496 research outputs found
-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?
This paper investigates the problem of signal estimation from undersampled
noisy sub-Gaussian measurements under the assumption of a cosparse model. Based
on generalized notions of sparsity, we derive novel recovery guarantees for the
-analysis basis pursuit, enabling highly accurate predictions of its
sample complexity. The corresponding bounds on the number of required
measurements do explicitly depend on the Gram matrix of the analysis operator
and therefore particularly account for its mutual coherence structure. Our
findings defy conventional wisdom which promotes the sparsity of analysis
coefficients as the crucial quantity to study. In fact, this common paradigm
breaks down completely in many situations of practical interest, for instance,
when applying a redundant (multilevel) frame as analysis prior. By extensive
numerical experiments, we demonstrate that, in contrast, our theoretical
sampling-rate bounds reliably capture the recovery capability of various
examples, such as redundant Haar wavelets systems, total variation, or random
frames. The proofs of our main results build upon recent achievements in the
convex geometry of data mining problems. More precisely, we establish a
sophisticated upper bound on the conic Gaussian mean width that is associated
with the underlying -analysis polytope. Due to a novel localization
argument, it turns out that the presented framework naturally extends to stable
recovery, allowing us to incorporate compressible coefficient sequences as
well
Dynamic Compressive Sensing of Time-Varying Signals via Approximate Message Passing
In this work the dynamic compressive sensing (CS) problem of recovering
sparse, correlated, time-varying signals from sub-Nyquist, non-adaptive, linear
measurements is explored from a Bayesian perspective. While there has been a
handful of previously proposed Bayesian dynamic CS algorithms in the
literature, the ability to perform inference on high-dimensional problems in a
computationally efficient manner remains elusive. In response, we propose a
probabilistic dynamic CS signal model that captures both amplitude and support
correlation structure, and describe an approximate message passing algorithm
that performs soft signal estimation and support detection with a computational
complexity that is linear in all problem dimensions. The algorithm, DCS-AMP,
can perform either causal filtering or non-causal smoothing, and is capable of
learning model parameters adaptively from the data through an
expectation-maximization learning procedure. We provide numerical evidence that
DCS-AMP performs within 3 dB of oracle bounds on synthetic data under a variety
of operating conditions. We further describe the result of applying DCS-AMP to
two real dynamic CS datasets, as well as a frequency estimation task, to
bolster our claim that DCS-AMP is capable of offering state-of-the-art
performance and speed on real-world high-dimensional problems.Comment: 32 pages, 7 figure
PAC-Bayes Compression Bounds So Tight That They Can Explain Generalization
While there has been progress in developing non-vacuous generalization bounds
for deep neural networks, these bounds tend to be uninformative about why deep
learning works. In this paper, we develop a compression approach based on
quantizing neural network parameters in a linear subspace, profoundly improving
on previous results to provide state-of-the-art generalization bounds on a
variety of tasks, including transfer learning. We use these tight bounds to
better understand the role of model size, equivariance, and the implicit biases
of optimization, for generalization in deep learning. Notably, we find large
models can be compressed to a much greater extent than previously known,
encapsulating Occam's razor. We also argue for data-independent bounds in
explaining generalization.Comment: NeurIPS 2022. Code is available at
https://github.com/activatedgeek/tight-pac-baye
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