Fernando Gama

Rethinking sketching as sampling: Linear transforms of graph signals

Sampling of bandlimited graph signals has well-documented merits for dimensionality reduction, affordable storage, and online processing of streaming network data. Most existing sampling methods are designed to minimize the error incurred when reconstructing the original signal from its samples. Oftentimes these parsimonious signals serve as inputs to computationally-intensive linear transformations (e.g., graph filters). Hence, interest shifts from reconstructing the signal itself towards instead approximating the output of the prescribed linear operator efficiently. In this context, we propose a novel sampling scheme that leverages the bandlimitedness of the input as well as the transformation whose output we wish to approximate. We formulate problems to jointly optimize sample selection and a sketch of the target linear transformation, so when the latter is affordably applied to the sampled input signal the result is close to the desired output. The developed sampling plus reduced-complexity processing pipeline is particularly useful for streaming data, where the linear transform has to be applied fast and repeatedly to successive inputs.

Deepest Minimum Criterion for Biased Affine Estimation

A new strategy called the Deepest Minimum Criterion (DMC) is presented for optimally obtaining an affine transformation of a given unbiased estimator, when a-priori information on the parameters is known. Here, it is considered that the samples are drawn from a distribution parametrized by an unknown deterministic vector parameter. The a-priori information on the true parameter vector is available in the form of a known subset of the parameter space to which the true parameter vector belongs. A closed form exact solution is given for the non-linear DMC problem in which it is known that the true parameter vector belongs to an ellipsoidal ball and the covariance matrix of the unbiased estimator does not depend on the parameters. A closed form exact solution is also given for the Min-Max strategy for this same case.

Weak law of large numbers for stationary graph processes

The ability to obtain accurate estimators from a set of measurements is a key factor in science and engineering. Typically, there is an inherent assumption that the measurements were taken in a sequential order, be it in space or time. However, data is increasingly irregular so this assumption of sequentially obtained measurements no longer holds. By leveraging notions of graph signal processing to account for these irregular domains, we propose an unbiased estimator for the mean of a wide sense stationary graph process based on the diffusion of a single realization. We also provide a bound on the estimation error and determine the conditions for a specific rate of convergence of the estimator to the mean, in a weak law of large numbers fashion.

Overlapping clustering of network data using cut metrics

We present a novel method to hierarchically cluster networked data allowing nodes to simultaneously belong to multiple clusters. Given a network, our method outputs a cut metric on the underlying node set, which can be related to data coverings at different resolutions. The cut metric is obtained by averaging a set of ultrametrics, which are themselves the output of (non-overlapping) hierarchically clustering noisy versions of the original network of interest. The resulting algorithm is illustrated in synthetic networks and is used to classify handwritten digits from the MNIST database.

Hierarchical Overlapping Clustering of Network Data Using Cut Metrics

A novel method to obtain hierarchical and overlapping clusters from network data—i.e., a set of nodes endowed with pairwise dissimilarities—is presented. The introduced method is hierarchical in the sense that it outputs a nested collection of groupings of the node set depending on the resolution or degree of similarity desired, and it is overlapping since it allows nodes to belong to more than one group. Our construction is rooted on the facts that a hierarchical (non-overlapping) clustering of a network can be equivalently represented by a finite ultrametric space and that a convex combination of ultrametrics results in a cut metric. By applying a hierarchical (non-overlapping) clustering method to multiple dithered versions of a given network, and then, convexly combining the resulting ultrametrics, we obtain a cut metric associated to the network of interest. We then show how to extract a hierarchical overlapping clustering structure from the aforementioned cut metric. Furthermore, the so-called overlapping function is presented as a tool for gaining insights about the data by identifying meaningful resolutions of the obtained hierarchical structure. Additionally, we explore hierarchical overlapping quasi-clustering methods that preserve the asymmetry of the data contained in directed networks. Finally, the presented method is illustrated via synthetic and real-world classification problems including handwritten digit classification and authorship attribution of famous plays.

Analysis and Comparison of Biased Affine Estimators

Affine biased estimation is particularly useful when there is some a-priori knowledge on the parameters that can be exploited in adverse situations (when the number of samples is low, or the noise is high). Three different affine estimation strategies are discussed, namely the Deepest Minimum Criterion (DMC), the Min-Max (MM), and the Linear Matrix Inequality (LMI) strategies, and closed form expressions are obtained for all of them, for the case when the a priori knowledge is given in the form of ellipsoidal constraints on the parameter space, and when the covariance matrix of the unbiased estimator is constant. A relationship between affine estimation and Bayesian estimation of the mean of a multivariate Gaussian distribution with Gaussian prior is established and it is shown how affine estimation theory can help in the choice of the Gaussian prior distribution.

A new approach for biased affine estimation

The problem of biased affine estimation is discussed in the present paper. Affine estimation is a technique for improving parameter estimation through the inclusion of a-priori information in a non-bayesian setting. Here, a new optimality criterion for affine estimation is presented and developed. The closed form optimal transformation for this criterion is obtained through the use of the KKT conditions, as the new criterion is posed as a convex optimization problem. This new criterion is compared with other affine estimation criteria through a numerical example.

Alternative Admissible Biased Estimators

In this work, a biased estimator obtained from an affine transformation of an unbiased estimator will be analyzed. In particular, the method proposed by Y. C. Eldar will be studied, and alternative approaches will be developed for the construction of biased estimators through transformations of unbiased ones. It is finally shown through simulations, that both estimators proposed in this work perform better than other estimators analyzed.