1,240 research outputs found
Lower Bounds for Two-Sample Structural Change Detection in Ising and Gaussian Models
The change detection problem is to determine if the Markov network structures
of two Markov random fields differ from one another given two sets of samples
drawn from the respective underlying distributions. We study the trade-off
between the sample sizes and the reliability of change detection, measured as a
minimax risk, for the important cases of the Ising models and the Gaussian
Markov random fields restricted to the models which have network structures
with nodes and degree at most , and obtain information-theoretic lower
bounds for reliable change detection over these models. We show that for the
Ising model, samples are
required from each dataset to detect even the sparsest possible changes, and
that for the Gaussian, samples are
required from each dataset to detect change, where is the smallest
ratio of off-diagonal to diagonal terms in the precision matrices of the
distributions. These bounds are compared to the corresponding results in
structure learning, and closely match them under mild conditions on the model
parameters. Thus, our change detection bounds inherit partial tightness from
the structure learning schemes in previous literature, demonstrating that in
certain parameter regimes, the naive structure learning based approach to
change detection is minimax optimal up to constant factors.Comment: Presented at the 55th Annual Allerton Conference on Communication,
Control, and Computing, Oct. 201
Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications
We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental trade-offs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdős-Rényi (n,p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomial-time (in n ) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction
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