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Register a new userThe spiked matrix model with generative priors
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Added by
Benjamin Aubin
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Using a low-dimensional parametrization of signals is a generic and powerful way to enhance performance in signal processing and statistical inference. A very popular and widely explored type of dimensionality reduction is sparsity; another type is generative modelling of signal distributions. Generative models based on neural networks, such as GANs or variational auto-encoders, are particularly performant and are gaining on applicability. In this paper we study spiked matrix models, where a low-rank matrix is observed through a noisy channel. This problem with sparse structure of the spikes has attracted broad attention in the past literature. Here, we replace the sparsity assumption by generative modelling, and investigate the consequences on statistical and algorithmic properties. We analyze the Bayes-optimal performance under specific generative models for the spike. In contrast with the sparsity assumption, we do not observe regions of parameters where statistical performance is superior to the best known algorithmic performance. We show that in the analyzed cases the approximate message passing algorithm is able to reach optimal performance. We also design enhanced spectral algorithms and analyze their performance and thresholds using random matrix theory, showing their superiority to the classical principal component analysis. We complement our theoretical results by illustrating the performance of the spectral algorithms when the spikes come from real datasets.
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We consider the problem of estimating a large rank-one tensor ${boldsymbol u}^{otimes k}in({mathbb R}^{n})^{otimes k}$, $kge 3$ in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio $lambda_{Bayes}= O(1)$ above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably no polynomial-time algorithm is known that achieved this goal unless $lambdage C n^{(k-2)/4}$ and even powerful semidefinite programming relaxations appear to fail for $1ll lambdall n^{(k-2)/4}$. In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-$k$ homogeneous polynomial over the unit sphere in $n$ dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions $n$, and give exact formulas for the exponential growth rate. We show that (for $lambda$ larger than a constant) critical points are either very close to the unknown vector ${boldsymbol u}$, or are confined in a band of width $Theta(lambda^{-1/(k-1)})$ around the maximum circle that is orthogonal to ${boldsymbol u}$. For local maxima, this band shrinks to be of size $Theta(lambda^{-1/(k-2)})$. These `uninformative local maxima are likely to cause the failure of optimization algorithms.
We consider the problem of compressed sensing and of (real-valued) phase retrieval with random measurement matrix. We derive sharp asymptotics for the information-theoretically optimal performance and for the best known polynomial algorithm for an ensemble of generative priors consisting of fully connected deep neural networks with random weight matrices and arbitrary activations. We compare the performance to sparse separable priors and conclude that generative priors might be advantageous in terms of algorithmic performance. In particular, while sparsity does not allow to perform compressive phase retrieval efficiently close to its information-theoretic limit, it is found that under the random generative prior compressed phase retrieval becomes tractable.
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