Rohan Varma

Discrete Signal Processing on Graphs: Sampling Theory

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erdös-Rényi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification of online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.

Signal Recovery on Graphs: Fundamental Limits of Sampling Strategies

This paper builds theoretical foundations for the recovery of a newly proposed class of smooth graph signals, approximately bandlimited graph signals, under three sampling strategies: uniform sampling, experimentally designed sampling, and active sampling. We then state minimax lower bounds on the maximum risk for the approximately bandlimited class under these three sampling strategies and show that active sampling cannot fundamentally outperform experimentally designed sampling. We propose a recovery strategy to compare uniform sampling with experimentally designed sampling. As the proposed recovery strategy lends itself well to statistical analysis, we derive the exact mean square error for each sampling strategy. To study convergence rates, we introduce two types of graphs and find that 1) the proposed recovery strategy achieves the optimal rates; and 2) the experimentally designed sampling fundamentally outperforms uniform sampling for Type-2 class of graphs. To validate our proposed recovery strategy, we test it on five specific graphs: a ring graph with k nearest neighbors, an Erdos-Rényi graph, a random geometric graph, a small-world graph, and a power-law graph and find that experimental results match the proposed theory well. This paper also presents a comprehensive explanation for when and why sampling for semi-supervised learning with graphs works.

Signal recovery on graphs: Random versus experimentally designed sampling

We study signal recovery on graphs based on two sampling strategies: random sampling and experimentally designed sampling. We propose a new class of smooth graph signals, called approximately bandlimited. We then propose two recovery strategies based on random sampling and experimentally designed sampling. The proposed recovery strategy based on experimentally designed sampling uses sampling scores, which is similar to the leverage scores used in the matrix approximation. We show that while both strategies are unbiased estimators for the low-frequency components, the convergence rate of experimentally designed sampling is much faster than that of random sampling when a graph is irregular1. We validate the proposed recovery strategies on three specific graphs: a ring graph, an Erdös-Rényi graph, and a star graph. The simulation results support the theoretical analysis.

Spectrum-blind signal recovery on graphs

We consider the problem of recovering a graph signal, sparse in the graph spectral domain from a few number of samples. In contrast to most previous work on the sampling of graph signals, the setting is “spectrum-blind” where we are unaware of the graph d support of the signal. We propose a class of spectrum-blind graph signals and study two recovery strategies based on random and experimentally designed sampling inspired by the compressed sensing paradigm. We further show sampling bounds for graphs, including Erdös-Rényi random graphs. We show that experimentally designed sampling significantly outperforms random sampling for some irregular graph families.

Representations of piecewise smooth signals on graphs

We study representations of piecewise-smooth signals on graphs. We first define classes for smooth, piecewise-constant, and piecewise-smooth graph signals, followed by a series of multiresolution local sets to analyze those signals by implementing a multiresolution analysis on graphs. Based on these local sets, we propose local-set-based piecewise-constant and piecewise-smooth dictionaries as graph signal representations that, in spirit, resemble the classical Haar wavelet basis and are naturally localized in both graph vertex and graph Fourier domains. Moreover, they promote sparsity when representing piecewise-smooth graph signals. In the experiments, we show that local-set-based dictionaries outperform graph Fourier domain based representations when approximating both simulated and real-world graph signals.

Reconstructing cancer drug response networks using multitask learning

BackgroundTranslating in vitro results to clinical tests is a major challenge in systems biology. Here we present a new Multi-Task learning framework which integrates thousands of cell line expression experiments to reconstruct drug specific response networks in cancer.ResultsThe reconstructed networks correctly identify several shared key proteins and pathways while simultaneously highlighting many cell type specific proteins. We used top proteins from each drug network to predict survival for patients prescribed the drug.ConclusionsPredictions based on proteins from the in-vitro derived networks significantly outperformed predictions based on known cancer genes indicating that Multi-Task learning can indeed identify accurate drug response networks.

A statistical perspective of sampling scores for linear regression

In this paper, we consider a statistical problem of learning a linear model from noisy samples. Existing work has focused on approximating the least squares solution by using leverage-based scores as an importance sampling distribution. However, no finite sample statistical guarantees and no computationally efficient optimal sampling strategies have been proposed. To evaluate the statistical properties of different sampling strategies, we propose a simple yet effective estimator, which is easy for theoretical analysis and is useful in multitask linear regression. We derive the exact mean square error of the proposed estimator for any given sampling scores. Based on minimizing the mean square error, we propose the optimal sampling scores for both estimator and predictor, and show that they are influenced by the noise-to-signal ratio. Numerical simulations match the theoretical analysis well.

Energy-efficient route planning for autonomous aerial vehicles based on graph signal recovery

We use graph signal sampling and recovery techniques to plan routes for autonomous aerial vehicles. We propose a novel method that plans an energy-efficient flight trajectory by considering the influence of wind. We model the weather stations as nodes on a graph and model wind velocity at each station as a graph signal. We observe that the wind velocities at two close stations are similar, that is, the graph signal of wind velocities is smooth. By taking advantages of the smoothness, we only query a small fraction of it and recover the rest by using a novel graph signal recovery algorithm, which solves an optimization problem. To validate the effectiveness of the proposed method, we first demonstrate the necessity to take wind into account when planning route for autonomous aerial vehicles, and then show that the proposed method produces a reliable and energy-efficient route.