3,056 research outputs found
Adaptive sensing performance lower bounds for sparse signal detection and support estimation
This paper gives a precise characterization of the fundamental limits of
adaptive sensing for diverse estimation and testing problems concerning sparse
signals. We consider in particular the setting introduced in (IEEE Trans.
Inform. Theory 57 (2011) 6222-6235) and show necessary conditions on the
minimum signal magnitude for both detection and estimation: if is a sparse vector with non-zero components then it
can be reliably detected in noise provided the magnitude of the non-zero
components exceeds . Furthermore, the signal support can be exactly
identified provided the minimum magnitude exceeds . Notably
there is no dependence on , the extrinsic signal dimension. These results
show that the adaptive sensing methodologies proposed previously in the
literature are essentially optimal, and cannot be substantially improved. In
addition, these results provide further insights on the limits of adaptive
compressive sensing.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ555 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Adaptive Sensing for Estimation of Structured Sparse Signals
In many practical settings one can sequentially and adaptively guide the
collection of future data, based on information extracted from data collected
previously. These sequential data collection procedures are known by different
names, such as sequential experimental design, active learning or adaptive
sensing/sampling. The intricate relation between data analysis and acquisition
in adaptive sensing paradigms can be extremely powerful, and often allows for
reliable signal estimation and detection in situations where non-adaptive
sensing would fail dramatically.
In this work we investigate the problem of estimating the support of a
structured sparse signal from coordinate-wise observations under the adaptive
sensing paradigm. We present a general procedure for support set estimation
that is optimal in a variety of cases and shows that through the use of
adaptive sensing one can: (i) mitigate the effect of observation noise when
compared to non-adaptive sensing and, (ii) capitalize on structural information
to a much larger extent than possible with non-adaptive sensing. In addition to
a general procedure to perform adaptive sensing in structured settings we
present both performance upper bounds, and corresponding lower bounds for both
sensing paradigms
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
Cliques in rank-1 random graphs: the role of inhomogeneity
We study the asymptotic behavior of the clique number in rank-1 inhomogeneous
random graphs, where edge probabilities between vertices are roughly
proportional to the product of their vertex weights. We show that the clique
number is concentrated on at most two consecutive integers, for which we
provide an expression. Interestingly, the order of the clique number is
primarily determined by the overall edge density, with the inhomogeneity only
affecting multiplicative constants or adding at most a
multiplicative factor. For sparse enough graphs the clique number is always
bounded and the effect of inhomogeneity completely vanishes.Comment: 29 page
Adaptive Selective Sampling for Online Prediction with Experts
We consider online prediction of a binary sequence with expert advice. For
this setting, we devise label-efficient forecasting algorithms, which use a
selective sampling scheme that enables collecting much fewer labels than
standard procedures, while still retaining optimal worst-case regret
guarantees. These algorithms are based on exponentially weighted forecasters,
suitable for settings with and without a perfect expert. For a scenario where
one expert is strictly better than the others in expectation, we show that the
label complexity of the label-efficient forecaster scales roughly as the square
root of the number of rounds. Finally, we present numerical experiments
empirically showing that the normalized regret of the label-efficient
forecaster can asymptotically match known minimax rates for pool-based active
learning, suggesting it can optimally adapt to benign settings
Anomaly Detection for a Large Number of Streams: A Permutation-Based Higher Criticism Approach
Anomaly detection when observing a large number of data streams is essential
in a variety of applications, ranging from epidemiological studies to
monitoring of complex systems. High-dimensional scenarios are usually tackled
with scan-statistics and related methods, requiring stringent modeling
assumptions for proper calibration. In this work we take a non-parametric
stance, and propose a permutation-based variant of the higher criticism
statistic not requiring knowledge of the null distribution. This results in an
exact test in finite samples which is asymptotically optimal in the wide class
of exponential models. We demonstrate the power loss in finite samples is
minimal with respect to the oracle test. Furthermore, since the proposed
statistic does not rely on asymptotic approximations it typically performs
better than popular variants of higher criticism that rely on such
approximations. We include recommendations such that the test can be readily
applied in practice, and demonstrate its applicability in monitoring the daily
number of COVID-19 cases in the Netherlands
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