2,573 research outputs found
Detecting a Vector Based on Linear Measurements
We consider a situation where the state of a system is represented by a
real-valued vector. Under normal circumstances, the vector is zero, while an
event manifests as non-zero entries in this vector, possibly few. Our interest
is in the design of algorithms that can reliably detect events (i.e., test
whether the vector is zero or not) with the least amount of information. We
place ourselves in a situation, now common in the signal processing literature,
where information about the vector comes in the form of noisy linear
measurements. We derive information bounds in an active learning setup and
exhibit some simple near-optimal algorithms. In particular, our results show
that the task of detection within this setting is at once much easier, simpler
and different than the tasks of estimation and support recovery
The Sparse Poisson Means Model
We consider the problem of detecting a sparse Poisson mixture. Our results
parallel those for the detection of a sparse normal mixture, pioneered by
Ingster (1997) and Donoho and Jin (2004), when the Poisson means are larger
than logarithmic in the sample size. In particular, a form of higher criticism
achieves the detection boundary in the whole sparse regime. When the Poisson
means are smaller than logarithmic in the sample size, a different regime
arises in which simple multiple testing with Bonferroni correction is enough in
the sparse regime. We present some numerical experiments that confirm our
theoretical findings
On the convergence of maximum variance unfolding
Maximum Variance Unfolding is one of the main methods for (nonlinear)
dimensionality reduction. We study its large sample limit, providing specific
rates of convergence under standard assumptions. We find that it is consistent
when the underlying submanifold is isometric to a convex subset, and we provide
some simple examples where it fails to be consistent
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