Geometry of Radial Basis Neural Networks for Safety Biased Approximation of Unsafe Regions Ahmad Abuaishy Mohit Srinivasanz Patricio A. Velay

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Geometry of Radial Basis Neural Networks for

Safety Biased Approximation of Unsafe Regions

Ahmad Abuaish†, Mohit Srinivasan‡, Patricio A. Vela†

Abstract— Barrier function-based inequality constraints are a

means to enforce safety speciﬁcations for control systems. When

used in conjunction with a convex optimization program, they

provide a computationally efﬁcient method to enforce safety

for the general class of control-afﬁne systems. One of the

main assumptions when taking this approach is the a priori

knowledge of the barrier function itself, i.e., knowledge of

the safe set. In the context of navigation through unknown

environments where the locally safe set evolves with time,

such knowledge does not exist. This manuscript focuses on the

synthesis of a zeroing barrier function characterizing the safe

set based on safe and unsafe sample measurements, e.g., from

perception data in navigation applications. Prior work formu-

lated a supervised machine learning algorithm whose solution

guaranteed the construction of a zeroing barrier function with

speciﬁc level-set properties. However, it did not explore the

geometry of the neural network design used for the synthesis

process. This manuscript describes the speciﬁc geometry of the

neural network used for zeroing barrier function synthesis, and

shows how the network provides the necessary representation

for splitting the state space into safe and unsafe regions.

I. INTRODUCTION

Invariant set-based safe control synthesis [1] has become

a favorable technique to enforce safety, as it provides the-

oretical guarantees when used to augment base controllers

[2]. For closed dynamical systems, the invariant set is

represented by the zero and positive level-sets of a con-

tinuously differentiable implicit function, typically termed

zeroing barrier function (ZBF). Subsequently, for controlled

systems, control barrier functions (CBFs) are introduced

to represent the invariant set. In safety-critical applications,

CBF-based controllers are designed to render safe regions

in the state space a positively invariant set. Prevailing use

of CBFs is in a point-wise optimization program solved via

quadratic programming (QP) [3]. CBF-based QPs have been

used in a wide range of applications in robotics such as

collision avoidance [4], [5], multi-robot coordination and task

satisfaction [6], [7], [8], and automotive applications [3].

Usually, a CBF is handcrafted based on a priori knowledge

of the safe regions in the state space. However, there are

applications where the safe regions evolve with time, with

navigation being one. In these applications, it is critical to

synthesize the CBF online using sensor measurements. Un-

fortunately, determining the true invariant region deﬁned by a

CBF is generally an NP-hard problem, but some techniques

This work was supported in part by the National Science Foundation

under Award S&AS #1849333, by DARPA PAI, and by KACST Fellowship.

†School of Electrical and Computer Engineering, Georgia In-

stitute of Technology, Atlanta, USA aabuaish@gatech.edu,

pvela@gatech.edu

‡Ford Motor Company, Dearborn, USA mohit.s@ieee.org

exist that can estimate the invariant region [9]. Since the

existence of a zeroing barrier function is needed to formally

conﬁrm the existence of a CBF, this manuscript solely

focuses on zeroing barrier function synthesis to separate state

space into safe and unsafe regions. The ZBF is constructed

from a two-layer kernel machine network trained from a

labeled dataset of safe and unsafe samples. Further, the

geometry of the kernel functions in the network is explored

to efﬁciently partition the space. The approach is motivated

by the Kolmogorov–Arnold representation theorem, which

implies that two-layer neural networks may be capable of

approximating continuous functions [10].

Recently, several data-driven approaches for constructing

a CBF were proposed to account for uncertainties in either

the system dynamics, unsafe regions, or both. One approach

category for learning CBF’s involves supervised ofﬂine learn-

ing. Instances include imitation learning where training data

is generated by an expert actor or optimal control simulations

[11], [12]. The ofﬂine nature lacks the ability to accommo-

date real-time changes in the environment. In contrast, self-

supervised approaches permit online learning. Initial work

on self-supervised Bayesian learning system of uncertain

dynamics [13] with known barrier functions was merged with

[14] to learn the system dynamics and an implicit function

representation of the unsafe region [15]. In [14], a signed

distance function representing obstacles is modeled as a deep

neural network trained from range sensor data via stochas-

tic gradient descent (SGD) with replay memory. Similarly,

[16] presented the construction of a probabilistic occupancy

map from a kernel-based logistic regression model trained

from range sensor data via SGD. However, there were no

hard constraints on misclassifying unsafe data points in the

underlying SGD optimization process, which nulliﬁes any

possible theoretical safety guarantee.

Our previous work focused on creating a ZBF for nav-

igation applications based on data collected from LiDAR

sensors [17]. A two-layer network with Gaussian radial basis

functions (GRBFs) was synthesized from this data. The ﬁrst

layer used sparsely distributed (over the domain) GRBF

centers, while the second GRBF layer was learnt during

the kernel support vector machine (kSVM) optimization pro-

cess. The kSVM optimization speciﬁcations provided formal

guarantees regarding the partitioning of the domain into safe

and unsafe regions. However, the work did not discuss the

geometry and associated properties of the GRBF network.

This work analyzes the geometry of the two-layer network

and the structure of the optimization problem to prove the

existence of a ZBF with known partitioning properties.

arXiv:2210.05596v2 [eess.SY] 28 Mar 2023

The manuscript organization is as follows: Section II

discusses the geometry of Gaussian kernel functions with

respect to the ﬁrst layer. Section III covers the second

layer construction, geometry, and optimization speciﬁcations.

Section IV discusses the qualiﬁcation of the two-layer kernel

machine network to be a zeroing barrier function along with

a kernel basis selection strategy. Sections V and VI present

case studies and concluding remarks, respectively.

II. GEOMETRY OF THE KERNEL HILBERT SPACE

GRBF Neural Networks (GRBF-NNs) have several prop-

erties ideal for zeroing barrier function creation. First they

are universal approximators [18], second they exhibit locality

[19], and third they partition space. Since the last property

is less frequently mentioned, but often used, this section

devotes attention to space partitioning, as it is essential for

data-driven safe set generation.

The focus of this section is on GRBF-NNs as kernel

machines whose kernel functions are radial basis functions

(RBFs), which generally take the following form,

k(xi, xj) = ϕ(||xi−xj|| ;σ),for ϕ:R+→R+,(1)

where xi, xj∈Rnd,||·|| is a norm and σis the scalar

bandwidth, which inﬂuences the sensitivity of the basis

function ϕ. A kernel machine based on radial basis functions

generates a mapping through the use of multiple kernel

function mappings with ﬁxed argument elements cjin a

center set C. For cj∈ C the kernel mapping is

k:Rnd→ Hnc, x 7→ [k(x, c1)· · · k(x, cnc)]T.(2)

where nc=|C| and H·indicates that the output space is

a Hilbert space. For this kernel machine to deﬁne a scalar

function requires specifying α∈Rncsuch that

fKM(x) = Dα , ~

k(x)E=αT~

k(x).(3)

Kernel machines permit more general function classes than

RBFs in Rnd(subsequent sections will use polynomial kernel

functions). That said, RBFs have geometric properties that

are implicitly exploited in kernel machine learning applica-

tions. Specializing to the case of the Gaussian radial basis

function (or Gaussian kernel), let the kernel function be

kG(x, c) = exp −||x−c||2

σ2!.(4)

Theorem 1. The kernel mapping, with ncGaussian kernels,

maps the input domain D ⊂ Rndinto a surface in the Hilbert

space Hnc⊂Rncwhen nc≥nd+ 1 and there are nd+ 1

centers capable of deﬁning a coordinate system in nd.

Proof. Showing that the kernel mapping is 1-1 establishes

this property. The pre-image, k−1(·, ci), of each coordinate’s

kernel mapping is a sphere in the input space. The intersec-

tion of all spheres for all coordinate mappings establishes the

pre-image point, which is unique only if the intersection is

unique. Finite solutions for the intersections has been proven

in the context of rigid body geometry for nc=nd[20],

with nc≥nd+ 1 necessary for a single valid solution.

The requirement for the centers is that they lead to a basis

of ndvectors when using one of the points as the origin

and using ndother points to obtain the basis vectors relative

to that origin. Each pre-image imposes a constraint on the

degrees of freedom of the input point x∈Rnd, such that the

nd+1 intersecting pre-image spheres do so at a single point.

When nc> nd+ 1, the additional pre-image constraints are

redundant and effectively impose no constraints. The rank

of the mapping is nd< nc, hence it maps to a surface of

dimension nd.

Theorem 1 applies to any RBF with inﬁnite support; an

RBF with ﬁnite support will have similar properties but will

include an -covering constraint. Basis functions generated

from other norms also have similar properties but may

require more centers to induce a 1-1 mapping. If the kernel

is differentiable, then the 1-1 mapping is an embedding.

Safe set generation with an appropriate kernel mapping will

involve deﬁning the concept of a kernel embedding1and its

associated kernel embedding inducing data set.

Deﬁnition 1. The kernel mapping ~

k:Rnd→ Hncis a

kernel embedding if the center set C ⊂ D generating the

kernel mapping are such that it is 1-1 and the kernel function

k:Rnd×Rnd→Ris differentiable. The set of centers is

called the kernel embedding inducing set (KEI set).

Corollary 1. Consider a ﬁnite set of points Xp⊂R2with

an associated triangulation. Under a kernel embedding, each

triangular region maps to a surface in Hnchomeomorphic

to a 2-simplex.

Corollary 2. Consider a ﬁnite set of points Xp⊂R3with

an associated tetrahedralization. Under a kernel embedding,

each tetrahedron maps to a surface in Hnchomeomorphic

to a 3-simplex.

If the domain D ⊂ Rndis a set of disconnected regions

excluding points at inﬁnity, then a kernel embedding will

map to a set of disconnected surfaces. Similarly, a collection

of non-intersecting triangulations/tetrahedralizations maps to

a set of disconnected surfaces under a kernel embedding.

1) Geometry of a Kernel Embedding: The Gaussian ker-

nel mapping outputs lie in the unit cube of Hnc⊂Rnc. Each

center ciin the set Cmaximizes its associated coordinate

(evaluates to 1), which means that there is a neighborhood

of ciin Dfor which this same coordinate is also maximal

for all points in the neighborhood. Points tending to inﬁnity

map to the origin in Hncsince the Gaussian radial basis

function tends to zero as the input radius tends to inﬁnity.

Corollary 3. For a kernel embedding deﬁned using the

Gaussian kernel, a compact domain Dmaps to a compact

surface whose points lie outside of a ball centered at the

origin. Furthermore, ~

k(Rnd)∪ {~

0}is a compact surface.

1Not to be confused with the kernel mean embedding which is for

probability distributions [21].

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GeometryofRadialBasisNeuralNetworksforSafetyBiasedApproximationofUnsafeRegionsAhmadAbuaishy,MohitSrinivasanz,PatricioA.VelayAbstractBarrierfunction-basedinequalityconstraintsareameanstoenforcesafetyspecicationsforcontrolsystems.Whenusedinconjunctionwithaconvexoptimizationprogram,theyprovideacomput...

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Geometry of Radial Basis Neural Networks for Safety Biased Approximation of Unsafe Regions Ahmad Abuaishy Mohit Srinivasanz Patricio A. Velay

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