Model Order Selection with Variational Autoencoding Michael Baur Franz Weißer Benedikt B ock and Wolfgang Utschick

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Model Order Selection with Variational

Autoencoding

Michael Baur, Franz Weißer, Benedikt B¨

ock, and Wolfgang Utschick

TUM School of Computation, Information and Technology, Technical University of Munich, Germany

Email: {mi.baur, franz.weisser, benedikt.boeck, utschick}@tum.de

Abstract—Classical methods for model order selection often

fail in scenarios with low SNR or few snapshots. Deep learning-

based methods are promising alternatives for such challenging

situations as they compensate lack of information in the available

observations with training on large datasets. This manuscript

proposes an approach that uses a variational autoencoder (VAE)

for model order selection. The idea is to learn a parameter-

ized conditional covariance matrix at the VAE decoder that

approximates the true signal covariance matrix. The method is

unsupervised and only requires a small representative dataset

for calibration after training the VAE. Numerical simulations

show that the proposed method outperforms classical methods

and even reaches or beats a supervised approach depending on

the considered snapshots.

Index Terms—Variational autoencoder, generative model,

model order, machine learning, direction of arrival estimation.

I. INTRODUCTION

Model order (MO) selection determines the number of

impinging wavefronts incident at a receiver. The MO is an es-

sential quantity for direction of arrival (DoA) estimation, both

for classical [1] and current deep learning (DL) methods [2].

Most well-known MO selection approaches utilize information

criteria (IC) [3]. More current treatment of model selection is

covered in [4], where DL methods are left out, however.

A popular IC-based method for MO selection reaches back

to the 80s [5]. The method is based on a subspace decompo-

sition of the sample covariance matrix and performs well in

cases with high signal-to-noise ratio (SNR) and many snap-

shots. In contrast, for low SNR or few snapshots, the sample

covariance matrix is a bad estimate of the true covariance

matrix. Consequently, the method fails in these cases. DL

methods are promising candidates to perform well in such

difﬁcult situations. As a result of the repeated presentation of

samples during the ofﬂine training phase, the DL model ex-

tracts overall prior information of the data and can compensate

for lack of knowledge in observations during the deployment

phase, e.g., if only a few snapshots are available. The strong

performance of DL-based methods is demonstrated in [6]–[9],

which use relatively simple neural network architectures to

determine the MO based on the (preprocessed) snapshots. The

methods are supervised, requiring access to a dataset, where

observations are labeled with their corresponding MO.

This work is funded by the Bavarian Ministry of Economic Affairs,

Regional Development and Energy within the project 6G Future Lab Bavaria.

The authors acknowledge the ﬁnancial support by the Federal Ministry of

Education and Research of Germany in the programme of “Souver¨

an. Digital.

Vernetzt.”. Joint project 6G-life, project identiﬁcation number: 16KISK002

If the exact signal model and the exact model of the

antenna array were available, it would be possible to generate

unlimited amounts of labeled data. This assumption, however,

only holds for idealistic circumstances, e.g., calibrated antenna

arrays, that do not hold in reality. Under realistic conditions,

data would only be available in the form of measurements

without any labels. These aspects motivate the investigation

of unsupervised learning methods because they do not require

labeling. Unsupervised methods additionally offer to include

model imperfections in the framework directly. The variational

autoencoder (VAE) is an unsupervised framework that learns

the data distribution by maximizing a lower bound to the

data log-likelihood [10]. It belongs to the class of generative

models, which means that the model can generate entirely new

samples from the learned distribution. Despite its popularity

in image processing and related disciplines, the VAE is rarely

employed in communications tasks. A current publication

investigates the generative modeling performance of the VAE

in a millimeter-wave UAV scenario [11]. Channel equalization

is another domain where the VAE is applied successfully [12]–

[14], as well as channel estimation [15].

Motivated by the performance of the VAE channel estimator

in [15], we propose a method for MO selection based on a

VAE. Our method is unsupervised, and only a small repre-

sentative dataset is required to distinctly assign the MO after

training of the VAE. The method is supposed to ﬁll the gaps

where classical methods for MO selection fail, i.e., at low

SNR and few snapshots. The contributions are as follows.

By parameterizing the covariance matrix of the conditionally

Gaussian distribution at the VAE decoder with an oversampled

discrete fourier transform (DFT) matrix, we can learn an

approximation of the eigenvalues of the true signal covariance

matrix. We leverage the approximation to determine the MO

with a custom evaluation routine based on entropy. Numerical

simulations show the advantage of the proposed framework

over IC-based methods in the considered scenarios. Moreover,

the proposed method can beat a supervised MO selection

method in a single snapshot scenario.

II. SYSTEM MODEL

An antenna array with Nelements receives signals from

Lsources in the far ﬁeld, which characterize the impinging

wavefronts. The received signal vector y(t)∈CNat snapshot

tof in total Tsnapshots is expressed as

y(t) = A(θ)s(t) + n(t), t = 1, . . . , T (1)

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arXiv:2210.15407v3 [eess.SP] 28 Jun 2023

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ModelOrderSelectionwithVariationalAutoencodingMichaelBaur,FranzWeißer,BenediktB¨ock,andWolfgangUtschickTUMSchoolofComputation,InformationandTechnology,TechnicalUniversityofMunich,GermanyEmail:{mi.baur,franz.weisser,benedikt.boeck,utschick}@tum.deAbstract—Classicalmethodsformodelorderselectionoftenfa...

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Model Order Selection with Variational Autoencoding Michael Baur Franz Weißer Benedikt B ock and Wolfgang Utschick

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