We consider the analysis of high dimensional data given in the form of a matrix with columns consisting of observations and rows consisting of features. Often the data is such that the observations do not reside on a regular grid, and the given order of the features is arbitrary and does not convey a notion of locality. Therefore, traditional transforms and metrics cannot be used for data organization and analysis. In this paper, our goal is to organize the data by defining an appropriate representation and metric such that they respect the smoothness and structure underlying the data. We also aim to generalize the joint clustering of observations and features in the case the data does not fall into clear disjoint groups. For this purpose, we propose multiscale data-driven transforms and metrics based on trees. Their construction is implemented in an iterative refinement procedure that exploits the co-dependencies between features and observations. Beyond the organization of a single dataset, our approach enables us to transfer the organization learned from one dataset to another and to integrate several datasets together. We present an application to breast cancer gene expression analysis: learning metrics on the genes to cluster the tumor samples into cancer sub-types and validating the joint organization of both the genes and the samples. We demonstrate that using our approach to combine information from multiple gene expression cohorts, acquired by different profiling technologies, improves the clustering of tumor samples.

This work studies the denoising of piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness over a graph, where the value at each node can be vector-valued. We extend the graph trend filtering framework to denoising vector-valued graph signals with a family of non-convex regularizers, which exhibit superior recovery performance over existing convex regularizers. Using an oracle inequality, we establish the statistical error rates of first-order stationary points of the proposed non-convex method for generic graphs. Furthermore, we present an ADMM-based algorithm to solve the proposed method and establish its convergence. Numerical experiments are conducted on both synthetic and real-world data for denoising, support recovery, event detection, and semi-supervised classification.

Graph signal processing (GSP) is a prominent framework for analyzing signals on non-Euclidean domains. The graph Fourier transform (GFT) uses the combinatorial graph Laplacian matrix to reveal the spectral decomposition of signals in the graph frequency domain. However, a common challenge in applying GSP methods is that in many scenarios the underlying graph of a system is unknown. A solution in such cases is to construct the unobserved graph from available data, which is commonly referred to as graph or network inference. Although different graph inference methods exist, they are restricted to learning from either smooth graph signals or simple additive Gaussian noise. Other types of noisy data, such as discrete counts or binary digits, are rather common in real-world applications, yet are underexplored in graph inference. In this paper, we propose a versatile graph inference framework for learning from graph signals corrupted by exponential family noise. Our framework generalizes previous methods from continuous smooth graph signals to various data types. We propose an alternating algorithm that jointly estimates the graph Laplacian and the unobserved smooth representation from the noisy signals. We also extend our approach to include an offset variable which models different levels of variation of the nodes. Since real-world graph signals are frequently non-independent and temporally correlated, we further adapt our original setting to a timevertex formulation. We demonstrate on synthetic and real-world data that our new algorithms outperform competing Laplacian estimation methods that suffer from noise model mismatch.

We study estimation of piecewise smooth signals over a graph. We propose a -norm penalized Graph Trend Filtering (GTF) model to estimate piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness across the nodes. We prove that the proposed GTF model is simultaneously a k-means clustering on the signal over the nodes and a minimum graph cut on the edges of the graph, where the clustering and the cut share the same assignment matrix. We propose two methods to solve the proposed GTF model: a spectral decomposition method and a method based on simulated annealing. In the experiment on synthetic and real-world datasets, we show that the proposed GTF model has a better performances compared with existing approaches on the tasks of denoising, support recovery and semi-supervised classification. We also show that the proposed GTF model can be solved more efficiently than existing models for the dataset with a large edge set.

Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: , where denote inputs and potentials, respectively, and the sparsity pattern of the Laplacian encodes the edge structure. Assuming to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of from temporally correlated samples of via an -regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size significantly exceeds the number of samples . We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on , we show that the ML estimate recovers the sparsity pattern of with high probability, where is the maximum degree of the graph underlying . We provide recovery guarantees for in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain).