Sampling and Recovery of Signals on a Simplicial Complex using Neighbourhood Aggregation

In this work, we focus on sampling and recovery of signals over simplicial complexes. In particular, we subsample a simplicial signal of a certain order and focus on recovering multi-order bandlimited simplicial signals of one order higher and one order lower. To do so, we assume that the simplicial signal admits the Helmholtz decomposition that relates simplicial signals of these different orders. Next, we propose an aggregation sampling scheme for simplicial signals based on the Hodge Laplacian matrix and a simple least squares estimator for recovery. We also provide theoretical conditions on the number of aggregations and size of the sampling set required for faithful reconstruction as a function of the bandwidth of simplicial signals to be recovered. Numerical experiments are provided to show the effectiveness of the proposed method

Forecasting Time Series with VARMA Recursions on Graphs

Graph-based techniques emerged as a choice to deal with the dimensionality issues in modeling multivariate time series. However, there is yet no complete understanding of how the underlying structure could be exploited to ease this task. This work provides contributions in this direction by considering the forecasting of a process evolving over a graph. We make use of the (approximate) time-vertex stationarity assumption, i.e., timevarying graph signals whose first and second order statistical moments are invariant over time and correlated to a known graph topology. The latter is combined with VAR and VARMA models to tackle the dimensionality issues present in predicting the temporal evolution of multivariate time series. We find out that by projecting the data to the graph spectral domain: (i) the multivariate model estimation reduces to that of fitting a number of uncorrelated univariate ARMA models and (ii) an optimal low-rank data representation can be exploited so as to further reduce the estimation costs. In the case that the multivariate process can be observed at a subset of nodes, the proposed models extend naturally to Kalman filtering on graphs allowing for optimal tracking. Numerical experiments with both synthetic and real data validate the proposed approach and highlight its benefits over state-of-the-art alternatives.Comment: submitted to the IEEE Transactions on Signal Processin

Sampling and Reconstruction of Signals on Product Graphs

In this paper, we consider the problem of subsampling and reconstruction of signals that reside on the vertices of a product graph, such as sensor network time series, genomic signals, or product ratings in a social network. Specifically, we leverage the product structure of the underlying domain and sample nodes from the graph factors. The proposed scheme is particularly useful for processing signals on large-scale product graphs. The sampling sets are designed using a low-complexity greedy algorithm and can be proven to be near-optimal. To illustrate the developed theory, numerical experiments based on real datasets are provided for sampling 3D dynamic point clouds and for active learning in recommender systems.Comment: 5 pages, 3 figure

Extended Definitions of Spectrum of a Sampled Signal

It is shown that a number of equivalent choices for the calculation of the spectrum of a sampled signal are possible. Two such choices are presented in this paper. It is illustrated that the proposed calculations are more physically relevant than the definition currently in use

Sparse Sampling for Inverse Problems with Tensors

We consider the problem of designing sparse sampling strategies for multidomain signals, which can be represented using tensors that admit a known multilinear decomposition. We leverage the multidomain structure of tensor signals and propose to acquire samples using a Kronecker-structured sensing function, thereby circumventing the curse of dimensionality. For designing such sensing functions, we develop low-complexity greedy algorithms based on submodular optimization methods to compute near-optimal sampling sets. We present several numerical examples, ranging from multi-antenna communications to graph signal processing, to validate the developed theory.Comment: 13 pages, 7 figure