[Paper Review] Compressive Sensing over Graphs

Weiyu Xu, Enrique Mallada+1 more|arXiv (Cornell University)|Aug 5, 2010

Sparse and Compressive Sensing Techniques16 references4 citations

TL;DR

This paper proposes compressive sensing over graphs to recover sparse link-level network parameters (e.g., congestion or delay) using only path-based measurements constrained by network topology. It shows that $O(k\log n)$ path measurements suffice to stably recover any $k$-sparse link vector, and that $\ell_1$ minimization provides theoretical guarantees for exact recovery under these constraints, even with non-negative, integer-valued measurement matrices derived from graph walks.

ABSTRACT

In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with $n$ nodes, it is shown that, using $O(k \log(n))$ path measurements, we are able to recover any $k$-sparse link vector (with no more than $k$ nonzero elements), even though the measurements have to follow the graph path constraints. We further show that the computationally efficient $\ell_1$ minimization can provide theoretical guarantees for inferring such $k$-sparse vectors with $O(k \log(n))$ path measurements from the graph.

Motivation & Objective

• To address the challenge of inferring sparse network link characteristics (e.g., delay, loss rate) from indirect, aggregate end-to-end measurements.
• To establish theoretical recovery guarantees for sparse signals over graphs when measurements are restricted to connected paths due to network topology.
• To demonstrate that $\ell_1$ minimization can provably recover $k$-sparse link vectors using only $O(k\log n)$ path measurements under graph constraints.
• To bridge compressive sensing theory with network tomography by modeling measurements as random walks over graphs.

Proposed method

• Model the network as an undirected graph $G = (V, E)$, where link parameters form a sparse vector $\mathbf{x} \in \mathbb{R}^{|E|}$.
• Construct measurement matrices $A$ from independent random walks of length $t$ over the graph, ensuring each measurement corresponds to a path sum over edges.
• Use $\ell_1$ minimization to recover the sparse vector $\mathbf{x}$ as $\arg\min_{\mathbf{z}} \|\mathbf{z}\|_1$ subject to $A\mathbf{z} = \mathbf{y}$, where $\mathbf{y}$ is the vector of path measurements.
• Leverage the $(k, \epsilon)$-expansion property of the measurement matrix $A$ to bound reconstruction error and establish recovery guarantees.
• Apply the triangular inequality for $\ell_1$-norm and analyze node connectivity and edge collision effects to derive bounds on $\|\mathbf{w}_{K'}\|_1$ for null-space vectors $\mathbf{w}$.
• Prove that if $\epsilon d_{\text{max}} / d_{\text{min}} < 1/12$, then $\ell_1$ minimization recovers any $k'$-sparse signal with $k' \leq k/2$ using $O(T(n)k\log n)$ measurements.

Experimental results

Research questions

• RQ1 Can $k$-sparse link vectors be recovered from $O(k\log n)$ path measurements when measurements are constrained to follow graph-connected paths?
• RQ2 Does $\ell_1$ minimization provide theoretical guarantees for exact recovery of sparse link parameters under graph-constrained measurements?
• RQ3 What is the minimal number of path measurements required to stably recover any $k$-sparse vector in a graph with $n$ nodes?
• RQ4 How do the expansion properties of the measurement matrix $A$ derived from random walks affect the recovery performance?

Key findings

• For a sufficiently connected graph with $n$ nodes, $O(k\log n)$ path measurements are sufficient to recover any $k$-sparse link vector, even under graph path constraints.
• $\ell_1$ minimization guarantees exact recovery of $k$-sparse vectors as long as $\epsilon d_{\text{max}} / d_{\text{min}} < 1/12$, with $O(T(n)k\log n)$ measurements.
• Numerical results show that $\ell_1$ minimization recovers real-valued sparse vectors with up to 17% nonzero elements and nonnegative sparse vectors with up to 24% nonzero elements using only 50% of the measurements in a complete graph of 50 nodes.
• In a random graph with 50 nodes and edge probability $p=0.5$, $\ell_1$ minimization achieves 99% recovery rate for $k$-sparse vectors when the number of measurements is sufficient to meet the $O(k\log n)$ threshold.
• The recovery performance is robust under non-negative constraints, with improved recovery rates for nonnegative sparse signals compared to real-valued ones.
• The theoretical framework extends to general graphs via the $(k, \epsilon)$-expansion property, ensuring stable recovery even with limited path diversity.