TY - JOUR

T1 - Matrix inference and estimation in multi-layer modelsThis article is an updated version of

T2 - Pandit P, Sahraee Ardakan M, Rangan S, Schniter P and Fletcher A K 2020 Matrix inference and estimation in multi-layer models Advances in Neural Information Processing Systems vol 33 ed H Larochelle, M Ranzato, R Hadsell, M F Balcan and H Lin (New York: Curran Associates) pp 22456–67. Code available at https://github.com/parthe/ML-Mat-VAMP.

AU - Pandit, Parthe

AU - Sahraee-Ardakan, Mojtaba

AU - Rangan, Sundeep

AU - Schniter, Philip

AU - Fletcher, Alyson K.

N1 - Publisher Copyright:
© 2021 IOP Publishing Ltd and SISSA Medialab srl.

PY - 2021/12

Y1 - 2021/12

N2 - We consider the problem of estimating the input and hidden variables of a stochastic multi-layer neural network (NN) 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 NNs. We extend a recently-developed algorithm—multi-layer vector approximate message passing, for this matrix-valued inference problem. It is shown that the performance of the proposed multi-layer matrix vector approximate message passing algorithm can be exactly predicted in a certain random large-system limit, where the dimensions N Ã d of the unknown quantities grow as N → ∞ 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 (NN) 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 NNs. We extend a recently-developed algorithm—multi-layer vector approximate message passing, for this matrix-valued inference problem. It is shown that the performance of the proposed multi-layer matrix vector approximate message passing algorithm can be exactly predicted in a certain random large-system limit, where the dimensions N Ã d of the unknown quantities grow as N → ∞ 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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U2 - 10.1088/1742-5468/ac3a75

DO - 10.1088/1742-5468/ac3a75

M3 - Article

AN - SCOPUS:85122501344

VL - 2021

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 12

M1 - 124004

ER -