6 research outputs found
Generalized Approximate Survey Propagation for High-Dimensional Estimation
In Generalized Linear Estimation (GLE) problems, we seek to estimate a signal
that is observed through a linear transform followed by a component-wise,
possibly nonlinear and noisy, channel. In the Bayesian optimal setting,
Generalized Approximate Message Passing (GAMP) is known to achieve optimal
performance for GLE. However, its performance can significantly degrade
whenever there is a mismatch between the assumed and the true generative model,
a situation frequently encountered in practice. In this paper, we propose a new
algorithm, named Generalized Approximate Survey Propagation (GASP), for solving
GLE in the presence of prior or model mis-specifications. As a prototypical
example, we consider the phase retrieval problem, where we show that GASP
outperforms the corresponding GAMP, reducing the reconstruction threshold and,
for certain choices of its parameters, approaching Bayesian optimal
performance. Furthermore, we present a set of State Evolution equations that
exactly characterize the dynamics of GASP in the high-dimensional limit
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Approximate Message Passing with Spectral Initialization for Generalized Linear Models.
We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms
with many appealing features: the performance of AMP in the high-dimensional limit
can be succinctly characterized under suitable model assumptions; AMP can also be
tailored to the empirical distribution of the
signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms.
However, a major issue of AMP is that in
many models (such as phase retrieval), it requires an initialization correlated with the
ground-truth signal and independent from
the measurement matrix. Assuming that
such an initialization is available is typically
not realistic. In this paper, we solve this
problem by proposing an AMP algorithm initialized with a spectral estimator. With such
an initialization, the standard AMP analysis fails since the spectral estimator depends
in a complicated way on the design matrix.
Our main contribution is a rigorous characterization of the performance of AMP with
spectral initialization in the high-dimensional
limit. The key technical idea is to define
and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the
iterates of the true AMP. We also provide numerical results that demonstrate the validity
of the proposed approach