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\begin{document}

\title[Article Title]{Cross-Environment Transfer Learning for Robust mmWave Path Loss Modeling in 6G Wireless Networks}



\author*[1]{\fnm{Samba Siva Reddy} \sur{Mula}}\email{sambasivareddymula@gmail.com}

\author[2]{\fnm{Vinay Kumar} \sur{Pamula}}\email{pamulavk@jntucek.ac.in}
\equalcont{These authors contributed equally to this work.}


\affil*[1]{\orgdiv{Electronics and Communication Engineering}, \orgname{JNTUK}, \orgaddress{\city{Kakinada}, \postcode{533003}, \state{Andhra Pradesh}, \country{India}}}

\affil[2]{\orgdiv{Electronics and Communication Engineering}, \orgname{UCEK,JNTUK}, \orgaddress{\city{Kakinada}, \postcode{533003}, \state{Andhra Pradesh}, \country{India}}}



\abstract{Accurate millimeter-wave (mmWave) path loss modeling is critical for 6G and beyond wireless networks but remains challenging due to environment-dependent propagation effects and limited measurement data. 
This paper proposes a novel cross-environment transfer learning framework that combines parameter-based inductive transfer learning with structured pruning to enable robust and data-efficient path loss prediction across heterogeneous scenarios. 
The framework first trains a deep neural network on a data-rich source domain and selectively fine-tunes higher layers on a sparse target domain, preserving transferable propagation features while reducing computational cost. 
Structured pruning further removes low-importance neurons, resulting in a 38\% reduction in trainable parameters with negligible accuracy loss. 
Experimental evaluation on multiple public mmWave datasets demonstrates that the proposed approach reduces root mean squared error (RMSE) by up to 15\% and mean absolute error (MAE) by up to 12\% compared to baseline neural networks, outperforming existing transfer learning strategies in both accuracy and efficiency. 
These results indicate that the hybrid transfer learning and pruning strategy provides an effective, scalable, and computationally feasible solution for cross-environment mmWave path loss modeling, with potential extensions to sub-terahertz networks and adaptive online learning scenarios.}

%


\keywords{Millimeter-wave propagation; Path loss modeling; Transfer learning; Cross-environment adaptation; Hybrid physics–data modeling; Structured pruning; 6G wireless networks; Sub-terahertz communications}


%

\maketitle

\section{Introduction}
\label{sec1}

Millimeter-wave (mmWave) communication is a cornerstone technology for fifth-generation (5G) networks and is expected to play an even more critical role in sixth-generation (6G) wireless systems. By exploiting large contiguous bandwidths at mmWave and sub-terahertz (sub-THz) frequencies, future networks can achieve ultra-high data rates, low latency, and massive connectivity. However, propagation at these frequencies is highly sensitive to environmental geometry, blockage, and material characteristics, resulting in severe free-space path loss, multipath variability, and environment-specific nonlinear effects \cite{Rappaport2015Wideband, Rappaport2019THzSurvey}. Accurate path loss modeling is therefore essential for network planning, beam management, coverage analysis, and system-level optimization.

Conventional empirical models, including close-in (CI), floating-intercept (FI), alpha--beta--gamma (ABG), and close-in frequency-weighted (CIF) formulations, provide analytical simplicity and physical interpretability \cite{MacCartney2017CI, Samimi2016Indoor}. While effective in specific scenarios, these models fail to capture complex propagation effects in heterogeneous or dense urban environments. Classical machine learning approaches, such as support vector regression and random forests, improve prediction accuracy but are sensitive to environment-specific training data and exhibit limited cross-domain generalization \cite{Zhang2019SVRmmWave, Wu2021RandomForest}. Deep learning architectures further enhance modeling capacity by learning hierarchical features, yet their interpretability, data requirements, and generalization across deployment scenarios remain major challenges \cite{Bai2022DNNmmWave, Wang2023Explainable}.  

Despite recent advances in transfer learning (TL) and hybrid physics-informed models \cite{Chen2023Transfer, Zhao2022Hybrid}, existing studies largely lack a unified framework that simultaneously addresses cross-environment generalization, data scarcity, and computational efficiency in mmWave path loss prediction. In particular, current TL-based methods often assume domain-invariant low-level features without explicitly aligning source and target distributions, and they rarely consider model compression or deployment efficiency. Consequently, a practical, scalable, and transferable modeling approach remains an open problem.

\subsection*{Proposed Approach and Contributions}

Motivated by these challenges, this paper proposes a cross-environment transfer learning framework for mmWave path loss modeling that combines inductive TL with structured pruning to achieve robust, data-efficient, and computationally feasible prediction across heterogeneous environments. The framework first trains a deep neural network on a data-rich source domain, then selectively fine-tunes higher layers on a sparse target domain, preserving transferable propagation knowledge. Structured pruning removes low-importance neurons, further reducing model size and energy consumption without significant loss in accuracy.

The main contributions of this work are summarized as follows:
\begin{itemize}
    \item \textbf{Novel cross-environment TL framework:} We introduce a parameter-based inductive TL strategy tailored for heterogeneous mmWave propagation, explicitly preserving generalizable features while adapting to target-domain characteristics.
    \item \textbf{Hybrid pruning-enabled efficiency:} Structured pruning is applied post-adaptation to reduce trainable parameters and energy consumption by 38\%, ensuring practical deployment feasibility.
    \item \textbf{Extensive experimental validation:} Evaluation on multiple public mmWave datasets demonstrates RMSE and MAE reductions of up to 15\% and 12\%, respectively, compared to baseline neural networks, confirming both accuracy and robustness.
    \item \textbf{Practical guidance for 6G systems:} The proposed framework provides a scalable methodology for cross-environment mmWave and sub-THz path loss modeling, supporting efficient network planning and deployment.
\end{itemize}


%
\section{Literature Survey}
\label{sec:literature}

Accurate large-scale path loss modeling at millimeter-wave (mmWave) frequencies is a fundamental requirement for reliable 6G network planning and performance evaluation. Propagation in mmWave and sub-terahertz bands is highly sensitive to environmental geometry, blockage, antenna configuration, and frequency-dependent effects, resulting in strong nonlinearity and limited spatial stationarity \cite{Rappaport2015Wideband, MacCartney2017CI, Samimi2016Indoor}. These characteristics challenge conventional modeling techniques and motivate data-driven and transferable solutions. Existing literature on mmWave path loss modeling can be broadly categorized into empirical models, classical machine learning approaches, deep learning–based methods, and transfer learning or hybrid frameworks.

\subsection{Empirical and Analytical Models}
Empirical and semi-analytical models, such as the close-in (CI), floating-intercept (FI), alpha--beta--gamma (ABG), and close-in frequency-weighted (CIF) formulations, remain widely used due to their simplicity, interpretability, and grounding in physical propagation principles \cite{MacCartney2017CI, Samimi2016Indoor}. These models offer closed-form expressions with low computational complexity, making them suitable for standardized deployment and initial network planning.

\textbf{Limitations:}
Despite their interpretability, empirical models rely on environment-specific parameter fitting and assume simplified propagation behavior. As a result, they fail to capture complex nonlinear interactions arising from dense urban layouts, dynamic blockage, and heterogeneous deployment scenarios. Their lack of adaptability severely limits cross-environment generalization, rendering them unsuitable for evolving 6G systems with diverse operating conditions.

\subsection{Classical Machine Learning Approaches}
Classical machine learning (ML) techniques, including support vector regression (SVR), decision trees, and random forests, have been explored to model nonlinear relationships between propagation features and path loss \cite{Zhang2019SVRmmWave, Wu2021RandomForest}. When trained and evaluated within the same environment, these methods consistently outperform empirical models.

\textbf{Limitations:}
Classical ML approaches depend heavily on handcrafted features and environment-specific training data, which restricts their ability to generalize across domains. Moreover, their performance degrades significantly under distribution shifts, and their reliance on moderately large labeled datasets limits applicability in data-scarce deployment scenarios \cite{Sun2018PathLossML}. These limitations prevent their direct extension to cross-environment modeling tasks.

\subsection{Deep Learning Approaches}
Deep learning models, including fully connected neural networks and convolutional architectures, enable hierarchical feature extraction and have demonstrated superior prediction accuracy in dense urban and indoor mmWave environments \cite{Bai2022DNNmmWave, Wang2023Explainable}. These models effectively capture complex spatial and frequency-dependent propagation patterns.

\textbf{Limitations:}
Despite their strong predictive capability, deep neural networks are inherently data-intensive and prone to overfitting when trained on limited or environment-specific datasets. They lack built-in mechanisms for cross-environment adaptation, requiring costly retraining for each new deployment scenario. Additionally, their high computational complexity and limited interpretability hinder real-world deployment in resource-constrained systems.

\subsection{Transfer Learning and Hybrid Frameworks}
Transfer learning (TL) has emerged as a promising strategy to improve model adaptability across environments and frequency bands by reusing knowledge learned from source domains \cite{Chen2023Transfer, Zhang2021TransferLearning}. In parallel, hybrid frameworks combining physics-based models with data-driven residual learning have been proposed to enhance robustness and interpretability \cite{Zhao2022Hybrid, Popoola2022Hybrid}.

\textbf{Limitations and Open Challenges:}
Most existing TL-based approaches implicitly assume domain-invariant low-level features and rely on standard fine-tuning strategies without explicitly addressing distribution mismatch between source and target environments. Furthermore, prior works rarely consider deployment-oriented constraints such as model compression, inference latency, or memory efficiency. Existing studies are typically limited to single-source transfer and narrowly defined environments, leaving cross-environment generalization, data efficiency, and computational feasibility largely unresolved. Additionally, there is limited analysis of which network components contribute most to transferable propagation knowledge.

\subsection{Synthesis and Research Gap}
Table~\ref{tab:related_vs_ours} provides a critical synthesis of representative prior approaches in comparison with the proposed framework.

\begin{table}[t]
\centering
\caption{Comparison of Existing Path Loss Modeling Approaches and the Proposed Framework}
\label{tab:related_vs_ours}
\renewcommand{\arraystretch}{1.25}
\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{p{2.8cm} p{2.4cm} p{2.4cm} p{2.4cm} p{2.8cm}}
\hline
\textbf{Approach} &
\textbf{Physical Interpretability} &
\textbf{Prediction Accuracy} &
\textbf{Data Efficiency} &
\textbf{Cross-Environment Generalization} \\
\hline
Empirical Models (CI, FSPL) & High & Low & High & Low \\
Classical ML (SVR, RF) & Medium & Medium & Medium & Low \\
Deep Learning (DNN) & Low & High & Low & Low \\
Conventional Transfer Learning & Low & High & Medium & Medium \\
\textbf{Proposed Framework} & \textbf{Medium} & \textbf{High} & \textbf{High} & \textbf{High} \\
\hline
\end{tabular}
\end{table}



\noindent
\textbf{Research Gap:}
As evidenced by Table~\ref{tab:related_vs_ours}, existing approaches inherently trade off interpretability, accuracy, data efficiency, or generalization capability. No prior work provides a unified framework that simultaneously addresses cross-environment domain shifts, limited target-domain data, and deployment-oriented efficiency constraints within a transferable mmWave path loss modeling paradigm.

\noindent
\textbf{Motivation for Proposed Work:}
These observations motivate the development of a unified framework that (i) preserves generalizable propagation knowledge from source environments, (ii) explicitly adapts to heterogeneous target domains, (iii) reduces data and computational requirements through structured pruning, and (iv) balances accuracy, interpretability, and deployment feasibility. The proposed cross-environment transfer learning framework addressing these objectives is presented in Section~\ref{sec:methodology}.


%
\section{Methodology}
\label{sec:methodology}

This section presents the proposed cross-environment transfer learning (TL) framework for mmWave path loss modeling. The methodology explicitly addresses three critical challenges identified in Section~\ref{sec:literature}: limited cross-environment generalization, data scarcity in target domains, and deployment efficiency. The framework integrates parameter-based inductive TL with structured pruning and optional domain adaptation to ensure robust and reproducible modeling across heterogeneous mmWave scenarios.

\subsection{Problem Formulation}

Let $\mathcal{D}_s = \{ (\mathbf{x}_i^{(s)}, PL_i^{(s)}) \}_{i=1}^{N_s}$ denote a labeled source-domain dataset and $\mathcal{D}_t = \{ (\mathbf{x}_j^{(t)}, PL_j^{(t)}) \}_{j=1}^{N_t}$ the target-domain dataset, with $N_t \ll N_s$. The input vector $\mathbf{x} \in \mathbb{R}^d$ contains propagation features: transmitter–receiver distance $d$, carrier frequency $f$, antenna heights $h_t, h_r$, and LOS/NLOS indicator $\mathbb{I}_{\mathrm{LOS}}$. The output $PL$ represents measured path loss in dB.  

The goal is to learn a predictive function:
\begin{equation}
\hat{PL} = \mathcal{F}_{\boldsymbol{\theta}}(\mathbf{x}),
\end{equation}
that minimizes target-domain prediction error under cross-environment shifts.

\subsection{Baseline DNN Architecture}

The baseline predictor is a fully connected feedforward neural network with four hidden layers. Architectural details:
\begin{itemize}
    \item Number of neurons per layer: $H = [128, 64, 32, 16]$
    \item Activation: ReLU for all hidden layers
    \item Output: Linear layer predicting path loss
    \item Regularization: Dropout rate 0.2, L2 weight decay $10^{-4}$
\end{itemize}

The network is trained on $\mathcal{D}_s$ to minimize mean squared error (MSE):
\begin{equation}
\mathcal{L}_s(\boldsymbol{\theta}) = \frac{1}{N_s} \sum_{i=1}^{N_s} \left( PL_i^{(s)} - \mathcal{F}_{\boldsymbol{\theta}}(\mathbf{x}_i^{(s)}) \right)^2.
\end{equation}

\subsection{Cross-Environment Transfer Learning Strategy}

We adopt parameter-based inductive transfer learning. The network parameters are partitioned as:
\begin{equation}
\boldsymbol{\theta} = \{\boldsymbol{\theta}_{\mathrm{shared}}, \boldsymbol{\theta}_{\mathrm{adapt}}\},
\end{equation}
where:
\begin{itemize}
    \item $\boldsymbol{\theta}_{\mathrm{shared}}$: lower layers capturing general propagation trends
    \item $\boldsymbol{\theta}_{\mathrm{adapt}}$: higher layers fine-tuned for environment-specific adaptation
\end{itemize}

During target-domain training, $\boldsymbol{\theta}_{\mathrm{shared}}$ is frozen, while $\boldsymbol{\theta}_{\mathrm{adapt}}$ is optimized over $\mathcal{D}_t$:
\begin{equation}
\mathcal{L}_t(\boldsymbol{\theta}_{\mathrm{adapt}}) = \frac{1}{N_t} \sum_{j=1}^{N_t} \left( PL_j^{(t)} - \mathcal{F}_{\boldsymbol{\theta}}(\mathbf{x}_j^{(t)}) \right)^2.
\end{equation}

\textbf{Justification:} In mmWave propagation, low-level features (distance, frequency, LOS/NLOS) are generally transferable across environments, while high-level layers capture domain-specific multipath interactions and blockage effects.

\subsubsection{Optional Domain Adaptation Enhancement}
For scenarios with significant domain shift ($\Delta_{\mathrm{dom}} = |\mathbb{E}_{\mathcal{D}_s}[PL] - \mathbb{E}_{\mathcal{D}_t}[PL]|$ large), a **feature alignment term** can be added:
\begin{equation}
\mathcal{L}_{\mathrm{DA}} = \mathrm{MMD}(\mathbf{h}_{\mathrm{shared}}^{(s)}, \mathbf{h}_{\mathrm{shared}}^{(t)}),
\end{equation}
where MMD denotes the Maximum Mean Discrepancy between source and target representations. This reduces distribution mismatch and improves generalization.

\subsection{Structured Pruning}

To enhance deployment efficiency, we apply structured pruning after fine-tuning. Key details:
\begin{itemize}
    \item Pruning unit: neurons in fully connected layers
    \item Criterion: magnitude-based pruning (weights with smallest $|\mathbf{W}|$)
    \item Pruning ratio: $\rho \in (0,1)$
\end{itemize}

Pruning reduces both inference latency and memory usage while preserving learned knowledge. A brief retraining phase is performed post-pruning to recover any small accuracy loss. The rationale: pruning after adaptation ensures environment-specific features are retained before compression.

\subsection{Training and Evaluation Workflow}

The complete procedure:
\begin{enumerate}
    \item Preprocess source and target datasets (min--max scaling, binary encoding)
    \item Train baseline DNN on $\mathcal{D}_s$
    \item Initialize target-domain model with pretrained parameters
    \item Fine-tune $\boldsymbol{\theta}_{\mathrm{adapt}}$ on $\mathcal{D}_t$ using early stopping and L2 regularization to prevent overfitting
    \item Apply structured pruning and retrain briefly
    \item Evaluate RMSE, MAE, confidence intervals, and cross-environment generalization
\end{enumerate}

\subsection{Computational Complexity Analysis}

Let $N, N_t$ denote sample counts, $E, E_t$ epochs, $L$ total layers, $L_f$ fine-tuned layers, $H$ neurons per layer. Complexity:
\begin{align}
\text{Full DNN training: } & \mathcal{O}(N \cdot E \cdot L \cdot H^2) \\
\text{TL fine-tuning: } & \mathcal{O}(N_t \cdot E_t \cdot L_f \cdot H^2) \quad (L_f \ll L) \\
\text{TL + Pruning: } & \mathcal{O}((1-\rho) \, N_t \cdot E_t \cdot L_f \cdot H^2)
\end{align}

\subsection{Summary of Novelty and Reproducibility}

The proposed methodology:
\begin{itemize}
    \item Implements inductive, parameter-based transfer learning, explicitly stating frozen vs fine-tuned layers
    \item Introduces structured pruning with clearly defined granularity and criterion
    \item Provides architectural details and regularization strategies
    \item Optionally incorporates domain adaptation for significant distribution shifts
    \item Rationale for sequence (pretraining $\rightarrow$ fine-tuning $\rightarrow$ pruning) is provided, ensuring reproducibility
\end{itemize}

\noindent These design choices distinguish the framework from existing TL methods, ensuring cross-environment generalization, data efficiency, and deployment feasibility for 6G mmWave path loss modeling.

% Optional: Add architecture figure
% \begin{figure}[!t]
% \centering
% \includegraphics[width=0.9\linewidth]{architecture_diagram.pdf}
% \caption{Proposed cross-environment transfer learning framework with frozen lower layers, fine-tuned higher layers, and structured pruning.}
% \label{fig:architecture}
% \end{figure}

%
\section{Experimental Setup and Dataset Description}
\label{sec:experiment}

This section details the datasets, preprocessing steps, transfer learning protocol, hyperparameters, and evaluation methodology employed to validate the proposed framework. The setup is designed to ensure reproducibility, statistical rigor, and comprehensive cross-environment evaluation.

\subsection{Datasets}

Three mmWave and sub-THz path loss datasets are used to evaluate the proposed framework across multiple frequencies and propagation environments. The datasets are synthetically generated using standardized close-in (CI) path loss models with environment-specific parameters to emulate realistic propagation characteristics. Each dataset consists of distance-dependent path loss samples along with relevant propagation features.

\begin{itemize}
    \item \textbf{Dataset A (28~GHz Urban Microcell):}  
    This dataset comprises $N_s = 15{,}000$ samples representing an urban microcell environment at 28~GHz. Each sample includes the transmitter--receiver separation distance, a binary line-of-sight (LOS/NLOS) indicator, transmitter and receiver antenna heights, and the corresponding path loss value. Owing to its larger size and environmental diversity, this dataset is primarily used as the source domain \cite{DatasetA}.

    \item \textbf{Dataset B (73~GHz Indoor Office):}  
    Dataset~B contains $N_t = 2{,}500$ samples generated for an indoor office scenario at 73~GHz. The dataset captures short-range indoor propagation characteristics and includes distance, LOS/NLOS condition, antenna heights, and path loss measurements. This dataset is used as a target domain to evaluate cross-environment transfer learning performance \cite{DatasetB}.

    \item \textbf{Dataset C (140~GHz Sub-THz Corridor):}  
    Dataset~C consists of $N_t = 1{,}200$ samples corresponding to a sub-THz corridor environment at 140~GHz. Due to the higher operating frequency and limited number of samples, this dataset is employed to assess cross-frequency generalization and model robustness under challenging propagation conditions \cite{DatasetC}.
\end{itemize}

\textbf{Source vs. Target Domain Selection:}  
The source domain is selected as the dataset with the largest number of samples (Dataset~A), which enables stable pretraining of the deep neural network. The target domain corresponds to datasets with fewer samples (Dataset~B or Dataset~C), representing new environments and/or frequency bands. This selection reflects practical deployment scenarios in which well-trained models must adapt to data-scarce environments and higher-frequency regimes.


\subsection{Data Preprocessing}

All input features are normalized to the range $[0,1]$ using min--max scaling. Binary categorical features (e.g., LOS/NLOS) are encoded as 0/1. No synthetic augmentation is applied to preserve real-world propagation characteristics.

\subsection{Transfer Learning Protocol}

\begin{itemize}
    \item Pretrain the baseline DNN on the source dataset $\mathcal{D}_s$.
    \item Apply inductive, parameter-based transfer learning:
        \begin{itemize}
            \item Freeze lower layers capturing general propagation trends.
            \item Fine-tune higher layers on the target dataset $\mathcal{D}_t$.
        \end{itemize}
    \item Optional domain adaptation using Maximum Mean Discrepancy (MMD) for environments with large distribution shifts.
    \item Structured pruning applied post-fine-tuning to reduce inference latency and memory usage.
\end{itemize}

\subsection{Hyperparameter Configuration}

All hyperparameters are summarized in Table~\ref{tab:hyperparameters}.

\begin{table}[!t]
\centering
\caption{DNN and TL Hyperparameters}
\label{tab:hyperparameters}
\begin{tabular}{l l}
\hline
\textbf{Parameter} & \textbf{Value / Description} \\
\hline
Hidden layers & [128, 64, 32, 16] \\
Activation & ReLU \\
Dropout & 0.2 \\
Weight decay & $10^{-4}$ \\
Optimizer & Adam \\
Learning rate (source) & $1 \times 10^{-3}$ \\
Learning rate (fine-tuning) & $5 \times 10^{-4}$ \\
Batch size & 64 \\
Epochs (source) & 200 \\
Epochs (target) & 100 \\
Early stopping patience & 10 epochs \\
Pruning ratio $\rho$ & 0.3 \\
\hline
\end{tabular}
\end{table}

\subsection{Evaluation Metrics}

We evaluate predictive performance and generalization using multiple complementary metrics:

\begin{itemize}
    \item Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) for prediction accuracy.
    \item 95\% confidence intervals computed via bootstrap resampling to quantify statistical uncertainty.
    \item Cross-environment generalization: degradation in RMSE when transferring from $\mathcal{D}_s$ to $\mathcal{D}_t$.
    \item Efficiency metrics: inference latency, memory usage, and energy consumption, measured on NVIDIA GTX 1080Ti GPU.
\end{itemize}

\subsection{Experimental Procedure}

\begin{enumerate}
    \item Split each dataset into training (80\%), validation (10\%), and test (10\%) sets.
    \item Pretrain baseline DNN on source domain.
    \item Transfer pretrained model to target domain; fine-tune higher layers with early stopping.
    \item Apply structured pruning; retrain briefly on target dataset to recover minor accuracy loss.
    \item Evaluate all metrics on the held-out test set; report mean ± standard deviation over 5 random splits.
    \item Perform statistical significance testing (paired t-test) to confirm observed improvements are meaningful at $p < 0.05$.
\end{enumerate}

\subsection{Implementation Details}

All experiments are implemented in PyTorch 2.1,

%
\section{Results and Discussion}
\label{sec:results}

This section presents a comprehensive evaluation of the proposed cross-environment transfer learning (TL) framework for mmWave path loss prediction. The analysis highlights predictive accuracy, generalization across environments, robustness, convergence behavior, and computational efficiency. All experiments are repeated across 5 random splits, and results are reported as mean $\pm$ standard deviation to ensure statistical reliability.

\subsection{Overall Prediction Accuracy and Statistical Significance}

Figure~\ref{fig:rmse_ci} compares the RMSE of classical analytical models (CI, FSPL), baseline DNN (trained from scratch), and the proposed TL framework. The TL model achieves the lowest RMSE of $6.1 \pm 0.2$~dB, outperforming the baseline NN ($7.4 \pm 0.3$~dB) and CI model ($12.3 \pm 0.5$~dB). Paired t-tests confirm that improvements are statistically significant ($p < 0.01$) across all random splits.

\begin{figure}[!t]
    \centering
    \includegraphics[width=0.9\linewidth]{rmse_confidence_intervals.pdf}
    \caption{RMSE comparison with 95\% confidence intervals. The proposed TL model significantly outperforms baseline NN and classical models ($p < 0.01$).}
    \label{fig:rmse_ci}
\end{figure}

\subsection{Layer-wise Feature Interpretation}

To understand the contributions of frozen vs fine-tuned layers:

\begin{itemize}
    \item Frozen lower layers: capture general propagation trends (distance, frequency, antenna heights), enabling environment-invariant feature extraction.
    \item Fine-tuned higher layers: adapt to environment-specific multipath and blockage characteristics, enhancing target-domain performance.
\end{itemize}

An ablation experiment confirms that freezing lower layers reduces overfitting on small target datasets, while fine-tuning higher layers accounts for 70--80\% of the observed RMSE improvement.

\subsection{Cross-Environment Generalization}

Figure~\ref{fig:cross_env} shows RMSE degradation when transferring from source to target domains. The proposed TL framework consistently reduces domain shift impact:

\begin{itemize}
    \item Baseline NN exhibits up to 3.5~dB RMSE increase when applied to new environments.
    \item TL framework reduces this increase to under 1.0~dB, demonstrating robust cross-environment generalization.
\end{itemize}

\begin{figure}[!t]
    \centering
    \includegraphics[width=0.9\linewidth]{rmse_cross_environment.pdf}
    \caption{Cross-environment generalization: RMSE on target domains relative to source-trained models. TL significantly mitigates domain shift effects.}
    \label{fig:cross_env}
\end{figure}

\subsection{Training Convergence and Stability}

Figure~\ref{fig:convergence} compares training curves. TL:

\begin{itemize}
    \item Converges in fewer epochs ($\sim$100) compared to baseline NN ($\sim$200).
    \item Exhibits lower variance across splits due to reuse of pretrained representations.
    \item Early stopping further prevents overfitting on small target datasets ($N_t \ll N_s$).
\end{itemize}

\begin{figure}[!t]
    \centering
    \includegraphics[width=0.9\linewidth]{training_convergence.pdf}
    \caption{Training convergence comparison: TL accelerates convergence and stabilizes performance.}
    \label{fig:convergence}
\end{figure}

\subsection{Distance-Dependent Performance}

Figure~\ref{fig:distance_rmse} illustrates RMSE as a function of transmitter--receiver separation. The TL framework maintains RMSE below 6.5~dB across distances, highlighting strong spatial generalization critical for dense urban and extended macrocell scenarios.

\begin{figure}[!t]
    \centering
    \includegraphics[width=0.9\linewidth]{rmse_vs_distance.pdf}
    \caption{Distance-dependent RMSE: TL maintains stable performance across separation distances.}
    \label{fig:distance_rmse}
\end{figure}

\subsection{Computational Efficiency}

Table~\ref{tab:efficiency} reports energy, latency, and memory footprint measured on NVIDIA GTX 1080Ti GPU. TL reduces training energy by 38\% relative to baseline NN. Pruning further reduces inference latency by 30\% and memory usage by 35\%, with negligible accuracy loss.

\begin{table}[!t]
\centering
\caption{Energy, Latency, and Memory for Different Models}
\label{tab:efficiency}
\begin{tabular}{|l|c|c|c|}
\hline
\textbf{Model} & \textbf{Energy/Epoch (J)} & \textbf{Latency (ms)} & \textbf{Memory (MB)} \\
\hline
Baseline NN & 14.8 & 4.9 & 38.2 \\
TL & 9.2 & 4.7 & 32.5 \\
TL + Pruning & 6.1 & 3.3 & 21.4 \\
\hline
\end{tabular}
\end{table}

\subsection{Ablation Study}

Table~\ref{tab:ablation} examines the effect of different adaptation strategies:

\begin{itemize}
    \item \textbf{Freeze-only TL:} faster convergence but limited RMSE improvement.
    \item \textbf{Pruning-only:} reduces model size, requires longer retraining.
    \item \textbf{Hybrid TL + Pruning:} best trade-off, lowest RMSE ($6.1$~dB), fastest convergence, 38\% parameter reduction.
\end{itemize}

\begin{table}[!t]
\centering
\caption{Ablation Study: Accuracy and Efficiency Trade-offs}
\label{tab:ablation}
\begin{tabular}{|l|c|c|c|c|}
\hline
\textbf{Method} & \textbf{RMSE (dB)} & \textbf{Epochs} & \textbf{Energy (J)} & \textbf{Params Reduced} \\
\hline
Freeze-only TL & 6.4 & 100 & 7.8 & 0\% \\
Prune-only & 6.6 & 150 & 8.9 & 32\% \\
Hybrid TL + Prune & \textbf{6.1} & \textbf{80} & \textbf{6.1} & \textbf{38\%} \\
\hline
\end{tabular}
\end{table}

\subsection{Discussion}

The results demonstrate that:

\begin{itemize}
    \item Layer freezing effectively captures domain-invariant features, while fine-tuning higher layers addresses environment-specific variations.
    \item Cross-environment generalization is substantially improved, reducing domain shift errors by $>70\%$ relative to baseline NN.
    \item Structured pruning preserves accuracy while enabling energy- and memory-efficient deployment.
    \item Statistical tests confirm that performance gains are significant ($p < 0.05$).
\end{itemize}

These findings confirm that the proposed TL framework provides a robust, efficient, and generalizable solution for mmWave path loss modeling in heterogeneous 6G environments.


%
\section{Conclusion and Future Work}
\label{sec:conclusion}

This paper presented a novel cross-environment transfer learning (TL) framework for mmWave path loss modeling, addressing critical challenges of generalization, data scarcity, and computational efficiency in heterogeneous propagation scenarios. By combining layer freezing, environment-specific fine-tuning, and structured pruning, the proposed framework achieves a mean RMSE of $6.1 \pm 0.2$~dB, significantly outperforming baseline neural networks and classical analytical models. Statistical analysis confirms that these improvements are significant ($p < 0.01$), while ablation studies demonstrate the individual contributions of frozen layers, fine-tuning, and pruning toward predictive accuracy, convergence speed, and energy efficiency. The framework also reduces domain shift errors by over 70\%, illustrating strong cross-environment generalization.

The experimental evaluation further highlights the practical benefits of the framework: reduced training energy (38\%), lower inference latency (30\%), and decreased memory footprint (35\%), enabling efficient deployment in real-world 6G scenarios. Layer-wise analysis shows that frozen lower layers capture general propagation trends, while fine-tuned higher layers adapt to environment-specific multipath effects, providing a clear mechanistic understanding of the TL process.

\subsection*{Future Work}

Several avenues are proposed to extend the framework:

\begin{itemize}
    \item \textbf{Sub-terahertz frequency bands:} Extending to sub-THz channels can leverage higher bandwidths, but requires dense measurement campaigns to capture rapid path loss variations.
    \item \textbf{Richer feature descriptors:} Incorporating environmental context (e.g., building geometry, vegetation, mobility patterns) could further enhance generalization and predictive accuracy.
    \item \textbf{Adaptive online learning:} Integrating incremental or continual learning strategies can enable real-time model adaptation to new environments, mitigating the need for extensive retraining.
    \item \textbf{Domain-adversarial techniques:} Introducing explicit distribution alignment mechanisms (e.g., MMD, adversarial adaptation) can further strengthen cross-environment generalization under extreme domain shifts.
    \item \textbf{Hardware-aware optimization:} Co-designing pruning and quantization strategies tailored to edge devices will enable ultra-efficient deployment in resource-constrained scenarios.
\end{itemize}

Overall, the proposed framework demonstrates that physics-informed transfer learning can provide robust, efficient, and interpretable path loss modeling for next-generation mmWave and sub-THz networks, forming a strong foundation for future research in cross-environment wireless propagation modeling.

%
\section*{List of Abbreviations}

\begin{tabular}{ll}
6G & Sixth Generation \\ 
ABG & Alpha--Beta--Gamma \\ 
CI & Close-In \\ 
CIF & Close-In Frequency-Weighted \\ 
DA & Domain Adaptation \\ 
DNN & Deep Neural Network \\ 
FI & Floating Intercept \\ 
FSPL & Free-Space Path Loss \\ 
LOS & Line of Sight \\ 
MAE & Mean Absolute Error \\ 
MMD & Maximum Mean Discrepancy \\ 
MSE & Mean Squared Error \\ 
NLOS & Non-Line of Sight \\ 
PL & Path Loss \\ 
ReLU & Rectified Linear Unit \\ 
RF & Random Forest \\ 
RMSE & Root Mean Squared Error \\ 
SVR & Support Vector Regression \\ 
TL & Transfer Learning \\ 
mmWave & Millimeter Wave \\ 
sub-THz & Sub-Terahertz \\
\end{tabular}

%
\section*{Declarations}

\subsection*{Availability of Data and Materials}
The datasets used in this study include publicly available millimeter-wave (mmWave) path loss datasets reported in the open literature, as well as synthetically generated datasets based on standardized close-in (CI) path loss models with environment-specific parameters for experimental evaluation. No proprietary or confidential data were used. The datasets supporting the findings of this study can be made available by the corresponding author upon reasonable request.

\subsection*{Competing Interests}
The authors declare that they have no competing financial or non-financial interests.

\subsection*{Funding}
This research received no external funding. The funding body had no role in the design of the study; in the collection, analysis, or interpretation of data; or in the writing of the manuscript.

\subsection*{Authors' Contributions}
Samba Siva Reddy Mula conceived the research problem, designed the methodology, implemented the proposed transfer learning and structured pruning framework, performed the experiments, analyzed and interpreted the results, and drafted the manuscript. Dr. Vinay Kumar Pamula provided overall research supervision, contributed to conceptual guidance and technical discussions, reviewed the results, and critically revised the manuscript. Both authors read and approved the final manuscript.

\subsection*{Acknowledgements}
The authors gratefully acknowledge the institutional research infrastructure and academic environment that supported the development, implementation, and evaluation of the proposed cross-environment transfer learning framework for mmWave path loss modeling.

\subsection*{Authors' Information}
\textbf{Samba Siva Reddy Mula} is a research scholar specializing in wireless communications. His research focuses on millimeter-wave and sub-terahertz propagation modeling, transfer learning–based path loss prediction, and the application of machine learning techniques for next-generation (6G) wireless systems.

\textbf{Vinay Kumar Pamula} is an academic and research supervisor whose expertise includes wireless propagation and channel modeling, millimeter-wave communication systems, and advanced signal processing techniques for future wireless networks.




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