A DISTRIBUTED ADAPTIVE ALGORITHM FOR NON-SMOOTH SPATIAL FILTERING PROBLEMS Charles Hovine and Alexander Bertrand

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A DISTRIBUTED ADAPTIVE ALGORITHM FOR NON-SMOOTH SPATIAL FILTERING

PROBLEMS

Charles Hovine and Alexander Bertrand

KU Leuven, Department of Electrical Engineering (ESAT)

STADIUS Center for Dynamical Systems, Signal Processing and Data Analytics

KU Leuven Institute for Artiﬁcial Intelligence (Leuven.AI)

Leuven, Belgium

{charles.hovine, alexander.bertrand}@esat.kuleuven.be

ABSTRACT

Computing the optimal solution to a spatial ﬁltering problems in a

Wireless Sensor Network can incur large bandwidth and computa-

tional requirements if an approach relying on data centralization is

used. The so-called distributed adaptive signal fusion (DASF) algo-

rithm solves this problem by having the nodes collaboratively solve

low-dimensional versions of the original optimization problem, re-

lying solely on the exchange of compressed views of the sensor data

between the nodes. However, the DASF algorithm has only been

shown to converge for ﬁltering problems that can be expressed as

smooth optimization problems. In this paper, we explore an exten-

sion of the DASF algorithm to a family of non-smooth spatial ﬁl-

tering problems, allowing the addition of non-smooth regularizers

to the optimization problem, which could for example be used to

perform node selection, and eliminate nodes not contributing to the

ﬁlter objective, therefore further reducing communication costs. We

provide a convergence proof of the non-smooth DASF algorithm and

validate its convergence via simulations in both a static and adaptive

setting.

Index Terms—Adaptive spatial ﬁltering, Wireless Sensor Net-

works, Non-smooth optimization, Distributed signal processing.

1. INTRODUCTION

A spatial ﬁltering problem usually consists in ﬁnding the linear com-

bination of a set of signals that is optimal with regards to some cri-

terion, and can therefore be expressed as the solution of an opti-

mization problem. Common examples include principal components

analysis [1], canonical correlation analysis [2], MAX-SNR beam-

forming, multichannel Wiener ﬁltering [3] and common spatial pat-

terns [4].

In the case of Wireless Sensor Networks (WSNs), where several

sensing nodes communicate via wireless links, signals are often only

short-term stationary, with statistics drifting over time. Being able to

adaptively compute ﬁlters therefore becomes an important require-

ment. The classical approach to computing spatial ﬁlters in WSNs

consists in designating a particular node as the fusion center (FC),

which will collect all the raw data and perform the ﬁlter computation

This project has received funding from the European Research Council

(ERC) under the European Union’s Horizon 2020 research and innovation

programme (grant agreement No. 802895), the FWO (Research Foundation

Flanders) for project G081722N and from the Flemish Government under

the ”Onderzoeksprogramma Artiﬁci¨

ele Intelligentie (AI) Vlaanderen” pro-

gramme.

centrally [5]. This approach is however not ideal, as the bandwidth

and computational power required at the FC scales poorly with the

number of both nodes and signals. Additionally, the FC constitutes

a single point of failure, which can be problematic for many de-

ployment scenarios. An alternative approach consists in solving the

ﬁltering problem in a distributed fashion, by sharing the work across

the sensor nodes.

The DASF algorithm [6] is a framework for solving adaptive

spatial ﬁltering problems in a distributed fashion. Instead of shar-

ing their raw observations, the nodes share efﬁciently crafted com-

pressed views of their sensor data, which are then used to locally

solve low-dimensional versions of the original optimization problem

at each node. In addition, the sensor signals’ statistics are allowed

to change during the course of the algorithm, such that the optimal

solution can be tracked adaptively.

The convergence and optimality of DASF in the case of ﬁlter-

ing problem expressible as smooth optimization problems, has been

studied in [7], but the applicability of the algorithm to non-smooth

problems is still unknown. In this paper, we show the convergence

and optimality of the algorithm for a family of non-smooth, and pos-

sibly non-convex optimization problems. In addition to allowing the

algorithm to be applied to well-known non-smooth problems such

as sparse signal recovery and compressed sensing [8,9], it allows the

use of non-smooth sparsity-promoting regularizers. In the context of

WSNs, such regularizers can for example be used to perform chan-

nel selection, and hence reduce both bandwidth and computational

stress on the sensor nodes.

2. PROBLEM STATEMENT

We consider a network consisting of Ksensor nodes, where each

node kcollects discrete observations of an Mk-channel signal yk(t).

We denote y(t)=[yT

1(t),...,yT

K(t)]Tthe network-wide multi-

channel sensor signal, where each observation is an element of RM

with M=PkMk.Our goal is to design a network-wide spatial

ﬁlter X∈RM×Qwhich fuses all the channels of y(t)into Qoutput

channels that satisfy a certain optimality criterion, which in generic

form can be written as

X?(t)∈argmin

f(XTy(t),XTB) + g(XTΓ)

s.t. [XT

kyk(t),XT

kBk]∈ Xk∀k.

(1)

where each block Xk∈RMk×Qis deﬁned according to the par-

titioning X= [XT

1,· · · ,XT

K]T.Γ=BlkDiag(Γ1,...,ΓK),

arXiv:2210.15243v2 [eess.SP] 28 Feb 2023

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ADISTRIBUTEDADAPTIVEALGORITHMFORNON-SMOOTHSPATIALFILTERINGPROBLEMSCharlesHovineandAlexanderBertrandKULeuven,DepartmentofElectricalEngineering(ESAT)STADIUSCenterforDynamicalSystems,SignalProcessingandDataAnalyticsKULeuvenInstituteforArticialIntelligence(Leuven.AI)Leuven,Belgiumfcharles.hovine,alexan...

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A DISTRIBUTED ADAPTIVE ALGORITHM FOR NON-SMOOTH SPATIAL FILTERING PROBLEMS Charles Hovine and Alexander Bertrand

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