65 research outputs found
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 Geometric View on Constrained M-Estimators
We study the estimation error of constrained M-estimators, and derive
explicit upper bounds on the expected estimation error determined by the
Gaussian width of the constraint set. Both of the cases where the true
parameter is on the boundary of the constraint set (matched constraint), and
where the true parameter is strictly in the constraint set (mismatched
constraint) are considered. For both cases, we derive novel universal
estimation error bounds for regression in a generalized linear model with the
canonical link function. Our error bound for the mismatched constraint case is
minimax optimal in terms of its dependence on the sample size, for Gaussian
linear regression by the Lasso
Phase Retrieval From Binary Measurements
We consider the problem of signal reconstruction from quadratic measurements
that are encoded as +1 or -1 depending on whether they exceed a predetermined
positive threshold or not. Binary measurements are fast to acquire and
inexpensive in terms of hardware. We formulate the problem of signal
reconstruction using a consistency criterion, wherein one seeks to find a
signal that is in agreement with the measurements. To enforce consistency, we
construct a convex cost using a one-sided quadratic penalty and minimize it
using an iterative accelerated projected gradient-descent (APGD) technique. The
PGD scheme reduces the cost function in each iteration, whereas incorporating
momentum into PGD, notwithstanding the lack of such a descent property,
exhibits faster convergence than PGD empirically. We refer to the resulting
algorithm as binary phase retrieval (BPR). Considering additive white noise
contamination prior to quantization, we also derive the Cramer-Rao Bound (CRB)
for the binary encoding model. Experimental results demonstrate that the BPR
algorithm yields a signal-to- reconstruction error ratio (SRER) of
approximately 25 dB in the absence of noise. In the presence of noise prior to
quantization, the SRER is within 2 to 3 dB of the CRB
A simple tool for bounding the deviation of random matrices on geometric sets
Let be an isotropic, sub-gaussian matrix. We prove that the
process has sub-gaussian increments. Using
this, we show that for any bounded set , the
deviation of around its mean is uniformly bounded by the Gaussian
complexity of . We also prove a local version of this theorem, which allows
for unbounded sets. These theorems have various applications, some of which are
reviewed in this paper. In particular, we give a new result regarding model
selection in the constrained linear model.Comment: 16 pages. Minor correction
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