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Compressive Sensing over Graphs

Weiyu Xu, Enrique Mallada, Ao Tang

arXiv:1008.0919v1cs.ITcs.NImath.IT

TL;DR

Network tomography seeks link characteristics from aggregate measurements when direct per-link monitoring is costly, but graph topology constrains which measurements can be taken. This paper studies sparse link recovery using path-based compressive sensing. It reports O(k log(n)) path measurements for recovering k-sparse link vectors and a theoretical recovery guarantee for ℓ1 minimization.

  • Problem

    The paper asks whether sparse link characteristics can be recovered from underdetermined aggregate observations when measurement paths are constrained by network topology.

    Source: §I · p. 1

  • Method

    The paper models link parameters as sparse vectors and uses additive measurements collected along graph paths, including matrices generated from regularized random walks.

    Source: §I · p. 1, §IV · pp. 4–5

  • Results

    O(k log(n)) path measurements suffice for any k-sparse link vector, and ℓ1 minimization has a theoretical recovery guarantee with that measurement order.

    Source: §I · p. 1

  • Takeaways & Limitations

    The paper concludes that sparse link vectors can be recovered under graph path constraints for sufficiently connected graphs.

    Source: §VI · p. 9

    Further research is needed to find efficient ways to construct measurement paths.

    Source: §VI · p. 9

Abstract

from arXiv · show

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.

I. INTRODUCTION

The paper asks whether sparse link characteristics can be recovered from indirect network measurements despite graph-imposed path constraints. It studies this as compressive sensing over graphs and presents recovery guarantees using O(k log(n)) path measurements, including an ℓ1 minimization guarantee.

I. INTRODUCTION

  • Network tomography uses aggregate end-to-end measurements to infer link characteristics when direct per-link monitoring is costly or operationally difficult.The paper situates this problem in network tomography, which is useful for traffic engineering and fault diagnosis.Source: §I · p. 1
  • The paper studies network tomography through compressive sensing, targeting sparse vectors that represent link parameters.A motivating example is that only a small fraction of network links may experience congestion or large packet loss rates.Source: §I · p. 1
  • The central question is whether individual link characteristics can be recovered from underdetermined observations constrained by network topology.Existing compressive-sensing assumptions may not hold because measurement matrices are restricted and measurements must obey path constraints.Source: §I · p. 1
  • O(k log(n)) path measurements suffice to identify any k-sparse link vector, such as one marking congested links.The measurements must follow paths over the graph.Source: §I · p. 1
  • ℓ1 minimization has a theoretical guarantee for recovering any k-sparse link vector from O(k log(n)) path measurements.The paper also describes this recovery as computationally efficient.Source: §I · p. 1

II. PROBLEM FORMULATION AND RELATED WORKS

The problem is to infer sparse link parameters from aggregate measurements whose routes must be connected paths in an undirected graph. The formulation makes each measurement an additive linear mixture of link values and contrasts these real-valued measurements with graph-constrained group testing.

II. PROBLEM FORMULATION AND RELATED WORKS

  • The network is modeled as an undirected graph, and each communication route must be a connected path over its edges.The graph has node set V and link set E.Source: §II · pp. 1–2
  • Each probe measures the sum of link parameters along its source-destination route, yielding a linear mixture y of the link vector x.The link vector can represent delay or −log(1−Pj) for link packet-loss rate Pj; y records the corresponding end-to-end measurement.Source: §II · p. 2
  • The path matrix A records whether each route traverses each link, while its nonzero entries must correspond to a connected path.Entries are 0 when a route omits a link and typically 1 when it traverses the link; traversals can be counted by a positive integer.Source: §II · p. 2
  • Although there are more links than probe paths, sparse link vectors may still be recoverable from the underdetermined measurements.The motivating case is that only a small fraction of links have substantially larger delays than other links.Source: §II · p. 2
  • The paper notes that link vectors of interest can be nonnegative, including delay vectors and inverse-log packet-loss vectors.Source: §II · p. 2
  • Unlike compressive sensing, graph-constrained group testing reports binary outcomes using logical OR rather than real-valued additive measurements.The paper's example gives y = (5, 0, 3, 0) for compressive sensing, while group testing reports (Y, N, Y, N).Source: §II · p. 2

III. WHEN IS COMPRESSIVE SENSING OVER GRAPHS POSSIBLE?

This section asks how many path observations suffice to recover any k-sparse vector of network edge failures. It presents progressively more demanding matrix conditions and then uses random-walk-generated measurement matrices to obtain an O(k log(n)) measurement bound.

III. WHEN IS COMPRESSIVE SENSING OVER GRAPHS POSSIBLE?

  • Random-walk-generated measurement matrices recover any k-sparse vector using O(k log(n)) measurements, according to the section's stated result.Source: §III · p. 2
  • The section gives three increasingly demanding conditions on measurement matrix A that guarantee recovery of k-sparse link vectors.These conditions are presented in Theorems 1, 2, and 3.Source: §III · p. 2
  • The section frames its analysis around the number of path observations needed to recover any k network edge failure.Source: §III · p. 2

A. Success Conditions for Compressive Sensing

The section gives conditions under which sparse solutions to y = Ax are unique, including criteria for nonnegative signals. It also gives a sufficient condition based on singleton rows in column subsets.

A. Success Conditions for Compressive Sensing

  • For nonnegative signals, uniqueness is characterized through the null space of A and the positive and negative nonzero counts of its vectors.The theorem also gives a converse: when its condition fails, a nonnegative k-sparse vector need not be the unique sparsest nonnegative solution.Source: A. Success Conditions for Compressive… · p. 3
  • The paper presents a stricter but easier-to-use recovery condition for arbitrary, not necessarily nonnegative, sparse vectors.Source: A. Success Conditions for Compressive… · p. 3
  • For arbitrary signals, a null-space condition ensures that x is the unique sparsest vector satisfying y = Ax.The supplied passage states the uniqueness consequence but does not include the full condition.Source: A. Success Conditions for Compressive… · p. 3
  • If every subset of at most h columns has a row with exactly one nonzero entry, the section gives a sufficient condition for unique recovery of sparse signals.The theorem states the sparsity bound as k < h+1; the following passage continues that bound incompletely.Source: A. Success Conditions for Compressive… · p. 3

B. How Many Measurement Paths are Needed?

The section asks how many measurement paths suffice to recover any k-sparse link vector on a sufficiently connected graph with n nodes.

B. How Many Measurement Paths are Needed?

1) Graph Assumptions:

The section specifies graph properties used in the measurement analysis, including degree uniformity and a random-walk mixing-time definition.

1) Graph Assumptions:

  • A (D, c) uniform graph has every vertex degree between D and cD, for a constant c.Source: 1) Graph Assumptions · p. 3
  • The section defines δ-mixing time as the smallest walk length after which a walk starting at any vertex has a distribution within δ in infinity norm of the stationary distribution.Source: 1) Graph Assumptions · p. 3

2) O(k log(n)) measurements are sufficient:

The analysis uses random walks to construct path-constrained measurements and derives a measurement bound for sparse-link recovery. Its argument uses rows that isolate a single edge within a chosen edge set.

2) O(k log(n)) measurements are sufficient:

IV. ℓ1 MINIMIZATION DECODING

This section establishes that ℓ1 minimization can recover sparse edge or link signals from graph-constrained path measurements when the measurement matrix has the required expansion property. It derives this recovery guarantee using expander properties and a null-space condition based on controlling collisions in the bipartite graph representation.

IV. ℓ1 MINIMIZATION DECODING

  • Regularized walks preserve the edges visited by the original walk while traversing each edge no more than twice, bounding the maximum entry of A by 2.The bipartite graph for A connects each edge node to the measurement nodes whose walks traverse that edge.Source: §IV · p. 5
  • For any edge set S of size k, the walk length can be chosen so the conditional probability of revisiting S after first entering it is at most η, arbitrarily close to zero.The same bound extends to probabilities of visiting successive edges from S.Source: §IV · p. 5
  • With m = O(T(n)k log(n)) good-start walks, the measurement matrix is a high-probability (k, 1 − (1 − η)/(1 + ǫ′)) expander.The expansion result follows from bounding how often walks revisit a small edge set and controlling the number of walks that visit each set.Source: §IV · pp. 6–7
  • With m = O(T(n)k log(n)) walks, every column of A has a number of nonzero entries between O(log(n)) and O(T(n) log(n)) with high probability.These column degrees bound the degrees of the corresponding edge nodes in the bipartite graph.Source: §IV · p. 7
  • With probability 1 − o(1), ℓ1 minimization recovers any Θ(k)-sparse edge vector using m = O(T(n)k log(n)) good-start regularized random walks of length t = O(|E|/k).The walks are independent, and their starting vertices are chosen with probability proportional to degree.Source: §IV · pp. 4, 8
  • The proof establishes the ℓ1 null-space requirement for |K′| = Θ(k) using a bipartite (k, ǫ) expander, with ǫ made arbitrarily small by choosing t and m.The expander property holds with high probability.Source: §IV · p. 8
  • The null-space proof follows [2] while accounting for uneven nonzero entries in A and unequal degrees of left-side nodes.Source: §IV · p. 8
  • Expansion bounds collisions as links are added in decreasing signal-amplitude order, controlling the resulting loss in the ℓ1 norm.The argument also uses that entries of A are upper bounded by 2.Source: §IV · p. 8
  • With the expansion property, ℓ1 minimization recovers k′-sparse signals using O(T(n)k log(n)) measurements.The stated condition is obtained by choosing t and m appropriately, as allowed by Theorems 7 and 8.Source: §IV · p. 8

V. NUMERICAL EXAMPLES

The simulations use random-walk measurements constrained by graph topology to test ℓ1 recovery. On a 50-node complete graph, recovery with half the measurements depends on whether edge values may be negative; a second example examines measurement count versus recoverable sparsity on a random graph.

Simulation setup

  • Across the simulations, the measurement matrix is generated from independent random walks whose lengths are constrained by the graph topology.Source: §V · p. 8

Example 1

  • With 50% of measurements saved, ℓ1 minimization recovers real-valued sparse vectors up to 17% nonzeros and nonnegative vectors up to about 24% on the complete graph.The simulation uses 612 random walks of length 612 on a 50-node complete graph with 1,225 edges.Source: §V · p. 8, Figure 3 · p. 9

Example 2

  • The random-graph example plots the relationship between measurement count and maximum recoverable sparsity, counting recovery when 99% of k-sparse vectors succeed.The graph has 50 nodes, with each possible edge included independently with probability 0.5; random-walk length is one third of |E|.Source: §V · p. 8, Figure 4 · p. 9

VI. CONCLUSION

The paper studies graph-constrained compressive sensing and establishes that O(k log(n)) path measurements suffice for recovering k-sparse link vectors on sufficiently connected graphs. It also gives ℓ1 minimization theoretical guarantees, while identifying noisy signals and more efficient path construction and algorithms as open directions.

Conclusion

  • For sufficiently connected graphs, O(k log(n)) path measurements suffice to recover any link vector with at most k nonzero elements.Source: §VI · p. 9
  • The paper demonstrates theoretical recovery guarantees using ℓ1 minimization for sparse link vectors under graph path constraints.Source: §VI · p. 9
  • The analysis covers ideally sparse signals; noisy measurements, efficient path construction, nonlinear measurements, low-rank recovery [26], and more efficient polynomial-time algorithms [23] remain future directions.Source: §VI · p. 9
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