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# Sensor Selection via GFlowNets: A Deep Generative Modeling Framework to
Navigate Combinatorial Complexity

Spilios Evmorfos, Zhaoyi Xu, and Athina Petropulu The authors are with the ECE
Department at Rutgers, the State University of New Jersey, New Brunswick, NJ
Work supported by ARO under Grants W911NF2110071 and W911NF2320103 and by NSF
under Grant ECCS-2335876

###### Abstract

The performance of sensor arrays in sensing and wireless communications
improves with more elements, but this comes at the cost of increased energy
consumption and hardware expense. This work addresses the challenge of
selecting $k$ sensor elements from a set of $m$ to optimize a generic Quality-
of-Service metric. Evaluating all $\binom{m}{k}$ possible sensor subsets is
impractical, leading to prior solutions using convex relaxations, greedy
algorithms, and supervised learning approaches. The current paper proposes a
new framework that employs deep generative modeling, treating sensor selection
as a deterministic Markov Decision Process where sensor subsets of size $k$
arise as terminal states. Generative Flow Networks (GFlowNets) are employed to
model an action distribution conditioned on the state. Sampling actions from
the aforementioned distribution ensures that the probability of arriving at a
terminal state is proportional to the performance of the corresponding subset.
Applied to a standard sensor selection scenario, the developed approach
outperforms popular methods which are based on convex optimization and greedy
algorithms. Finally, a multiobjective formulation of the proposed approach is
adopted and applied on the sparse antenna array design for Integrated Sensing
and Communication (ISAC) systems. The multiobjective variation is shown to
perform well in managing the trade-off between radar and communication
performance.

###### Index Terms:

Sensor Selection, GFlowNets, Combinatorial Complexity, Deep Generative
Modeling, Sparse Array Design, ISAC Systems

##  I Introduction

In a sensor array, the measurement captured by each sensor is a function of
the quantity of interest. The objective is to accurately determine that
quantity from the diverse set of measurements. Generally, a larger number of
measurements leads to more accurate estimates of the quantity of interest. In
cases like MIMO radar [1], a larger array leads to enhanced sensing resolution
[2], reflecting overall improved performance with more elements. However,
operating a large number of sensor elements escalates operational costs (power
consumption, monetary expenses, etc.). To address this, a common strategy is
to activate only a subset of deployed sensor elements, aiming to optimize a
specific performance objective. Given $m$ sensor elements and a desire to
activate only $k<m$ of them, the number of possible subsets is $\binom{m}{k}$.
When both $m$ and $k$ are substantial, exhaustive search becomes impractical.
As a result, numerous research endeavors have concentrated on devising
computationally efficient methods to choose the optimal subset, addressing the
hurdles presented by the combinatorial explosion within the solution space.

Convex Optimization: Given that the original selection problem entails integer
constraints, certain approaches utilize convex relaxations to create surrogate
problems that are convex in continuous domains. They employ convex
optimization techniques, which have demonstrated efficacy in addressing such
problems. Subsequently, the solutions of the convex surrogates are transformed
into feasible solutions corresponding to the selection of a specific sensor
subset. The approach outlined in [3] selects a sensor subset aimed at
minimizing the estimation error of a vector of interest. Each sensor
measurement corresponds to a linear function of the vector of interest. The
methodology in [3] involves a convex relaxation followed by the application of
an Interior Point Method (IPM) [4] approach. A similar method is proposed in
[5]. The work by [6] examines the problem of sensor selection in the context
of broadband receiver beamforming, where the goal is to choose the subset that
minimizes the $L_{2}$ distance between the resulting beampattern and a desired
one and the solution is obtained via a convex relaxation and a Branch-and-
Bound approach [7]. The work of [8] introduces a semidefinite programming
method to choose a transmitter-receiver antenna pair that maximizes the
separation between desired and undesired directions of arrival. The authors in
[9] demonstrate that designing a sparse beamforming array with unequally
spaced antennas, achieved by selecting specific elements from a uniform array,
can be formulated as a quasi-norm minimization problem. Additionally, they
introduce a simplex search method as a proposed solution.

Greedy Methods: Greedy approaches are characterized by their sequential and
iterative selection of sensors for the final subset. These methods often
leverage the specific structure of the optimization objective to devise a
near-optimal greedy selection strategy, subject to certain assumptions. A
greedy approach is presented in [10] to select a subset that minimizes the
Cramer-Rao Bound of target estimates within the context of distributed
multiple-radar systems. The authors in [11] examine the same setting as the
work in [2] and formulate the sensor selection problem as the maximization of
a monotone submodular function over uniform matroids. They develop an
efficient greedy selection approach that yields a near-optimal solution to the
selection problem. Similar analysis and approach are provided in [12], albeit
in the context of heterogeneous sensors, where sensors exhibit varying
precision levels and operational costs.

Supervised Learning: Supervised machine learning approaches frame the sensor
selection problem as a multilabel classification task, training a function
approximator on a dataset of annotated examples. In the work of [13], Support
Vector Machines [14] are proposed for sparse antenna array design in wireless
communications. Similarly, in [15], the antenna selection problem is
reformulated as a classification problem, and a Convolutional Neural Network
[16] serves as the classifier. Meanwhile, [17] introduces a supervised deep
learning method for learning optimal antenna selection and precoding matrices.
For massive MIMO systems, [18] proposes a supervised deep learning approach to
antenna selection based on a channel state information extrapolation metric.
In the context of MIMO transmit beamforming, [19] advocates a supervised
learning approach utilizing a neural network composed of elementary operations
resembling self-attention [20]. A similar strategy is employed in [21] for
sensor selection in Integrated Sensing and Communication (ISAC) systems.

Pitfalls of existing approaches: Analytical approaches, which include both
convex optimization methods and greedy selection strategies, have demonstrated
notable success in various sensor selection scenarios. However, they have
certain drawbacks. Firstly, they are susceptible to getting stuck at local
optima. Additionally, they are criterion-dependent, relying on specific
properties of the optimization objective to facilitate the application of
convex relaxations or greedy selection. This dependency on the optimization
criterion poses challenges in sensor selection applications when the objective
does not align with the assumptions inherent in these analytical methods. For
example, the sensor selection objective discussed in [21], tailored for ISAC
systems, does not adhere to the submodular function property, making the
development of a greedy approach difficult. Moreover, the application of
convex relaxations and approximations for employing off-the-shelf convex
optimization algorithms is not straightforward. On a different note,
supervised learning methods demand a significant amount of annotated data, a
resource notably scarce within the pertinent domains of interest.

Contributions: The current paper introduces a novel framework for addressing
sensor selection problems, aiming to overcome the limitations associated with
existing paradigms. This proposed framework is distinctive in its unsupervised
nature, eliminating the reliance on annotated data, and it is criterion-
agnostic, thus removing the necessity for specific assumptions regarding the
selection objective (e.g., convexity, differentiability, submodularity, etc.).

The innovation of the proposed method lies in its modeling of the selection
process as a deterministic Markov Decision Process (MDP) with a single root.
Each sensor subarray selection of diserable size corresponds to a root-to-
terminal-node trajectory in the MDP. Non-terminal states are assigned a zero
reward, while the reward for each terminal node is determined by evaluating
the objective for the corresponding subset. This formulation sets the goal of
learning a distribution over actions, conditioned on the state, ensuring the
cumulative probability of reaching a terminal state is proportional to the
state’s reward (the performance of the corresponding subset).

The distribution over actions is parametrized by the Generative Flow Networks
(GFlowNets) paradigm. GFlowNets introduce a flow quantity to amortize the
learning cost of distributions over composite objects like MDPs. This approach
transforms flow-matching equations into a learning objective, shifting the
distribution learning problem from the combinatorial action space to the
continuous parameter space of a function approximator. Stochastic gradient
descent methods, recognized for their generalization performance to unseen
examples, can be leveraged to adjust the parameters of the function
approximator.

Furthermore, the GFlowNet paradigm is applied to the sensor selection setting
examined in previous works [3, 11], showcasing superior performance compared
to the convex optimization approach of [3] and the greedy approach of [11].
Specifically, our approach was implemented for selecting $15$ sensors from a
pool of $100$. With approximately $2\times 10^{17}$ possible subsets within
the feasible set, we employ a small neural network featuring $3$ layers with
$150$ neurons each layer, trained over $40,000$ root-to-leaf trajectories.
This signifies that out of the $2\times 10^{17}$ subsets, we utilize at most
$40,000$ (which corresponds to less than $5\times 10^{-10}\%$ of the possible
ones) for training. Furthermore, the GFlowNet paradigm for sensor selection is
modified to accommodate a multiobjective setting for sensor selection and is
applied to the ISAC sensor selection scenario that is outlined in [21].

The contributions of the current work are outlined below:

  * •

The current paper introduces an innovative approach to sensor selection
problems that is both unsupervised and criterion-agnostic, allowing for a
broad applicability across different settings without the need for specific
assumptions and annotated data.

  * •

The proposed approach models the sensor selection process as a deterministic
MDP with a singular root and sensor subsets of desirable size arising as
terminal states.

  * •

The GFlowNet paradigm is employed to parametrize the distribution over actions
in the MDP. It is trained such that the cumulative probability of arriving at
a terminal state is proportional to the performance of the corresponding
subset.

  * •

The proposed approach is applied on a sensor selection setting for linear
estimation and showcased to surpass the performance of a convex optimization
method and a greedy selection approach.

  * •

The paper extends the proposed approach to a multiobjective formulation and
showcases its versatility to a sparse array design problem for ISAC systems.

Notation: We denote the matrices and vectors by bold uppercase and bold
lowercase letters, respectively. The operators
$\left(\cdot\right)^{\mathsf{T}}$ and $\left(\cdot\right)^{H}$ denote
transposition and conjugate transposition respectively. Caligraphic letters
will be used to denote sets. The $\ell_{p}$-norm of $\bm{x}\in\mathbb{R}^{n}$
is
$\left\|\bm{x}\right\|_{p}\triangleq\left(\sum_{i=1}^{n}\left|x\left(i\right)\right|^{p}\right)^{1/p}$,
for all $\mathbb{N}\ni p\geq 1$. The expectation of a random vector
$\mathbf{x}$ is denoted as $\mathbb{E}\left(\mathbf{x}\right)$. Continuous
sets are denoted by $[\cdot]$ while discrete sets are denoted by
$\\{\cdot\\}$.

##  II Generative Flow Networks

This section offers an overview of the GFlowNet framework [22]. Consider a
deterministic MDP with $\mathcal{S}$ as the set of states and
$\mathcal{X}\subset\mathcal{S}$ as the set of terminal states. Let
$\mathcal{A}$ represent the set of discrete actions, and
$\mathcal{A}(\mathbf{s})$ denote the set of permissible actions at state
$\mathbf{s}$. The MDP is represented as a Directed Acyclic Graph (DAG). All
leaf nodes possess positive rewards, all intermediate states bear zero reward
($\mathcal{R}(\mathbf{s})=0\hskip
2.84544pt\forall\mathbf{s}\notin\mathcal{X}$), and a unique root
$\mathbf{s}_{0}$ exists. The DAG is non-injective, implying that different
action sequences (starting from the root) can lead to the same state. The
primary objective is to learn an action selection policy such that the
probability of reaching a terminal state is proportional to the terminal
state’s reward.

The GFlowNet conceptualizes the MDP as a flow network, where the flow
originates from the root, and each leaf node serves as a flow sink. Assuming
that action $\mathbf{a}$ is executed at state $\mathbf{s}$, the subsequent
state $\mathbf{s}^{\prime}$ is denoted as
$T(\mathbf{s},\mathbf{a})=\mathbf{s}^{\prime}$. Given the deterministic nature
of the MDP, $T(\mathbf{s},\mathbf{a})$ is uniquely determined for each pair.
The flow along edge $(\mathbf{s},\mathbf{a})$ is represented as
$F(\mathbf{s},\mathbf{a})$, and the total flow passing through state
$\mathbf{s}$ is denoted as $F(\mathbf{s})$. To adhere to flow balance
conditions, the incoming flow of each state must equal the outgoing flow. For
any node $\mathbf{s}^{\prime}$, the in-flow is:

$F(\mathbf{s}^{\prime})=\mkern-24.0mu\sum_{\mathbf{s},\mathbf{a}\mathrel{\mathop{\ordinarycolon}}T(\mathbf{s},\mathbf{a})=\mathbf{s}^{\prime}}\mkern-24.0muF(\mathbf{s},\mathbf{a}).$
(1)

On the other hand, the out-flow can be defined as:

$F(\mathbf{s}^{\prime})=\mkern-12.0mu\sum_{\mathbf{a}^{\prime}\in\mathcal{A}(\mathbf{s}^{\prime})}\mkern-12.0muF(\mathbf{s}^{\prime},\mathbf{a}^{\prime}).$
(2)

The flow of each state is the sum of the reward of the state and the flow of
all outgoing edges:

$F(\mathbf{s})=R(\mathbf{s})+\sum_{\mathbf{a}\in\mathcal{A}(\mathbf{s})}F(\mathbf{s},\mathbf{a}).$
(3)

This implies that the flow of each terminal state is the reward of the state:

$F(\mathbf{x})=R(\mathbf{x})>0.$ (4)

Since the flow of each state is also equal to the flow of all incoming edges,
the flow-matching equation that holds on each node $\mathbf{s}^{{}^{\prime}}$
is:

$\sum_{\mathbf{s},\mathbf{a}\mathrel{\mathop{\ordinarycolon}}T(\mathbf{s},\mathbf{a})=\mathbf{s}^{\prime}}F(\mathbf{s},\mathbf{a})=R(\mathbf{s}^{\prime})+\sum_{\mathbf{a}^{\prime}\in\mathcal{A}(\mathbf{s}^{\prime})}F(\mathbf{s}^{\prime},\mathbf{a}^{\prime}).$
(5)

Under the assumption that the flow of each state $F(\mathbf{s})$ and the flow
of each edge $F(\mathbf{s},\mathbf{a})$ are known and satisfy the above
equation, if actions are sampled on each state (starting from the root) based
on the ratio

$\pi(\mathbf{a}|\mathbf{s})=\frac{F(\mathbf{s},\mathbf{a})}{F(\mathbf{s})},$
(6)

the following results hold [23]:

  1. 1.

The flow of the root (also known as the partition function of the DAG) is
equal to the sum of all the rewards of the terminal nodes:

$F(\mathbf{s}_{0})=\sum_{\mathbf{x}\in\mathcal{X}}R(\mathbf{x}).$ (7)

  2. 2.

The probability of reaching a terminal state $\mathbf{x}$ is the ratio of the
reward of the state over the partition function:

$\pi(\mathbf{x})=\frac{R(\mathbf{x})}{\sum_{\mathbf{x}^{\prime}\in\mathcal{X}}R(\mathbf{x}^{\prime})}=\frac{R(\mathbf{x})}{F(\mathbf{s}_{0})}.$
(8)

.

The GFlowNet paradigm involves parameterizing the flow $F$ using a function
approximator $F_{\mathbf{w}}$ selected from a class with sufficient
expressivity, such as a deep neural network. Trajectories of the MDP are
sampled, and for each state $\mathbf{s}^{\prime}$, the following objective,
referred to as flow-matching objective, is minimized via gradient descent on
$\mathbf{w}$:

$\displaystyle
L_{\mathbf{w}}(\mathbf{s}^{\prime})=\mkern-24.0mu\sum_{\mathbf{s},\mathbf{a}\mathrel{\mathop{\ordinarycolon}}T(\mathbf{s},\mathbf{a})=\mathbf{s}^{\prime}}\mkern-24.0muF_{\mathbf{w}}(\mathbf{s},\mathbf{a})-R(\mathbf{s}^{\prime})-\mkern-18.0mu\sum_{\mathbf{a}^{\prime}\in\mathcal{A}(\mathbf{s}^{\prime})}\mkern-15.0muF_{\mathbf{w}}(\mathbf{s}^{\prime},\mathbf{a}^{\prime}).$
(9)

##  III Sensor Selection via GFlowNets

The problem of sensor selection entails the task of choosing a subset of $k$
active sensors from a sensor array containing $m$ elements to optimize a
performance metric $Q$. Given the $\binom{m}{k}$ possible subsets to consider,
the primary challenge involves identifying the subset that maximizes (or
minimizes) the performance metric.

To effectively leverage the GFlowNet paradigm, let us transform the sensor
selection problem into the task of sampling terminal states from a MDP, which
is defined as follows: The state space is discrete, with each state
$\mathbf{s}$ is represented as a binary vector of $m$ elements. Within this
vector, “1” elements denote active sensor positions, while “0” elements
indicate inactive sensors. The initial state, denoted as $\mathbf{s}_{0}$, is
the zero vector where all elements are inactive. The action space is also
discrete, allowing actions at each state to involve adding one extra active
element from the set of inactive ones.

Terminal states in the MDP are characterized by state vectors with exactly $k$
active elements ($\mathbf{s}\equiv\mathbf{x}\in\mathcal{X}$ iff
$\lVert\mathbf{s}\rVert_{0}=k$). Intermediate states, where
$\lVert\mathbf{s}\rVert_{0}<k$, yield zero rewards.

For all terminal states $\mathbf{x}$, the reward is determined based on the
evaluation of a function of the performance objective, represented as
$f(Q(\mathbf{x}))$. The corresponding general graph for the sensor selection
MDP is visually depicted in Fig. 1. The choice of the function of the
performance metric is application-dependent. In the simplest case, assuming
$Q(\mathbf{x})>0\hskip 2.84544pt\forall\mathbf{x}$, the reward function is
defined as $R(\mathbf{x})=Q(\mathbf{x})$ for maximization goals, or as
$R(\mathbf{x})=\frac{1}{Q(\mathbf{x})}$ for minimization objectives.

Figure 1: The Sensor Selection MDP pertains to the task of selecting 2 sensor
elements from a total of 3. In the visual representation, the red circles
signify active elements, while the white circles indicate inactive elements.

The algorithm proposed for sensor selection, identified as GFlowNet for Sensor
Selection (GFLOW-SS), is outlined in Algorithm 1. GFLOW-SS parameterizes the
MDP flow using a neural network, and trajectories are sampled on-policy.
Stochastic gradient descent is employed to update the network’s weights based
on the flow-matching loss (eq. (9)). Following training, subsets can be
sampled as terminal states utilizing the flow-induced action sampling policy
defined in eq. (6).

Two important aspects should be highlighted regarding the proposed approach.
First, the training is unsupervised, eliminating the need for offline data
preprocessing and annotation. This sets the proposed approach apart from
previous machine learning methods addressing similar problems [13, 15, 18].
Secondly, a notable feature of this method is its objective-agnostic nature,
making it independent of annotations and free from assumptions regarding the
smoothness, differentiability, or convexity of the system’s performance
objective. The only requirement is the capability to evaluate the objective
for a given subset of sensor elements. This contrasts with analytical methods,
as exemplified in [10, 3, 11, 8, 24], which rely on application-specific
properties of the objective for employing convex relaxations or greedy
selection.

Algorithm 1 GFLOW - SS

Initialize $F_{\mathbf{w}}$, $\zeta\in[0,1]$ for exploration, learning rate
$\eta$

for all root-to-leaf trajectories do

$\mathbf{s}=\mathbf{s}_{0}=[0]^{m}$

for $k-1$ transitions do

Sample $z\sim\mathcal{U}(0,1)$ (Uniform distribution)

If $z<\zeta$ choose $\mathbf{a}\in\mathcal{A}(\mathbf{s})$ randomly

If $z>=\zeta$ choose
$\mathbf{a}=\operatorname*{arg\,max}_{\mathbf{a}^{\prime}}F_{\mathbf{w}}(\mathbf{s},\mathbf{a}^{\prime})$

Apply action $\mathbf{a}$, compute sub. state $\mathbf{s}^{\prime}$

$\mathbf{w}^{\prime}\rightarrow\mathbf{w}-\eta\nabla_{\mathbf{w}}L_{\mathbf{w}}(\mathbf{s}^{\prime})$
eq. (9) $R(\mathbf{s}^{\prime})=0$

$\mathbf{s}$ = $\mathbf{s}^{\prime}$

end for

Sample $z\sim\mathcal{U}(0,1)$

If $z>\zeta$ choose $\mathbf{a}\in\mathcal{A}(\mathbf{s})$ randomly

If $z\leq\zeta$ choose
$\mathbf{a}=\operatorname*{arg\,max}_{\mathbf{a}^{\prime}}F_{\mathbf{w}}(\mathbf{s},\mathbf{a}^{\prime})$

Apply action $\mathbf{a}$, compute terminal state $\mathbf{x}$

$\mathbf{w}^{\prime}\rightarrow\mathbf{w}-\eta\nabla_{\mathbf{w}}L_{\mathbf{w}}(\mathbf{x})$
$R(\mathbf{x})=Q(\mathbf{x})\text{ or }\frac{1}{Q(\mathbf{x})}$

end for

##  IV Sensor Selection for Linear Estimation

The method delineated in Section III is inherently versatile. In this section,
we tailor it specifically to address the sensor selection problem, previously
tackled through convex optimization in [3] and through greedy selection in
[11]. Subsequently, we evaluate its performance in direct comparison to the
two existing methodologies.

Consider $m$ linear sensor measurements $y_{i}$ corresponding to the vector of
interest $\mathbf{z}\in\mathbb{R}^{n}$. Each measurement is corrupted by
independent noise:

$y_{i}=\mathbf{a}_{i}^{\mathsf{T}}\mathbf{z}+v_{i},\hskip 8.5359pti=1,\dots,m$
(10)

The noise scalars $v_{1},\dots,v_{m}$ are independent identically distributed
$\mathcal{N}(0,\sigma^{2})$ random variables. The measurement vectors
$\mathbf{a}_{1},\dots,\mathbf{a}_{m}$ span $\mathbb{R}^{n}$.

The sensor selection problem is to choose $k<m$ out of $m$ sensor vectors
$\mathbf{a}_{i}$ such that the log of the determinant of the sum of outer
products is maximized:

$\displaystyle\text{maximize}_{\mathcal{S}}$
$\displaystyle\log\text{det}\left(\sum_{i\in\mathcal{S}}\mathbf{a}_{i}\mathbf{a}_{i}^{\mathsf{T}}\right)$
(11) subject to $\displaystyle\|\mathcal{S}\|=k$

where $\mathcal{S}\subseteq\\{1,\dots,m\\}$ and $\|\mathcal{S}\|$ denotes the
cardinality of set $\mathcal{S}$. The solution of the above optimization
problem minimizes the estimation error of the maximum likelihood estimate of
$\mathbf{z}$ [11].

###  IV-A Convex Optimization

The above problem is categorized as Boolean-convex, since the objective
function is convex and the constraints are Boolean [25]. In the work of [3],
it is proposed to relax the boolean constraints into convex constraints,
thereby facilitating the solution of the problem through convex optimization
techniques. Specifically, the authors introduce a variable
$\mathbf{x}\in\mathbb{R}^{m}$ and reformulate the original optimization
problem into the relaxed version presented below:

$\displaystyle\text{maximize}_{\mathbf{x}}$
$\displaystyle\log\text{det}\left(\sum_{i=1}^{m}x_{i}\mathbf{a}_{i}\mathbf{a}_{i}^{\mathsf{T}}\right)$
(12) subject to $\displaystyle\|\mathbf{x}\|_{1}=k$ $\displaystyle
x_{i}\in[0,1]\hskip 5.69046pti=1,\dots,m$

The above problem can be effectively addressed using IPMs [4]. The solution
derived from solving the relaxed problem serves as an upper bound for the
optimal solution of the original Boolean-convex problem [3]. To obtain a
solution for the initial problem, one can derive it from the solution of the
relaxed version by arranging the elements in descending order, selecting the
first $k$ elements (those with the highest magnitude), and setting their
corresponding indices to $1$, while the rest are set to $0$.

We denote this approach as CVX-OPT-SS throughout the manuscript. While there
is no established theoretical result defining the optimality gap between the
solution of the relaxed problem and the feasible (binary) solution derived
from it, it is widely acknowledged that the binary solution is generally
suboptimal. The authors of [3] advocate for the use of local search methods
([26, 27]) to refine the binary solution of the IPM, with such local methods
demonstrating an additional increase in performance. In the experiments
conducted in this paper, we intentionally omit these local approaches as they
are orthogonal to the convex optimization approach and can be independently
applied on top of the solution of any approach (including the subsequent
greedy approach and the proposed GFLOW-SS).

###  IV-B Greedy Selection

In [11], the optimization problem is posed as follows:

$\displaystyle\text{maximize}_{\mathbf{x}}$
$\displaystyle\log\text{det}\left(\sum_{i=1}^{m}x_{i}\mathbf{a}_{i}\mathbf{a}_{i}^{\mathsf{T}}+\epsilon\mathbf{I}\right)$
(13) subject to $\displaystyle\|\mathbf{x}\|_{1}=k$
$\displaystyle\mathbf{x}_{i}\in\\{0,1\\}\hskip 5.69046pti=1,\dots,m$

where $\mathbf{I}$ denotes the identity matrix and $\epsilon$ is a small
positive constant. The authors show that the objective function is a monotone
submodular function, and the feasible set constitutes a uniform matroid [28].
Building upon this, [11] introduces a straightforward greedy algorithm,
commencing with a zero vector $\mathbf{x}_{0}$ and incrementally adding “1”
elements in accordance with a greedy criterion. Throughout the manuscript, we
refer to this approach as GREEDY-SS.

###  IV-C GFlowNet Approach

Here we adapt the proposed approach, outlined in Section III, to the original
problem formulation of eq. (11). This involves defining the reward function
for each terminal node (corresponding to the performance of the respective
subset) and determining the parameterization of the flow network.

Given that the GFlowNet framework is akin to the amortization of an energy
function to a sampling distribution [23], it is essential for the reward of
each terminal node to be nonnegative. The goal is to identify the subset-
terminal node that maximizes the resulting determinant of the sum of outer
products. As the determinant of a matrix falls within the range
$[-\infty,\infty]$, the sigmoid function, being strictly increasing, is
employed to compress the output into the range $[0,1]$. The sigmoid function
is defined as follows:

$\sigma(x)=\frac{1}{1+e^{-x}}$

If we assume that a terminal node $\mathbf{x}\in\mathcal{X}$ corresponds to a
determinant
$D(\mathbf{x})=\text{det}\left(\sum_{i=1}^{m}x_{i}\mathbf{a}_{i}\mathbf{a}_{i}^{\mathsf{T}}\right)$,
the reward assigned to node $\mathbf{x}$ is the sigmoid of the determinant
$D(\mathbf{x})$, i.e.,

$R(\mathbf{x})=\sigma\left(D(\mathbf{x})\right).$ (14)

As the reward signal is the sole learning signal during updates and influences
the estimation of the flow for each node through bootstrapping, we opt to
scale the output of the sigmoid by a large constant $c$. This multiplication
transforms the range of the reward signal into $[0,c]$:

$R(\mathbf{x})=\sigma\left(D(\mathbf{x})\right)c.$ (15)

Regarding the parametrization of the flow, a straightforward option is to use
a multilayer perceptron neural network, where each layer comprises an affine
transformation and Rectified Linear (ReLU) activation. However, recent
findings in deep learning for function approximation [29] suggest that
multilayer perceptrons with ReLU activations struggle to capture high-
frequency components of a target signal, particularly when dealing with binary
inputs. Hence, we adopt an alternative approach proposed in [30, 31, 29, 32].
In this approach, the initial layer of the perceptron is replaced with a
learnable Fourier features kernel, consisting of an affine transformation
followed by a sinusoidal activation function. The elements of the Fourier
kernel are initialized from a zero-mean Gaussian distribution, with the
variance treated as a tunable hyperparameter.

###  IV-D Experiments

In this subsection, we perform a series of experiments to compare the two
previously established methods (CVX-OPT-SS and GREEDY-SS) with our proposed
approach, which leverages the GFlowNet paradigm (GFLOW-SS).

We consider an instance of the setting, where $m=100$ and the number of
selected sensors ranges from $5$ to $15$. The projection (measurement) vectors
$\mathbf{a}_{1},\dots,\mathbf{a}_{m}$ are sampled independently, in a random
fashion, from a Gaussian distribution $\mathcal{N}\left(0,1\right)$.

Concerning the GREEDY-SS approach, we set the $\epsilon$ parameter to
$1$e-$12$. This parameter is essential for ensuring that the objective
function remains monotone submodular and is expected to be kept small.

Figure 2: Comparison of the performance of GFLOW-SS, GREEDY-SS and CVX-OPT-SS
for the problem of selecting $k$ sensors out of $100$. The parameter $k$
ranges from $5$ to $15$. Each point corresponds to the average performance of
the respective approach over $8$ different instantiations. The GFLOW-SS
approach is trained for $40000$ root-to-leaf MDP trajectories for every
instantiation of the problem. This corresponds to $40000\times k$ gradient
descent steps.

Concerning the proposed GFLOW-SS approach, we opt for a $3$-layer dense neural
network as the parametrization for the flow. Each layer is configured with
$150$ neurons. The activation function for the second layer is ReLU, and the
first layer corresponds to the Fourier kernel. The elements of this kernel are
i.i.d samples from a zero-mean Gaussian distribution with a standard deviation
of $0.1$ at initialization. The reward scaling parameter $c$ (eq. 15) is
chosen to be $1000$. The Adam optimizer [33] is used as the stochastic
optimization algorithm with a learning rate of $2$e-4.

The results are illustrated in Fig. 2. Each dot in the plot represents the
performance of a solution (a $k$-size subset) generated by the respective
approach. For each value of $k$ and each algorithm, the problem is solved from
scratch. Each plotted point reflects the average over $8$ different
instantiations of the problem. For every value of $k$, the GFLOW-SS method
undergoes training with a total of $40000$ root-to-leaf trajectories of the
MDP. The training involves approximately $40000\times k$ gradient steps.
Consequently, the GFLOW-SS method explores a maximum of $40000$ different
subsets. It’s crucial to note that for the smallest experiment
($\binom{100}{5}$), the number of potential subsets is approximately $75$
million. In contrast, for the largest experiment ($\binom{100}{15}$), the
cardinality of the solution set escalates to $2\times 10^{17}$. Clearly,
$40000$ subsets represent only a minute fraction of the possible solutions in
each case. The GFLOW-SS method consistently outperforms both previous methods,
occasionally yielding subsets with determinants up to twice as large.

The trained GFlowNet explicitly models an action sampling distribution
conditioned on the state of the MDP of interest. Assuming that the training
process has identified a point in the flow approximator parameter space
satisfying the flow-matching objective, sampling actions from the
corresponding distribution ensures that each terminal state of the MDP is
sampled with a probability proportional to its reward. In the context of
sensor selection, each k-size subset is sampled with a probability
proportional to the corresponding system determinant (after being transformed
by the sigmoid function and scaled). To sample the best subset from the
trained GFlowNet, one starts from the MDP root (the state represented as all
“0”s) and greedily selects the action that maximizes the estimated flow at
each state. The performance of GFLOW-SS, depicted in Fig. 2, corresponds to
the subset sampled in that greedy fashion.

####  IV-D1 Diversity in Generation

In contrast to GREEDY-SS and CVX-OPT-SS, which offer a method to identify only
the best possible subset, the GFLOW-SS approach implicitly establishes a
ranking among the set of potential subsets. This introduces the flexibility to
select not only the best subset but also the second-best, third-best, and so
on. To illustrate, to choose the second-best defined subset, one can initiate
from the root state and sequentially opt for the action corresponding to the
second-largest estimated edge flow.

Figure 3: Comparison of the performance of GFLOW-SS (both for the best and the
$2$nd best subset), GREEDY-SS and CVX-OPT-SS for the problem of selecting $k$
sensors out of $50$. The parameter $k$ ranges from $10$ to $15$. Each point
corresponds to the average performance of the respective approach over $8$
different instantiations. In order to select the $2$nd best subset, the action
that corresponds to the $2$nd best flow is chosen at each state. The GFLOW-SS
approach is trained for $40000$ root-to-leaf MDP trajectories for every
instantiation of the problem. This corresponds to $40000\times k$ gradient
descent steps.

Fig. 3 presents a comparison of the performance between the best and the
second-best subsets obtained from the GFLOW-SS approach, alongside the
performances of GREEDY-SS and CVX-OPT-SS. The scenario involves selecting $k$
sensors out of a total of $m=50$, where $k$ ranges from $10$ to $15$. As
illustrated in the figure, the second-best subset consistently outperforms
both GREEDY-SS and CVX-OPT-SS approaches across all depicted values of $k$.

##  V Sensor Selection in ISAC

The integration of sensing capabilities into communications stands as a
pivotal direction in the evolution of the next generation of wireless systems
[34]. This convergence leads to the contemporary framework of ISAC systems
[35]. Antenna selection emerges as a viable solution aimed at curbing power
consumption and system costs in ISAC systems, all while upholding adequate
performance for both sensing and communication modules. In [36], a supervised
learning approach is proposed to select active antennas and optimize the
transmit beamforming matrix, aiming to optimize a linear combination of the
CRB of target estimates and the overall communication rate. Additionally, [37]
examines an ISAC setting with quantized phase shifters and introduces a deep
reinforcement learning approach to dynamically select a subset of antennas for
transmission in each channel, thereby mitigating antenna coupling.

###  V-A System Model

We adopt the system model of [36]. We consider a hybrid beamforming MIMO ISAC
system that possesses $N_{t}$ transmit antennas, $N_{r}$ receive antenna
elements and $N_{s}$ RF chains. The system performs two tasks: it is actively
tracking a distributed target and is communicating with a single user which is
equipped with $N_{s}$ antennas. The transmitter performs hybrid beamforming
with beamforming matrix $\mathbf{F}\in\mathbb{C}^{N_{t}\times N{s}}$. The
transmitted signal is:

$\mathbf{V}=\mathbf{F}\mathbf{X},$ (16)

where $\mathbf{X}\in\mathbb{C}^{N_{s}\times N_{t}}$ consists of $N_{s}$ unit
power streams of length $L$, which translates to
$\frac{1}{L}\mathbb{E}\left[\mathbf{X}\mathbf{X}^{H}\right]=\mathbf{I}_{N_{s}}$.
Furthermore, the channel matrix is denoted as
$\mathbf{H}\in\mathbb{C}^{N_{s}\times N_{t}}$. The objective is to choose
$N_{s}$ out of $N_{t}$ transmission antenna elements. To achieve this, a
selection matrix $\mathbf{S}\in\\{0,1\\}^{N_{s}\times N_{t}}$ is introduced.
Initially, all elements of $\mathbf{S}$ are set to $0$. Subsequently, each
index and row are processed sequentially. During processing, each row of
$\mathbf{S}$ remains $0$ except for the corresponding index of the selected
element, which is set to $1$. Consequently, the matrix
$\mathbf{S}^{H}\mathbf{S}$ is a diagonal matrix where each diagonal element is
$0$ if the corresponding antenna is not selected, or $1$ if the corresponding
antenna element is selected to be active.

The CRB of the radar estimates of the sparse array is [36]:

$\text{CRB}(\mathbf{S},\mathbf{F})=\frac{N_{r}}{L}\text{trace}\left((\mathbf{S}\mathbf{F}\mathbf{F}^{H}\mathbf{S}^{H})^{-1}\right).$
(17)

Correspondingly, the communication rate of the selected sensors is [36]:

$G_{\text{rate}}(\mathbf{S},\mathbf{F})=\text{log}_{2}|\mathbf{I}_{N_{r}}+\frac{1}{N_{s}}\mathbf{H}\mathbf{S}^{H}\mathbf{S}\mathbf{F}\mathbf{F}^{H}\mathbf{S}^{H}\mathbf{S}\mathbf{H}^{H}|.$
(18)

The objective is to select the antenna elements (therefore implicitly select
the selection matrix $\mathbf{S}$) and the beamforming matrix $\mathbf{F}$
such that a convex combination of the communication rate and the inverse of
the CRB is maximized.

$\displaystyle\text{maximize}_{\mathbf{S},\mathbf{F}}$
$\displaystyle\beta_{\text{CRB}}\frac{1}{\text{CRB}(\mathbf{S},\mathbf{F})}+\beta_{\text{rate}}G_{\text{rate}}(\mathbf{S},\mathbf{F})$
(19) subject to $\displaystyle\mathbf{S}\text{ is selection matrix, selecting
}N_{s}\text{ out of }N_{t}$

We denote as $\bm{\beta}=[\beta_{\text{CRB}},\beta_{\text{rate}}]$ the
$2$-dimensional vector of the coefficients of the two terms of the objective.
Both scalar parameters should be nonnegative, $\beta_{\text{CRB}}\geq 0$,
$\beta_{\text{rate}}\geq 0$. Furthermore they should add to 1,
$\beta_{\text{CRB}}+\beta_{\text{rate}}=1$. We refer to $\bm{\beta}$ as the
preference vector for the rest of the manuscript.

###  V-B A Multiobjective GFlowNet Approach

The proposed GFlowNet method, outlined in Section III, readily extends to the
specific ISAC scenario. Specifically, each terminal state $\mathbf{x}$ of a
Sensor Selection MDP can be directly mapped to a selection matrix
$\mathbf{S}$. If the selected antenna element positions (i.e., the indices of
the 1 elements in $\mathbf{x}$) are $l_{1},\dots,l_{N_{s}}$, then the
corresponding selection matrix $\mathbf{S}(\mathbf{x})$ is defined as follows:

$s_{ij}=\begin{cases}1&\text{if }l_{i}=j\\\ 0&\text{if
}\text{otherwise}\end{cases}$ (20)

Given a terminal state of the MDP and a fixed valid preference vector
$\bm{\beta}$, the reward of the terminal state can be computed as:

$R(\mathbf{x})=\beta_{\text{CRB}}\frac{1}{\text{CRB}(\mathbf{S}(\mathbf{x}),\mathbf{F})}+\beta_{\text{rate}}G_{\text{rate}}(\mathbf{S}(\mathbf{x}),\mathbf{F}).$
(21)

Considering that both the CRB and the communication rate depend not only on
the selected antennas but also on the precoding matrix $\mathbf{F}$, upon
reaching the terminal state $\mathbf{x}$, we can derive the corresponding
optimal precoding matrix $\mathbf{F}^{*}$ by maximizing the objective function
outlined in eq. (19) via a fixed number of gradient descent steps. Given that
the convex combination of the inverse CRB and the communication rate is a
function of the complex matrix $\mathbf{F}$, the Wirtinger conjugate
derivative [38] can be utilized. Hence, the reward function of the MDP is
defined as follows:

$R(\mathbf{x})=\beta_{\text{CRB}}\frac{1}{\text{CRB}(\mathbf{S}(\mathbf{x}),\mathbf{F}^{*})}+\beta_{\text{rate}}G_{\text{rate}}(\mathbf{S}(\mathbf{x}),\mathbf{F}^{*}).$
(22)

###  V-C Multiobjective Formulation

The limitation of directly employing the proposed GFlowNet method in the
context of the ISAC system is that it is applicable only for an appriori
determined vector $\bm{\beta}$. Should there be any alterations to the
coefficient terms, the reward function would consequently change,
necessitating the problem to be resolved anew. In this subsection, we present
an alternative formulation that tackles the challenge of multiobjective
optimization, thereby circumventing the necessity of retraining the GFlowNet
for different values of the preference vector, each corresponding to distinct
design system specifications.

The flow-matching objective outlined in eq. (9) pertains to the direct
parameterization of the optimal flow within the relevant MDP. Along the lines
of [39] , we introduce three additional quantities derived from the optimal
flow $F$: the forward sampling policy denoted as
$P^{F}(\mathbf{s}^{{}^{\prime}}=T(\mathbf{s},\mathbf{a})|\mathbf{s})=\frac{F(s,a)}{F(s)}$,
the backward sampling policy represented by
$P^{B}(\mathbf{s}|\mathbf{s}^{{}^{\prime}}=T(\mathbf{s},\mathbf{a}))=\frac{F(\mathbf{s},\mathbf{a})}{F(\mathbf{s}^{{}^{\prime}})}$,
and the partition function of the MDP, denoted as
$Z=\sum_{\mathbf{x}\in\mathcal{X}}R(\mathbf{x})$.

We then use a function approximator, such as a neural network, for the forward
policy $P_{\mathbf{w}}^{F}(\mathbf{s}^{{}^{\prime}}|\mathbf{s})$, a function
approximator for the backward policy
$P_{\mathbf{w}}^{B}(\mathbf{s}|\mathbf{s}^{{}^{\prime}})$, and a learnable
parameter for the partition function $Z_{\mathbf{w}}$. Then, for a root-to-
leaf trajectory
$\tau=(\mathbf{s}_{0}\rightarrow\mathbf{s}_{1}\rightarrow\dots\rightarrow\mathbf{s}_{n}=\mathbf{x})$,
the trajectory balance objective is:

$L_{\mathbf{w}}^{TB}(\tau)=\left(\text{log}\frac{Z_{\mathbf{w}}\prod_{t=1}^{n}P_{\mathbf{w}}^{F}(\mathbf{s}_{t}|\mathbf{s}_{t-1})}{R(\mathbf{x})\prod_{t=1}^{n}P_{\mathbf{w}}^{B}(\mathbf{s}_{t-1}|\mathbf{s}_{t})}\right)^{2}.$
(23)

By identifying a parameter vector $\mathbf{w}^{*}$ that minimizes the
aforementioned objective across all root-to-leaf trajectories, sampling
actions according to the converged forward policy $P_{\mathbf{w}^{*}}^{F}$
guarantees that terminal states are sampled proportionally to their
corresponding rewards.

The trajectory balance variation of the GFlowNet learning objective directly
parameterizes the partition function of the MDP, thereby enabling the
parametrization of a range of action-sampling distributions corresponding to
various reward functions within the MDP’s structure. These reward functions
can be conditioned on multiple variables [40]. In the context of sensor
selection for ISAC, the conditioning variable for the reward is the preference
vector $\bm{\beta}$ representing the coefficients of the two objective terms.
Hence, the reward of a terminal state $\mathbf{x}$ also depends on the choice
of $\bm{\beta}$, denoted as $R(\mathbf{x},\bm{\beta})$. Consequently, the
estimations of forward and backward policies are influenced by $\bm{\beta}$ as
well, expressed as
$P_{\mathbf{w}}^{B}(\mathbf{s}_{t-1}|\mathbf{s}_{t};\bm{\beta})$ and
$P_{\mathbf{w}}^{F}(\mathbf{s}_{t}|\mathbf{s}_{t-1};\bm{\beta})$. In this more
generalized formulation, the partition function is not a single learnable
scalar but rather a function approximator that takes the preference vector
$\bm{\beta}$ as input, $Z_{\mathbf{w}}(\bm{\beta})$.

The main difference between the GFlowNet approach that is outlined in Section
III and the multiobjective GFlowNet approach, besides the form of the
objective, is that before sampling each root-to-leaf trajectory for gradient
descent, a $\bm{\beta}$ vector is sampled from a set of predefined values.

Algorithm 2 MOGFLOW - SS

Initialize
$P_{\mathbf{w}}^{B}(\mathbf{s}^{{}^{\prime}}|\mathbf{s};\bm{\beta})$,$P_{\mathbf{w}}^{F}(\mathbf{s}|\mathbf{s}^{{}^{\prime}};\bm{\beta})$,
$Z_{\mathbf{w}}(\bm{\beta})$, $\zeta\in[0,1]$ for exploration, learning rate
$\eta$, dictionary of $\bm{\beta}$ values
$\\{\bm{\beta}^{1},\dots,\bm{\beta}^{d}\\}$, $N_{\text{Wir}}$ the number of
Wirtinger conjugate gradient steps for optimizing the beamforming matrix
$\mathbf{F}$

for all root-to-leaf trajectories do

$\mathbf{s}=\mathbf{s}_{0}=[0]^{N_{t}}$

Sample $\bm{\beta}$ randomly from the dictionary

Compute $Z_{\mathbf{w}}(\bm{\beta})$

for $N_{s}-1$ transitions do

Sample $z\sim\mathcal{U}(0,1)$ (Uniform distribution)

If $z<\zeta$ choose $\mathbf{s}^{{}^{\prime}}$ randomly

If $z>=\zeta$ sample $\mathbf{s}^{{}^{\prime}}$ from
$P_{\mathbf{w}}^{F}(\mathbf{s}|\mathbf{s}^{{}^{\prime}};\bm{\beta})$

Compute $P_{\mathbf{w}}^{B}(\mathbf{s}^{{}^{\prime}}|\mathbf{s};\bm{\beta})$

$\mathbf{s}$ = $\mathbf{s}^{\prime}$

end for

Sample $z\sim\mathcal{U}(0,1)$

If $z<\zeta$ choose choose $\mathbf{s}^{{}^{\prime}}$ randomly randomly

If $z>=\zeta$ choose $\mathbf{x}$ from
$P_{\mathbf{w}}^{F}(\mathbf{s}|\mathbf{x};\bm{\beta})$

Compute $P_{\mathbf{w}}^{B}(\mathbf{x}|\mathbf{s};\bm{\beta})$

Compute $\mathbf{F}^{*}$ by $N_{\text{Wir}}$ gradient steps and learning rate
$\eta$

Compute $R(\mathbf{x},\bm{\beta})$

$\tau=(\mathbf{s}_{0}\rightarrow\mathbf{s}_{1}\rightarrow\dots\rightarrow\mathbf{s}_{N_{s}}=\mathbf{x})$

$\mathbf{w}^{\prime}\rightarrow\mathbf{w}-\eta\nabla_{\mathbf{w}}L_{\mathbf{w}}^{TB}(\tau)$

end for

The multiobjective reformulation of the GFlowNet approach for sensor selection
in the ISAC system is referred to as MOGFLOW-SS, and its procedural framework
is outlined in Algorithm 2. A key practical advantage of MOGFLOW-SS over
GFLOW-SS lies in its ability to learn a spectrum of sampling distributions
conditioned not only on the current state of the MDP but also on the design
choice of the preference vector $\bm{\beta}$. This capability facilitates
generalization across values of the preference vector that may not have been
encountered during the training phase.

###  V-D Experiments

We consider a practical setting where the number of transmit antennas is
$N_{t}=80$ and the number of receive antennas is also $N_{r}=80$. The number
of RF chains is $N_{s}=10$. The length of the ISAC symbol is chosen to be
$L=100$. Every element of the channel matrix $\mathbf{H}$ is sampled from a
standard complex normal distribution $H_{ij}\sim\mathcal{C}\mathcal{N}(0,1)$.

The backward policy $P^{B}(\mathbf{s}^{{}^{\prime}}|\mathbf{s};\bm{\beta})$
serves as a prior over the parent nodes for each MDP state. It is essentially
an artifact of the training process and is discarded post-convergence. While
it could be explicitly parameterized, following the methodology outlined in
[39], we opt for a uniform distribution over the potential parent states of a
given state. Specifically, for every state in the Sensor Selection MDP
(excluding the root), the backward policy is selected to be inversely
proportional to the number of “1”s present in the vector representation of the
state, $P^{B}(-|\mathbf{s};\bm{\beta})=1/||\mathbf{s}||_{0}$.

Figure 4: The architecture of the two parametrizations
($Z_{\mathbf{w}}(\bm{\beta})$,
$P_{\mathbf{w}}^{F}(\cdot|\mathbf{s};\bm{\beta})$) employed for MOGFLOW-SS.

The forward policy is realized through a $3$-layer dense neural network, each
layer comprising $150$ neurons. Remarkably, the first layer incorporates the
Fourier kernel, as introduced by [30], initialized with elements drawn from a
zero-mean Gaussian distribution with a variance of $0.1$. Similarly, the
partition function $Z_{\mathbf{w}}(\bm{\beta})$ is parameterized in the log-
domain, employing a $3$-layer neural network structure identical to that of
the forward policy. The variance of the initialization for the Fourier kernel
elements is set to $0.001$. Despite employing two distinct parametrizations,
both are referenced using the same parameter vector $\mathbf{w}$, facilitating
joint optimization of both networks with a consistent objective and optimizer.
Specifically, we utilize the Adam optimizer [33] with a learning rate of
$10^{-5}$.

Regarding the preference vector
$\bm{\beta}=[\beta_{\text{CRB}},\beta_{\text{rate}}]$, where both coefficients
are nonnegative and sum up to $1$, we opt to represent them using a single
scalar variable $n$. Specifically, we set $\beta_{\text{CRB}}=n$ and
$\beta_{\text{rate}}=1-n$, where $n\in(0,1)$. For the training process, we
adopt a strategy of updating on three distinct values of $\bm{\beta}$. Each
value corresponds to a different value of $n$, namely ${0.1,0.5,0.9}$.

In formulating the reward function for a given terminal state-subset
$\mathbf{x}$ and a preference variable $n$, we encounter the challenge that
the two objective terms span different value ranges. To address this
discrepancy, we introduce an additional scalar parameter $c_{\text{scale}}$,
which we use to scale the inverse of the CRB term. This scaling ensures that
both objective terms exhibit roughly comparable ranges of values. To determine
the parameter $c_{\text{scale}}$, we compute the product
$\text{CRB}(\mathbf{S}(\mathbf{x}),\mathbf{F})\times
G_{\text{rate}}(\mathbf{S}(\mathbf{x}),\mathbf{F})$ for $10$ randomly sampled
subsets of size $N_{s}$. Finally, the reward function is formulated as
follows:

$R(\mathbf{x},n)=n\frac{c_{\text{scale}}}{\text{CRB}(\mathbf{S}(\mathbf{x}),\mathbf{F}^{*})}+(1-n)G_{\text{rate}}(\mathbf{S}(\mathbf{x}),\mathbf{F}^{*}),$
(24)

where $c_{\text{scale}}$ is chosen to be $20000$ for the setting of interest.

The training process encompasses a total of $60,000$ episodes (root-to-leaf
trajectories) and involves $N_{\text{Wir}}=50$ Wirtinger conjugate gradient
steps for estimating $\mathbf{F}^{*}$ at the conclusion of each episode.

Figure 5: The trajectory balance loss for MOGFLOW-SS is computed over $60000$
episodes, representing root-to-leaf trajectories.

Fig. 5 depicts the trajectory balance loss (eq. 23) over the course of
training. The plot illustrates a rapid decrease in the loss value, ultimately
converging to values very close to zero. This convergence signifies that the
forward policy effectively captures the action-sampling policy, ensuring that
terminal states are sampled proportionally to the performances of their
respective subsets.

Figure 6: The value of $\text{log}Z$ during training for the $3$ different
values of $n$.

Fig. 6 illustrates the evolution of the partition function (in the log domain)
throughout the training process, considering all three values of $n$ ($0.1$,
$0.5$, and $0.9$). Each $n$ value corresponds to a distinct preference vector
$\bm{\beta}$. While the partition function typically remains constant for a
single MDP [22], in this scenario, it varies with $n$. Consequently, as each
$n$ signifies a different reward function, the final values of $\text{log}Z$
upon convergence differ across different $n$ values. Lower values of $n$
prioritize the objective term linked to the communication system
(communication rate), whereas higher values of $n$ emphasize the term
associated with radar performance (the inverse of the CRB). This nuance allows
for flexible optimization tailored to diverse system requirements.

Upon completing the training phase, we select five distinct values of $n$:
$0.1$, $0.3$, $0.5$, $0.7$, and $0.9$. Notably, three of these ($0.1$, $0.5$,
$0.9$) were employed during training, while the remaining two ($0.3$, $0.7$)
were not utilized for training purposes. For each $n$ value, we implement a
greedy strategy to sample the terminal state, selecting the action that
maximizes the estimated forward policy network at each state, starting from
the root. This approach results in a subset containing $N_{s}$ active antenna
elements for each $n$ value.

Figure 7: The CRB values (lower values indicate better performance) associated
with subsets selected by both the MOGFLOW-SS and the L2S methods for five
different values of $n$. Stars represent the performance of subsets recovered
by MOGFLOW-SS, while circles represent subsets recovered by L2S. Black stars
denote values of $n$ used during training, whereas Red stars denote values of
$n$ not included in the training process. Every point in the plot corresponds
to the average over $15$ different seeds.

Fig. 7 illustrates the CRB values associated with the optimal sensor subset
and beamforming matrix $\mathbf{F}^{*}$ obtained through the MOGFLOW-SS
method, across the five selected values of $n$. This plot provides a
comparison between the CRB performance achieved by MOGFLOW-SS and the method
presented in the work of [21], denoted as L2S in their original paper.
Notably, the approach of [21] examines the same setting. The results
demonstrate MOGFLOW-SS’s clear superiority over L2S. Consistently, MOGFLOW-SS
yields subsets and corresponding beamforming matrices that lead to lower CRB
values across all $n$ values. Here, lower CRB values obtained through MOGFLOW-
SS signify enhanced radar localization performance.

Figure 8: The communication rate values (higher values indicate better
performance) associated with subsets selected by both the MOGFLOW-SS and the
L2S methods for five different values of $n$. Stars represent the performance
of subsets recovered by MOGFLOW-SS, while circles represent subsets recovered
by L2S. Black stars denote values of $n$ used during training, whereas Red
stars denote values of $n$ not included in the training process. Every point
in the plot corresponds to the average over $15$ different seeds.

Fig. 8 illustrates the communication rate values corresponding to the optimal
sensor subset and beamforming matrix $\mathbf{F}^{*}$ obtained through
MOGFLOW-SS across five selected values of $n$. The communication rate
performance achieved by MOGFLOW-SS is compared with that of subsets derived
from L2S. Notably, MOGFLOW-SS surpasses L2S by generating subsets and
corresponding beamforming matrices associated with higher communication rates
across all $n$ values.

The parameter $n$ serves as a coefficient that scales the inverse of the CRB
term in the reward function. With the coefficients of both reward terms
summing to $1$, higher $n$ values prioritize the CRB term, while lower values
prioritize the communication rate term. This distinction is crucial in
understanding the trends observed in the derived subsets by MOGFLOW-SS. As $n$
increases, emphasizing the CRB term, MOGFLOW-SS produces subsets with smaller
CRB values and correspondingly larger inverse CRB values, as depicted in Fig.
7. Conversely, larger $n$ values result in subsets with lower communication
rates, as observed in Fig. 8. Moreover, for each $n$ value, MOGFLOW-SS
consistently outperforms L2S in terms of both radar performance (lower CRB
values) and communication performance (higher communication rates). This
superiority is particularly notable considering that L2S requires training
from scratch for each $n$ value, whereas MOGFLOW-SS is trained once for all
$n$ values. Despite this, MOGFLOW-SS effectively balances the trade-off
between radar and communication performance, making it superior in solving the
selection problem across varying $n$ values.

The pivotal aspect of MOGFLOW-SS performance is elucidated by the red points
in Figures 7 and 8. Specifically, these red points represent the performance
of MOGFLOW-SS for the two $n$ values excluded from the training process.
Essentially serving as a test set, these values validate the model’s
generalization capability beyond its training data. Remarkably, the trained
GFlowNet exhibits robust generalization to preference variable values absent
during gradient descent updates. This underscores the method’s ability to
effectively adapt to the entire spectrum of system configurations without
necessitating additional training or resources, thereby bolstering its
practical applicability.

####  V-D1 Thorough Assessment of Generalization

While the performance of MOGFLOW-SS on unseen values of $n$ (specifically
$n=0.3$ and $n=0.7$) highlights its generalization capability, questions arise
regarding its adaptability to diverse system settings and requirements.
Notably, each subset’s performance depends not only on the active sensor
selection but also on the estimation of the beamforming matrix $\mathbf{F}$.
Since $\mathbf{F}$ is optimized via gradient descent on the objective
function, different $n$ values may yield varying CRB and communication rate
outcomes for the same subset. Consequently, distinct $n$ values may result in
different $\mathbf{F}^{*}$ values, even for identical terminal states
$\mathbf{x}$. Therefore, genuine generalization would entail MOGFLOW-SS
selecting different subsets for different $n$ values, particularly for those
($0.3$ and $0.7$) absent during the training phase.

TABLE I: Antennas chosen by MOGFLOW-SS for different values of $n$ $n$ |  Indices of Antenna Positions Chosen
---|---
$0.1$ |  [$6$, $12$, $14$, $15$, $18$, $32$, $45$, $46$, $62$, $77$]
$\mathbf{0.3}$ |  [$\mathbf{6}$, $\mathbf{14}$, $\mathbf{35}$, $\mathbf{38}$, $\mathbf{46}$, $\mathbf{53}$, $\mathbf{57}$, $\mathbf{62}$, $\mathbf{66}$, $\mathbf{68}$]
$0.5$ |  [$0$, $4$, $10$, $27$, $33$, $38$, $39$, $51$, $60$, $68$]
$\mathbf{0.7}$ |  [$\mathbf{0}$, $\mathbf{10}$, $\mathbf{19}$, $\mathbf{22}$, $\mathbf{29}$, $\mathbf{33}$, $\mathbf{38}$, $\mathbf{40}$, $\mathbf{51}$, $\mathbf{68}$]
$0.9$ |  [$0$, $10$, $19$, $29$, $33$, $38$, $40$, $51$, $60$, $67$]

Table I displays the indices of the selected antennas, determined by MOGFLOW-
SS, across different $n$ values for a single random seed, representing a sole
iteration of the training process. Notably, while significant overlap exists
among the antenna subsets, each subset corresponding to a specific $n$ value
remains distinct. This distinctiveness is particularly pronounced for $n=0.3$
and $n=0.7$, which were excluded from the training process. The uniqueness of
each subset underscores the robust generalization capacity inherent in the
GFlowNet paradigm, facilitating the accurate estimation of well-performing
antenna configurations across the entire range of $n\in(0,1)$. Each $n$ value
embodies a distinct trade-off between the radar and communication systems.
Notably, this uniqueness persists across all $15$ seeds utilized in the
experimentation.

##  VI Remarks

The challenge of sensor selection problems lies in the combinatorial explosion
of the solution set. While greedy selection methods offer efficiency by
narrowing down the search space, they often yield optimal or near-optimal
solutions only under highly specific problem instances. Motivated by this
limitation, our work seeks to pioneer a paradigm that can reason about the
quality of a vast majority of potential solutions without the need for
exhaustive evaluation. Initially, the idea of modeling selection as an MDP may
seem counterintuitive, as it ostensibly expands the search space. This arises
from the fact that the number of possible state-action pairs in the sensor
selection MDP exceeds the number of possible subsets (terminal states).
However, this approach enables us to leverage GFlowNets to implicitly
parameterize the action sampling distribution. By doing so, we transition the
search space from discrete subsets or state-action pairs to the continuous
parameter space of the function approximator class. Operating within this
framework, we can employ stochastic gradient descent methods. Notably, the
same parameter vector governs estimations for all state-action pairs,
facilitating implicit generalization. Consequently, updates based on specific
subsets influence estimations for all other subsets. This inherent
generalization capability allows us to reason about subsets without the need
for exhaustive evaluation.

In the multiobjective formulation, this generalization extends to the space of
preference vectors. This aspect is enabled by the explicit parametrization of
the partition function of the MDP (eq. (23)).

##  VII Conclusion

The current study has investigated the broad domain of problems that are
prevalent in signal processing and focus on the selection of a fixed-size
subset of sensors from a deployed set in order to optimize a metric related to
detection and estimation. These problems resemble knapsack-type scenarios,
inheriting the combinatorial complexity stemming from the exponential growth
of the solution set. Prior research has proposed numerous methodologies to
address such problems, broadly categorized into two main groups: analytical
approaches (such as convex optimization and greedy selection) and supervised
machine learning methods. Analytical approaches, while offering insights, are
often reliant on specific properties of the objective function and tend to
provide suboptimal solutions. On the other hand, supervised machine learning
methods necessitate substantial amounts of annotated data for effective
training. In light of these limitations, the current work has presented a
novel general framework predicated on generative AI to tackle sensor selection
problems, aiming to overcome the drawbacks of previous paradigms. The proposed
approach formulates the sensor selection problem as a deterministic MDP, where
subsets of desired size emerge as terminal states. The objective is to learn
an action sampling distribution that ensures that the cumulative probability
of sampling a terminal state is proportional to the performance of the
corresponding subset. Leveraging the deep generative modeling paradigm of
GFlowNet, the proposed approach amortizes the cost of learning the action
sampling distribution. Empirical evaluation has demonstrated that the proposed
approach outperforms both convex optimization methods and greedy selection
approaches. Notably, the approach is trained on a significantly reduced subset
of the solution set. Furthermore, a multiobjective formulation of the approach
has been introduced and applied to a sensor selection problem for ISAC
systems. This problem entails a trade-off between radar and communication
system performance, encapsulated by a preference vector. The proposed approach
adeptly manages this trade-off and demonstrates robust generalization even to
preference vector values not used during the GFlowNet training process.

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