5,917,436 research outputs found
Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach
This paper develops theoretical results regarding noisy 1-bit compressed
sensing and sparse binomial regression. We show that a single convex program
gives an accurate estimate of the signal, or coefficient vector, for both of
these models. We demonstrate that an s-sparse signal in R^n can be accurately
estimated from m = O(slog(n/s)) single-bit measurements using a simple convex
program. This remains true even if each measurement bit is flipped with
probability nearly 1/2. Worst-case (adversarial) noise can also be accounted
for, and uniform results that hold for all sparse inputs are derived as well.
In the terminology of sparse logistic regression, we show that O(slog(n/s))
Bernoulli trials are sufficient to estimate a coefficient vector in R^n which
is approximately s-sparse. Moreover, the same convex program works for
virtually all generalized linear models, in which the link function may be
unknown. To our knowledge, these are the first results that tie together the
theory of sparse logistic regression to 1-bit compressed sensing. Our results
apply to general signal structures aside from sparsity; one only needs to know
the size of the set K where signals reside. The size is given by the mean width
of K, a computable quantity whose square serves as a robust extension of the
dimension.Comment: 25 pages, 1 figure, error fixed in Lemma 4.
Dimension reduction by random hyperplane tessellations
Given a subset K of the unit Euclidean sphere, we estimate the minimal number
m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense
that the fraction of the hyperplanes separating any pair x, y in K is nearly
proportional to the Euclidean distance between x and y. Random hyperplanes
prove to be almost ideal for this problem; they achieve the almost optimal
bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map
that sends x in K to the sign vector with respect to the hyperplanes, we
conclude that every bounded subset K of R^n embeds into the Hamming cube {-1,
1}^m with a small distortion in the Gromov-Haussdorf metric. Since for many
sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension
reduction for sets in Euclidean spaces.Comment: 17 pages, 3 figures, minor update
One-bit compressed sensing by linear programming
We give the first computationally tractable and almost optimal solution to
the problem of one-bit compressed sensing, showing how to accurately recover an
s-sparse vector x in R^n from the signs of O(s log^2(n/s)) random linear
measurements of x. The recovery is achieved by a simple linear program. This
result extends to approximately sparse vectors x. Our result is universal in
the sense that with high probability, one measurement scheme will successfully
recover all sparse vectors simultaneously. The argument is based on solving an
equivalent geometric problem on random hyperplane tessellations.Comment: 15 pages, 1 figure, to appear in CPAM. Small changes based on referee
comment
The generalized Lasso with non-linear observations
We study the problem of signal estimation from non-linear observations when
the signal belongs to a low-dimensional set buried in a high-dimensional space.
A rough heuristic often used in practice postulates that non-linear
observations may be treated as noisy linear observations, and thus the signal
may be estimated using the generalized Lasso. This is appealing because of the
abundance of efficient, specialized solvers for this program. Just as noise may
be diminished by projecting onto the lower dimensional space, the error from
modeling non-linear observations with linear observations will be greatly
reduced when using the signal structure in the reconstruction. We allow general
signal structure, only assuming that the signal belongs to some set K in R^n.
We consider the single-index model of non-linearity. Our theory allows the
non-linearity to be discontinuous, not one-to-one and even unknown. We assume a
random Gaussian model for the measurement matrix, but allow the rows to have an
unknown covariance matrix. As special cases of our results, we recover
near-optimal theory for noisy linear observations, and also give the first
theoretical accuracy guarantee for 1-bit compressed sensing with unknown
covariance matrix of the measurement vectors.Comment: 21 page
A probabilistic and RIPless theory of compressed sensing
This paper introduces a simple and very general theory of compressive
sensing. In this theory, the sensing mechanism simply selects sensing vectors
independently at random from a probability distribution F; it includes all
models - e.g. Gaussian, frequency measurements - discussed in the literature,
but also provides a framework for new measurement strategies as well. We prove
that if the probability distribution F obeys a simple incoherence property and
an isotropy property, one can faithfully recover approximately sparse signals
from a minimal number of noisy measurements. The novelty is that our recovery
results do not require the restricted isometry property (RIP) - they make use
of a much weaker notion - or a random model for the signal. As an example, the
paper shows that a signal with s nonzero entries can be faithfully recovered
from about s log n Fourier coefficients that are contaminated with noise.Comment: 36 page
Average-case Hardness of RIP Certification
The restricted isometry property (RIP) for design matrices gives guarantees
for optimal recovery in sparse linear models. It is of high interest in
compressed sensing and statistical learning. This property is particularly
important for computationally efficient recovery methods. As a consequence,
even though it is in general NP-hard to check that RIP holds, there have been
substantial efforts to find tractable proxies for it. These would allow the
construction of RIP matrices and the polynomial-time verification of RIP given
an arbitrary matrix. We consider the framework of average-case certifiers, that
never wrongly declare that a matrix is RIP, while being often correct for
random instances. While there are such functions which are tractable in a
suboptimal parameter regime, we show that this is a computationally hard task
in any better regime. Our results are based on a new, weaker assumption on the
problem of detecting dense subgraphs
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