7,556 research outputs found
Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices
Super-resolution is a fundamental task in imaging, where the goal is to
extract fine-grained structure from coarse-grained measurements. Here we are
interested in a popular mathematical abstraction of this problem that has been
widely studied in the statistics, signal processing and machine learning
communities. We exactly resolve the threshold at which noisy super-resolution
is possible. In particular, we establish a sharp phase transition for the
relationship between the cutoff frequency () and the separation ().
If , our estimator converges to the true values at an inverse
polynomial rate in terms of the magnitude of the noise. And when no estimator can distinguish between a particular pair of
-separated signals even if the magnitude of the noise is exponentially
small.
Our results involve making novel connections between {\em extremal functions}
and the spectral properties of Vandermonde matrices. We establish a sharp phase
transition for their condition number which in turn allows us to give the first
noise tolerance bounds for the matrix pencil method. Moreover we show that our
methods can be interpreted as giving preconditioners for Vandermonde matrices,
and we use this observation to design faster algorithms for super-resolution.
We believe that these ideas may have other applications in designing faster
algorithms for other basic tasks in signal processing.Comment: 19 page
Limits on Support Recovery with Probabilistic Models: An Information-Theoretic Framework
The support recovery problem consists of determining a sparse subset of a set
of variables that is relevant in generating a set of observations, and arises
in a diverse range of settings such as compressive sensing, and subset
selection in regression, and group testing. In this paper, we take a unified
approach to support recovery problems, considering general probabilistic models
relating a sparse data vector to an observation vector. We study the
information-theoretic limits of both exact and partial support recovery, taking
a novel approach motivated by thresholding techniques in channel coding. We
provide general achievability and converse bounds characterizing the trade-off
between the error probability and number of measurements, and we specialize
these to the linear, 1-bit, and group testing models. In several cases, our
bounds not only provide matching scaling laws in the necessary and sufficient
number of measurements, but also sharp thresholds with matching constant
factors. Our approach has several advantages over previous approaches: For the
achievability part, we obtain sharp thresholds under broader scalings of the
sparsity level and other parameters (e.g., signal-to-noise ratio) compared to
several previous works, and for the converse part, we not only provide
conditions under which the error probability fails to vanish, but also
conditions under which it tends to one.Comment: Accepted to IEEE Transactions on Information Theory; presented in
part at ISIT 2015 and SODA 201
Secure State Estimation: Optimal Guarantees against Sensor Attacks in the Presence of Noise
Motivated by the need to secure cyber-physical systems against attacks, we
consider the problem of estimating the state of a noisy linear dynamical system
when a subset of sensors is arbitrarily corrupted by an adversary. We propose a
secure state estimation algorithm and derive (optimal) bounds on the achievable
state estimation error. In addition, as a result of independent interest, we
give a coding theoretic interpretation for prior work on secure state
estimation against sensor attacks in a noiseless dynamical system.Comment: A shorter version of this work will appear in the proceedings of ISIT
201
The capacity of non-identical adaptive group testing
We consider the group testing problem, in the case where the items are
defective independently but with non-constant probability. We introduce and
analyse an algorithm to solve this problem by grouping items together
appropriately. We give conditions under which the algorithm performs
essentially optimally in the sense of information-theoretic capacity. We use
concentration of measure results to bound the probability that this algorithm
requires many more tests than the expected number. This has applications to the
allocation of spectrum to cognitive radios, in the case where a database gives
prior information that a particular band will be occupied.Comment: To be presented at Allerton 201
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