists of undirected and bidirected edges. We name this new family maximal covariance-concentration graphs (MCCGs) because it includes both covariance and concentration graphs as subfamilies. However, every MCCG can be seen as the result of marginalizing out some nodes in an AMP CG. We describe global, local and pairwise Markov properties for MCCGs and prove their equivalence. We characterize when two MCCGs are Markov equivalent, and show that every Markov equivalence class of MCCGs has a distinguished member. We present a constraint based algorithm for learning a MCCG a given probability distribution is faithful to. Finally, we present a graphical criterion for reading dependencies from a MCCG of a probability distribution that satisfies the graphoid properties, weak transitivity and composition. We prove that the criterion is sound and complete in certain sense.