17 research outputs found
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
Multi-channel Sampling on Graphs and Its Relationship to Graph Filter Banks
In this paper, we consider multi-channel sampling (MCS) for graph signals. We
generally encounter full-band graph signals beyond the bandlimited one in many
applications, such as piecewise constant/smooth and union of bandlimited graph
signals. Full-band graph signals can be represented by a mixture of multiple
signals conforming to different generation models. This requires the analysis
of graph signals via multiple sampling systems, i.e., MCS, while existing
approaches only consider single-channel sampling. We develop a MCS framework
based on generalized sampling. We also present a sampling set selection (SSS)
method for the proposed MCS so that the graph signal is best recovered.
Furthermore, we reveal that existing graph filter banks can be viewed as a
special case of the proposed MCS. In signal recovery experiments, the proposed
method exhibits the effectiveness of recovery for full-band graph signals
Model selection-inspired coefficients optimization for polynomial-kernel graph learning
Graph learning has been extensively investigated for over a decade, in which the graph structure can be learnt from multiple graph signals (e.g., graphical Lasso) or node features (e.g., graph metric learning). Given partial graph signals, existing node feature-based graph learning approaches learn a pair-wise distance metric with gradient descent, where the number of optimization variables dramatically scale with the node feature size. To address this issue, in this paper, we propose a low-complexity model selection-inspired graph learning (MSGL) method with very few optimization variables independent with feature size, via leveraging on recent advances in graph spectral signal processing (GSP). We achieve this by 1) interpreting a finite-degree polynomial function of the graph Laplacian as a positive-definite precision matrix, 2) formulating a convex optimization problem with variables being the polynomial coefficients, 3) replacing the positive-definite cone constraint for the precision matrix with a set of linear constraints, and 4) solving efficiently the objective using the Frank-Wolfe algorithm. Using binary classification as an application example, our simulation results show that our proposed MSGL method achieves competitive performance with significant speed gains against existing node feature-based graph learning methods
Graph Signal Sampling Under Stochastic Priors
We propose a generalized sampling framework for stochastic graph signals.
Stochastic graph signals are characterized by graph wide sense stationarity
(GWSS) which is an extension of wide sense stationarity (WSS) for standard
time-domain signals. In this paper, graph signals are assumed to satisfy the
GWSS conditions and we study their sampling as well as recovery procedures. In
generalized sampling, a correction filter is inserted between sampling and
reconstruction operators to compensate for non-ideal measurements. We propose a
design method for the correction filters to reduce the mean-squared error (MSE)
between original and reconstructed graph signals. We derive the correction
filters for two cases: The reconstruction filter is arbitrarily chosen or
predefined. The proposed framework allows for arbitrary sampling methods, i.e.,
sampling in the vertex or graph frequency domain. We also show that the graph
spectral response of the resulting correction filter parallels that for
generalized sampling for WSS signals if sampling is performed in the graph
frequency domain. Furthermore, we reveal the theoretical connection between the
proposed and existing correction filters. The effectiveness of our approach is
validated via experiments by comparing its MSE with existing approaches