13 research outputs found
Sequential Changepoint Approach for Online Community Detection
We present new algorithms for detecting the emergence of a community in large
networks from sequential observations. The networks are modeled using
Erdos-Renyi random graphs with edges forming between nodes in the community
with higher probability. Based on statistical changepoint detection
methodology, we develop three algorithms: the Exhaustive Search (ES), the
mixture, and the Hierarchical Mixture (H-Mix) methods. Performance of these
methods is evaluated by the average run length (ARL), which captures the
frequency of false alarms, and the detection delay. Numerical comparisons show
that the ES method performs the best; however, it is exponentially complex. The
mixture method is polynomially complex by exploiting the fact that the size of
the community is typically small in a large network. However, it may react to a
group of active edges that do not form a community. This issue is resolved by
the H-Mix method, which is based on a dendrogram decomposition of the network.
We present an asymptotic analytical expression for ARL of the mixture method
when the threshold is large. Numerical simulation verifies that our
approximation is accurate even in the non-asymptotic regime. Hence, it can be
used to determine a desired threshold efficiently. Finally, numerical examples
show that the mixture and the H-Mix methods can both detect a community quickly
with a lower complexity than the ES method.Comment: Submitted to 2014 INFORMS Workshop on Data Mining and Analytics and
an IEEE journa
Sequential Sensing with Model Mismatch
We characterize the performance of sequential information guided sensing,
Info-Greedy Sensing, when there is a mismatch between the true signal model and
the assumed model, which may be a sample estimate. In particular, we consider a
setup where the signal is low-rank Gaussian and the measurements are taken in
the directions of eigenvectors of the covariance matrix in a decreasing order
of eigenvalues. We establish a set of performance bounds when a mismatched
covariance matrix is used, in terms of the gap of signal posterior entropy, as
well as the additional amount of power required to achieve the same signal
recovery precision. Based on this, we further study how to choose an
initialization for Info-Greedy Sensing using the sample covariance matrix, or
using an efficient covariance sketching scheme.Comment: Submitted to IEEE for publicatio
Info-Greedy sequential adaptive compressed sensing
We present an information-theoretic framework for sequential adaptive
compressed sensing, Info-Greedy Sensing, where measurements are chosen to
maximize the extracted information conditioned on the previous measurements. We
show that the widely used bisection approach is Info-Greedy for a family of
-sparse signals by connecting compressed sensing and blackbox complexity of
sequential query algorithms, and present Info-Greedy algorithms for Gaussian
and Gaussian Mixture Model (GMM) signals, as well as ways to design sparse
Info-Greedy measurements. Numerical examples demonstrate the good performance
of the proposed algorithms using simulated and real data: Info-Greedy Sensing
shows significant improvement over random projection for signals with sparse
and low-rank covariance matrices, and adaptivity brings robustness when there
is a mismatch between the assumed and the true distributions.Comment: Preliminary results presented at Allerton Conference 2014. To appear
in IEEE Journal Selected Topics on Signal Processin
On the Power of Adaptivity in Matrix Completion and Approximation
We consider the related tasks of matrix completion and matrix approximation
from missing data and propose adaptive sampling procedures for both problems.
We show that adaptive sampling allows one to eliminate standard incoherence
assumptions on the matrix row space that are necessary for passive sampling
procedures. For exact recovery of a low-rank matrix, our algorithm judiciously
selects a few columns to observe in full and, with few additional measurements,
projects the remaining columns onto their span. This algorithm exactly recovers
an rank matrix using observations,
where is a coherence parameter on the column space of the matrix. In
addition to completely eliminating any row space assumptions that have pervaded
the literature, this algorithm enjoys a better sample complexity than any
existing matrix completion algorithm. To certify that this improvement is due
to adaptive sampling, we establish that row space coherence is necessary for
passive sampling algorithms to achieve non-trivial sample complexity bounds.
For constructing a low-rank approximation to a high-rank input matrix, we
propose a simple algorithm that thresholds the singular values of a zero-filled
version of the input matrix. The algorithm computes an approximation that is
nearly as good as the best rank- approximation using
samples, where is a slightly different coherence parameter on the matrix
columns. Again we eliminate assumptions on the row space
Adaptive Compressed Sensing for Support Recovery of Structured Sparse Sets
This paper investigates the problem of recovering the support of structured
signals via adaptive compressive sensing. We examine several classes of
structured support sets, and characterize the fundamental limits of accurately
recovering such sets through compressive measurements, while simultaneously
providing adaptive support recovery protocols that perform near optimally for
these classes. We show that by adaptively designing the sensing matrix we can
attain significant performance gains over non-adaptive protocols. These gains
arise from the fact that adaptive sensing can: (i) better mitigate the effects
of noise, and (ii) better capitalize on the structure of the support sets.Comment: to appear in IEEE Transactions on Information Theor
Recovering Graph-Structured Activations using Adaptive Compressive Measurements
<p>We study the localization of a cluster of activated vertices in a graph, from adaptively designed compressive measurements. We propose a hierarchical partitioning of the graph that groups the activated vertices into few partitions, so that a top-down sensing procedure can identify these partitions, and hence the activations, using few measurements. By exploiting the cluster structure, we are able to provide localization guarantees at weaker signal to noise ratios than in the unstructured setting. We complement this performance guarantee with an information theoretic lower bound, providing a necessary signal-to-noise ratio for any algorithm to successfully localize the cluster. We verify our analysis with some simulations, demonstrating the practicality of our algorithm.</p