TY - JOUR

T1 - Matrix inference and estimation in multi-layer models

AU - Pandit, Parthe

AU - Sahraee-Ardakan, Mojtaba

AU - Rangan, Sundeep

AU - Schniter, Philip

AU - Fletcher, Alyson K.

N1 - Funding Information:
The work of P. Schniter was supported by NSF grant 1716388. The work of P. Pandit, M. Saharee-Ardakan and A. K. Fletcher was supported in part by the NSF Grants 1738285 and 1738286, ONR Grant N00014-15-1-2677. The work of S. Rangan was supported in part by NSF grants 1116589, 1302336, and 1547332, NIST, SRC and the the industrial affiliates of NYU Wireless.
Publisher Copyright:
© 2020 Neural information processing systems foundation. All rights reserved.

PY - 2020

Y1 - 2020

N2 - We consider the problem of estimating the input and hidden variables of a stochastic multi-layer neural network from an observation of the output. The hidden variables in each layer are represented as matrices with statistical interactions along both rows as well as columns. This problem applies to matrix imputation, signal recovery via deep generative prior models, multi-task and mixed regression, and learning certain classes of two-layer neural networks. We extend a recently-developed algorithm – Multi-Layer Vector Approximate Message Passing (ML-VAMP), for this matrix-valued inference problem. It is shown that the performance of the proposed Multi-Layer Matrix VAMP (ML-Mat-VAMP) algorithm can be exactly predicted in a certain random large-system limit, where the dimensions N ×d of the unknown quantities grow as N ? 8 with d fixed. In the two-layer neural-network learning problem, this scaling corresponds to the case where the number of input features as well as training samples grow to infinity but the number of hidden nodes stays fixed. The analysis enables a precise prediction of the parameter and test error of the learning.

AB - We consider the problem of estimating the input and hidden variables of a stochastic multi-layer neural network from an observation of the output. The hidden variables in each layer are represented as matrices with statistical interactions along both rows as well as columns. This problem applies to matrix imputation, signal recovery via deep generative prior models, multi-task and mixed regression, and learning certain classes of two-layer neural networks. We extend a recently-developed algorithm – Multi-Layer Vector Approximate Message Passing (ML-VAMP), for this matrix-valued inference problem. It is shown that the performance of the proposed Multi-Layer Matrix VAMP (ML-Mat-VAMP) algorithm can be exactly predicted in a certain random large-system limit, where the dimensions N ×d of the unknown quantities grow as N ? 8 with d fixed. In the two-layer neural-network learning problem, this scaling corresponds to the case where the number of input features as well as training samples grow to infinity but the number of hidden nodes stays fixed. The analysis enables a precise prediction of the parameter and test error of the learning.

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M3 - Conference article

AN - SCOPUS:85108431180

VL - 2020-December

JO - Advances in Neural Information Processing Systems

JF - Advances in Neural Information Processing Systems

SN - 1049-5258

T2 - 34th Conference on Neural Information Processing Systems, NeurIPS 2020

Y2 - 6 December 2020 through 12 December 2020

ER -