Emission from a class of benzene-based molecules known as Polycyclic Aromatic Hydrocarbons (PAHs) dominates the infrared spectrum of star-forming regions. The observed emission appears to arise from the combined emission of numerous PAH species, each with its unique spectrum. Linear superposition of the PAH spectra identifies this problem as a source separation problem. It is, however, of a formidable class of source separation problems given that different PAH sources potentially number in the hundreds, even thousands, and there is only one measured spectral signal for a given astrophysical site. Fortunately, the source spectra of the PAHs are known, but the signal is also contaminated by other spectral sources. We describe our ongoing work in developing Bayesian source separation techniques relying on nested sampling in conjunction with an ON/OFF mechanism enabling simultaneous estimation of the probability that a particular PAH species is present and its contribution to the spectrum.

On many social networking web sites such as Facebook and Twitter, resharing or reposting functionality allows users to share others' content with their own friends or followers. As content is reshared from user to user, large cascades of reshares can form. While a growing body of research has focused on analyzing and characterizing such cascades, a recent, parallel line of work has argued that the future trajectory of a cascade may be inherently unpredictable. In this work, we develop a framework for addressing cascade prediction problems. On a large sample of photo reshare cascades on Facebook, we find strong performance in predicting whether a cascade will continue to grow in the future. We find that the relative growth of a cascade becomes more predictable as we observe more of its reshares, that temporal and structural features are key predictors of cascade size, and that initially, breadth, rather than depth in a cascade is a better indicator of larger cascades. This prediction performance is robust in the sense that multiple distinct classes of features all achieve similar performance. We also discover that temporal features are predictive of a cascade's eventual shape. Observing independent cascades of the same content, we find that while these cascades differ greatly in size, we are still able to predict which ends up the largest.

Clustering is a fundamental problem in many scientific applications. Standard methods such as $k$-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of $k$-means and hierarchical clustering shrink cluster centroids toward one another and ensure a unique global minimizer. In this work we present two splitting methods for solving the convex clustering problem. The first is an instance of the alternating direction method of multipliers (ADMM); the second is an instance of the alternating minimization algorithm (AMA). In contrast to previously considered algorithms, our ADMM and AMA formulations provide simple and unified frameworks for solving the convex clustering problem under the previously studied norms and open the door to potentially novel norms. We demonstrate the performance of our algorithm on both simulated and real data examples. While the differences between the two algorithms appear to be minor on the surface, complexity analysis and numerical experiments show AMA to be significantly more efficient.

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing smooth functions, even when search directions are computed inexactly. Many popular methods tailored to problems arising in bioinformatics, signal processing, and statistical learning are special cases of proximal Newton-type methods, and our analysis yields new convergence results for some of these methods.

We present a nonparametric prior over reversible Markov chains. We use completely random measures, specifically gamma processes, to construct a countably infinite graph with weighted edges. By enforcing symmetry to make the edges undirected we define a prior over random walks on graphs that results in a reversible Markov chain. The resulting prior over infinite transition matrices is closely related to the hierarchical Dirichlet process but enforces reversibility. A reinforcement scheme has recently been proposed with similar properties, but the de Finetti measure is not well characterised. We take the alternative approach of explicitly constructing the mixing measure, which allows more straightforward and efficient inference at the cost of no longer having a closed form predictive distribution. We use our process to construct a reversible infinite HMM which we apply to two real datasets, one from epigenomics and one ion channel recording.

Mach Learn manuscript No. (will be inserted by the editor) Abstract We present a new application and covering number bound for the framework of "Machine Learning with Operational Costs (MLOC)," which is an exploratory form of decision theory. The MLOC framework incorporates knowledge about how a predictive model will be used for a subsequent task, thus combining machine learning with the decision that is made afterwards. In this work, we use the MLOC framework to study a problem that has implications for power grid reliability and maintenance, called the Machine Learning and Traveling Repairman Problem (ML&TRP). The goal of the ML&TRP is to determine a route for a "repair crew," which repairs nodes on a graph. The repair crew aims to minimize the cost of failures at the nodes, but as in many real situations, the failure probabilities are not known and must be estimated. The MLOC framework allows us to understand how this uncertainty influences the repair route. Keywords decision theory · generalization bound · constrained linear function classes · covering numbers · traveling repairman · mixed-integer programming 1 Introduction In many domains, it is essential to understand how uncertainty in predictions influences decision-making. Funding for Theja Tulabandhula was provided by a Fulbright Fellowship and Xerox Fellowship. Cynthia Rudin's work on this project was funded in part by Con Edison, by the MIT Energy Initiative Seed Fund, and NSF grant IIS-1053407. The new framework of Machine Learning with Operational Costs (MLOC) (Tulabandhula and Rudin, 2013) provides a mechanism to do this, and is a type of exploratory decision theory. Where usual decision theories provide a single policy that minimizes expected costs, the MLOC framework is able to produce a range of reasonable policies that span the full set of reasonable costs. To do this, the operational cost becomes a regularization term within the machine learning model, and adjusting the regularization constant allows us to explore solutions for all reasonable costs. This gives decision makers a way to understand the uncertainty in their predictive model in terms of something they can grasp - uncertainty in the cost to solve the problem. The MLOC framework can also be used in another way, namely to incorporate prior knowledge about the cost to produce a better predictive model.

Experimental life sciences like biology or chemistry have seen in the recent decades an explosion of the data available from experiments. Laboratory instruments become more and more complex and report hundreds or thousands measurements for a single experiment and therefore the statistical methods face challenging tasks when dealing with such high dimensional data. However, much of the data is highly redundant and can be efficiently brought down to a much smaller number of variables without a significant loss of information. The mathematical procedures making possible this reduction are called dimensionality reduction techniques; they have widely been developed by fields like Statistics or Machine Learning, and are currently a hot research topic. In this review we categorize the plethora of dimension reduction techniques available and give the mathematical insight behind them.

There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for the ridge regression problem within the framework of abstract Wiener spaces and show how the support vector machine solution to such problems can be interpreted in terms of the Gaussian Radon transform.

Scientific practice typically involves repeatedly studying a system, each time trying to unravel a different perspective. In each study, the scientist may take measurements under different experimental conditions (interventions, manipulations, perturbations) and measure different sets of quantities (variables). The result is a collection of heterogeneous data sets coming from different data distributions. In this work, we present algorithm COmbINE, which accepts a collection of data sets over overlapping variable sets under different experimental conditions; COmbINE then outputs a summary of all causal models indicating the invariant and variant structural characteristics of all models that simultaneously fit all of the input data sets. COmbINE converts estimated dependencies and independencies in the data into path constraints on the data-generating causal model and encodes them as a SAT instance. The algorithm is sound and complete in the sample limit. To account for conflicting constraints arising from statistical errors, we introduce a general method for sorting constraints in order of confidence, computed as a function of their corresponding p-values. In our empirical evaluation, COmbINE outperforms in terms of efficiency the only pre-existing similar algorithm; the latter additionally admits feedback cycles, but does not admit conflicting constraints which hinders the applicability on real data. As a proof-of-concept, COmbINE is employed to co-analyze 4 real, mass-cytometry data sets measuring phosphorylated protein concentrations of overlapping protein sets under 3 different interventions.

Multi-label classification has attracted an increasing amount of attention in recent years. To this end, many algorithms have been developed to classify multi-label data in an effective manner. However, they usually do not consider the pairwise relations indicated by sample labels, which actually play important roles in multi-label classification. Inspired by this, we naturally extend the traditional pairwise constraints to the multi-label scenario via a flexible thresholding scheme. Moreover, to improve the generalization ability of the classifier, we adopt a boosting-like strategy to construct a multi-label ensemble from a group of base classifiers. To achieve these goals, this paper presents a novel multi-label classification framework named Variable Pairwise Constraint projection for Multi-label Ensemble (VPCME). Specifically, we take advantage of the variable pairwise constraint projection to learn a lower-dimensional data representation, which preserves the correlations between samples and labels. Thereafter, the base classifiers are trained in the new data space. For the boosting-like strategy, we employ both the variable pairwise constraints and the bootstrap steps to diversify the base classifiers. Empirical studies have shown the superiority of the proposed method in comparison with other approaches.