Here, we propose a clustering technique for general clustering problems including those that have non-convex clusters. For a given desired number of clusters $K$, we use three stages to find a clustering. The first stage uses a hybrid clustering technique to produce a series of clusterings of various sizes (randomly selected). They key steps are to find a $K$-means clustering using $K_\ell$ clusters where $K_\ell \gg K$ and then joins these small clusters by using single linkage clustering. The second stage stabilizes the result of stage one by reclustering via the `membership matrix' under Hamming distance to generate a dendrogram. The third stage is to cut the dendrogram to get $K^*$ clusters where $K^* \geq K$ and then prune back to $K$ to give a final clustering. A variant on our technique also gives a reasonable estimate for $K_T$, the true number of clusters. We provide a series of arguments to justify the steps in the stages of our methods and we provide numerous examples involving real and simulated data to compare our technique with other related techniques.

We demonstrate that a popular class of nonparametric mutual information (MI) estimators based on k-nearest-neighbor graphs requires number of samples that scales exponentially with the true MI. Consequently, accurate estimation of MI between two strongly dependent variables is possible only for prohibitively large sample size. This important yet overlooked shortcoming of the existing estimators is due to their implicit reliance on local uniformity of the underlying joint distribution. We introduce a new estimator that is robust to local non-uniformity, works well with limited data, and is able to capture relationship strengths over many orders of magnitude. We demonstrate the superior performance of the proposed estimator on both synthetic and real-world data.

We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this problem from a geometry perspective as the spanning set of a subspace arrangement, and focus on understanding the case when the underlying hypergraph of the subspace arrangement is specified. For this Fitted Dictionary Learning problem, we completely characterize the combinatorics of the associated subspace arrangements (i.e.\ their underlying hypergraphs). Specifically, a combinatorial rigidity-type theorem is proven for a type of geometric incidence system. The theorem characterizes the hypergraphs of subspace arrangements that generically yield (a) at least one dictionary (b) a locally unique dictionary (i.e.\ at most a finite number of isolated dictionaries) of the specified size. We are unaware of prior application of combinatorial rigidity techniques in the setting of Dictionary Learning, or even in machine learning. We also provide a systematic classification of problems related to Dictionary Learning together with various algorithms, their assumptions and performance.

The presence of a sparse "truth" has been a constant assumption in the theoretical analysis of sparse PCA and is often implicit in its methodological development. This naturally raises questions about the properties of sparse PCA methods and how they depend on the assumption of sparsity. Under what conditions can the relevant variables be selected consistently if the truth is assumed to be sparse? What can be said about the results of sparse PCA without assuming a sparse and unique truth? We answer these questions by investigating the properties of the recently proposed Fantope projection and selection (FPS) method in the high-dimensional setting. Our results provide general sufficient conditions for sparsistency of the FPS estimator. These conditions are weak and can hold in situations where other estimators are known to fail. On the other hand, without assuming sparsity or identifiability, we show that FPS provides a sparse, linear dimension-reducing transformation that is close to the best possible in terms of maximizing the predictive covariance.

Bayesian optimization is a sample-efficient method for black-box global optimization. How- ever, the performance of a Bayesian optimization method very much depends on its exploration strategy, i.e. the choice of acquisition function, and it is not clear a priori which choice will result in superior performance. While portfolio methods provide an effective, principled way of combining a collection of acquisition functions, they are often based on measures of past performance which can be misleading. To address this issue, we introduce the Entropy Search Portfolio (ESP): a novel approach to portfolio construction which is motivated by information theoretic considerations. We show that ESP outperforms existing portfolio methods on several real and synthetic problems, including geostatistical datasets and simulated control tasks. We not only show that ESP is able to offer performance as good as the best, but unknown, acquisition function, but surprisingly it often gives better performance. Finally, over a wide range of conditions we find that ESP is robust to the inclusion of poor acquisition functions.

We present a method to quantify uncertainty in the predictions made by simulations of mathematical models that can be applied to a broad class of stochastic, discrete, and differential equation models. Quantifying uncertainty is crucial for determining how accurate the model predictions are and identifying which input parameters affect the outputs of interest. Most of the existing methods for uncertainty quantification require many samples to generate accurate results, are unable to differentiate where the uncertainty is coming from (e.g., parameters or model assumptions), or require a lot of computational resources. Our approach addresses these challenges and opportunities by allowing different types of uncertainty, that is, uncertainty in input parameters as well as uncertainty created through stochastic model components. This is done by combining the Karhunen-Loeve decomposition, polynomial chaos expansion, and Bayesian Gaussian process regression to create a statistical surrogate for the stochastic model. The surrogate separates the analysis of variation arising through stochastic simulation and variation arising through uncertainty in the model parameterization. We illustrate our approach by quantifying the uncertainty in a stochastic ordinary differential equation epidemic model. Specifically, we estimate four quantities of interest for the epidemic model and show agreement between the surrogate and the actual model results.

In this paper, we are concerned with obtaining distribution-free concentration inequalities for mixture of independent Bernoulli variables that incorporate a notion of variance. Missing mass is the total probability mass associated to the outcomes that have not been seen in a given sample which is an important quantity that connects density estimates obtained from a sample to the population for discrete distributions. Therefore, we are specifically motivated to apply our method to study the concentration of missing mass - which can be expressed as a mixture of Bernoulli - in a novel way. We not only derive - for the first time - Bernstein-like large deviation bounds for the missing mass whose exponents behave almost linearly with respect to deviation size, but also sharpen McAllester and Ortiz (2003) and Berend and Kontorovich (2013) for large sample sizes in the case of small deviations which is the most interesting case in learning theory. In the meantime, our approach shows that the heterogeneity issue introduced in McAllester and Ortiz (2003) is resolvable in the case of missing mass in the sense that one can use standard inequalities but it may not lead to strong results. Thus, we postulate that our results are general and can be applied to provide potentially sharp Bernstein-like bounds under some constraints.

We study a probability distribution d on the truth assignments to a uniform binary AND-OR tree. Liu and Tanaka [2007, Inform. Process. Lett.] showed the following: If d achieves the equilibrium among independent distributions (ID) then d is an independent identical distribution (IID). We show a stronger form of the above result. Given a real number r such that 0 < r < 1, we consider a constraint that the probability of the root node having the value 0 is r. Our main result is the following: When we restrict ourselves to IDs satisfying this constraint, the above result of Liu and Tanaka still holds. The proof employs clever tricks of induction. In particular, we show two fundamental relationships between expected cost and probability in an IID on an OR-AND tree: (1) The ratio of the cost to the probability (of the root having the value 0) is a decreasing function of the probability x of the leaf. (2) The ratio of derivative of the cost to the derivative of the probability is a decreasing function of x, too.

Canonical correlation analysis (CCA) is a valuable method for interpreting cross-covariance across related datasets of different dimensionality. There are many potential applications of CCA to neuroimaging data analysis. For instance, CCA can be used for finding functional similarities across fMRI datasets collected from multiple subjects without resampling individual datasets to a template anatomy. In this paper, we introduce Pyrcca, an open-source Python module for executing CCA between two or more datasets. Pyrcca can be used to implement CCA with or without regularization, and with or without linear or a Gaussian kernelization of the datasets. We demonstrate an application of CCA implemented with Pyrcca to neuroimaging data analysis. We use CCA to find a data-driven set of functional response patterns that are similar across individual subjects in a natural movie experiment. We then demonstrate how this set of response patterns discovered by CCA can be used to accurately predict subject responses to novel natural movie stimuli.

We introduce local expectation gradients which is a general purpose stochastic variational inference algorithm for constructing stochastic gradients through sampling from the variational distribution. This algorithm divides the problem of estimating the stochastic gradients over multiple variational parameters into smaller sub-tasks so that each sub-task exploits intelligently the information coming from the most relevant part of the variational distribution. This is achieved by performing an exact expectation over the single random variable that mostly correlates with the variational parameter of interest resulting in a Rao-Blackwellized estimate that has low variance and can work efficiently for both continuous and discrete random variables. Furthermore, the proposed algorithm has interesting similarities with Gibbs sampling but at the same time, unlike Gibbs sampling, it can be trivially parallelized.