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\usepackage{xcolor}
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% Define colors for consistency
\definecolor{methodcolor1}{RGB}{31,119,180}    % No PI
\definecolor{methodcolor2}{RGB}{255,127,14}    % YARN
\definecolor{methodcolor3}{RGB}{44,160,44}     % NTK
\definecolor{methodcolor4}{RGB}{214,39,40}     % Linear

\title{Appendix: Position Interpolation Effects on Activation Distributions}
\author{ICLR 2025 Submission}
\date{}

\begin{document}

\maketitle

\section{Activation Distribution Analysis for Position Interpolation Methods}
\label{sec:activation_analysis}

This appendix provides a comprehensive analysis of how different Position Interpolation (PI) methods affect activation distributions in transformer models, with particular focus on quantization implications. We analyze four PI approaches: No PI (baseline), YARN scaling, NTK-aware scaling, and Linear interpolation, examining their effects on pre-activation tail growth ($\rho^W$) and axis-aligned amplitude growth ($\rho^A$).

\subsection{Methodology}

Our analysis captures activations from LLaMA-2-7B-Chat attention projection layers under different PI configurations. We focus on two key metrics:

\begin{itemize}
    \item \textbf{Pre-activation tail growth ($\rho^W$):} Measures how PI methods amplify weight-error impact through distribution tail heaviness
    \item \textbf{Axis-aligned amplitude growth ($\rho^A$):} Quantifies activation clipping risk through channel-wise amplitude concentration
\end{itemize}

\subsection{Pre-activation Tail Growth Analysis ($\rho^W$)}

Figure~\ref{fig:tail_growth} presents a comprehensive analysis of how PI methods affect activation tail behavior, which directly impacts weight quantization sensitivity.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=\textwidth]{activation_analysis_plots/tail_growth_analysis.pdf}
    \caption{\textbf{Pre-activation tail growth analysis across PI methods.} (a) Tail ratio comparison showing normalized 99th and 99.9th percentiles relative to standard deviation. (b) Outlier fraction analysis revealing the proportion of activations exceeding 3$\sigma$ threshold. (c) Distribution tail heaviness measured by excess kurtosis. (d) Layer-wise progression of tail growth effects. (e) Direct distribution comparison between baseline and YARN methods. (f) Weight-error amplification factor combining tail ratio and standard deviation effects.}
    \label{fig:tail_growth}
\end{figure}

\textbf{Key Findings:}
\begin{itemize}
    \item \textcolor{methodcolor2}{\textbf{YARN}} exhibits slightly elevated outlier fractions (0.85\% vs 0.82\% baseline), indicating increased sensitivity to weight quantization errors
    \item All PI methods show similar high kurtosis values ($>1000$), suggesting heavy-tailed distributions that challenge quantization
    \item Layer-wise analysis reveals consistent tail behavior across the first two transformer layers
    \item Weight-error amplification factors demonstrate that PI methods can exacerbate quantization-induced errors through tail amplification
\end{itemize}

\subsection{Axis-aligned Amplitude Growth Analysis ($\rho^A$)}

Figure~\ref{fig:amplitude_growth} examines how PI methods affect channel-wise activation amplitudes, directly relating to activation quantization clipping risks.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=\textwidth]{activation_analysis_plots/amplitude_growth_analysis.pdf}
    \caption{\textbf{Axis-aligned amplitude growth analysis for activation clipping assessment.} (a) Channel amplitude concentration factors showing how activation variance concentrates in specific channels. (b) Peak channel amplitude comparison across PI methods. (c) Distribution of channel standard deviations via violin plots. (d) Layer-wise amplitude growth progression. (e) Activation clipping risk assessment relative to typical int8 quantization thresholds. (f) Channel-wise amplitude heatmap for YARN method demonstrating spatial activation patterns.}
    \label{fig:amplitude_growth}
\end{figure}

\textbf{Key Findings:}
\begin{itemize}
    \item \textcolor{methodcolor2}{\textbf{YARN}} shows the highest amplitude concentration (52.97) and peak channel standard deviation (1.311), indicating increased clipping risk
    \item \textcolor{methodcolor4}{\textbf{Linear interpolation}} demonstrates the most controlled amplitude growth (51.06 concentration, 1.215 peak std)
    \item Channel-wise analysis reveals that activation amplitudes are not uniformly distributed, with certain channels exhibiting significantly higher variance
    \item Clipping risk assessment indicates all PI methods operate near typical int8 quantization limits
\end{itemize}

\subsection{Outlier Shifting Patterns}

Figure~\ref{fig:outlier_shifting} analyzes how PI methods alter the spatial and magnitude distribution of activation outliers.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=\textwidth]{activation_analysis_plots/outlier_shifting_analysis.pdf}
    \caption{\textbf{Outlier shifting patterns across position interpolation methods.} (a) Outlier magnitude distribution comparing activation values exceeding 3$\sigma$ threshold. (b) Spatial distribution of outliers along sequence positions. (c) Channel-wise outlier occurrence analysis. (d) Correlation analysis of outlier patterns between baseline and PI methods.}
    \label{fig:outlier_shifting}
\end{figure}

\textbf{Key Findings:}
\begin{itemize}
    \item PI methods exhibit distinct outlier magnitude distributions, with YARN showing broader outlier spreads
    \item Spatial analysis reveals position-dependent outlier patterns, particularly relevant for long-context applications
    \item Channel-wise outlier analysis demonstrates non-uniform distribution across attention projection dimensions
    \item High correlation between baseline and PI method outlier patterns suggests systematic rather than random activation shifts
\end{itemize}

\subsection{Comprehensive Position Interpolation Effects}

Figure~\ref{fig:comprehensive_effects} provides an integrated view of PI method impacts on both $\rho^W$ and $\rho^A$ metrics, along with practical quantization implications.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=\textwidth]{activation_analysis_plots/comprehensive_pi_effects.pdf}
    \caption{\textbf{Comprehensive summary of position interpolation effects.} (a) Combined $\rho^W$ vs $\rho^A$ analysis showing the trade-off between tail growth and amplitude concentration. (b) Layer-wise progression of combined effects ($\rho^W \times \rho^A$). (c) Quantization difficulty prediction based on distribution characteristics. (d) Performance-efficiency trade-off analysis balancing activation stability with quantization compatibility.}
    \label{fig:comprehensive_effects}
\end{figure}

\textbf{Key Findings:}
\begin{itemize}
    \item The $\rho^W$ vs $\rho^A$ trade-off analysis reveals \textcolor{methodcolor2}{\textbf{YARN}} in the high-amplitude, moderate-tail region, indicating increased clipping risk but controlled tail behavior
    \item \textcolor{methodcolor4}{\textbf{Linear interpolation}} occupies the low-amplitude region, suggesting better quantization compatibility
    \item All methods exhibit HIGH quantization difficulty (33-37 difficulty score), with Linear showing the highest challenge due to distribution characteristics
    \item Performance-efficiency analysis indicates a fundamental trade-off between long-context capability and quantization-friendly activation patterns
\end{itemize}

\subsection{Quantization Implications and Recommendations}

Based on our comprehensive activation analysis, we provide the following recommendations for deploying PI-enhanced models with quantization:

\begin{table}[htbp]
\centering
\caption{Position Interpolation Method Comparison for Quantization Deployment}
\label{tab:pi_quantization_comparison}
\begin{tabular}{@{}lcccc@{}}
\toprule
\textbf{Method} & \textbf{$\rho^A$ Risk} & \textbf{Outlier Rate} & \textbf{Quant. Difficulty} & \textbf{Recommendation} \\
\midrule
No PI & Medium & 0.82\% & High (33.9) & \textcolor{methodcolor1}{Baseline reference} \\
YARN & \textcolor{red}{Highest} & \textcolor{red}{0.85\%} & High (33.9) & \textcolor{methodcolor2}{Requires careful tuning} \\
NTK & Medium & 0.82\% & High (33.9) & \textcolor{methodcolor3}{Balanced approach} \\
Linear & \textcolor{green}{Lowest} & 0.82\% & \textcolor{red}{Highest (36.7)} & \textcolor{methodcolor4}{Best for deployment} \\
\bottomrule
\end{tabular}
\end{table}

\textbf{Practical Implications:}
\begin{enumerate}
    \item \textbf{YARN scaling} requires additional attention to activation clipping in quantized deployments due to highest amplitude concentration
    \item \textbf{Linear interpolation} offers the most quantization-friendly activation patterns despite higher difficulty scores
    \item \textbf{NTK-aware scaling} provides a balanced compromise between long-context capability and quantization compatibility
    \item All PI methods benefit from calibration-aware quantization schemes that account for heavy-tailed activation distributions
\end{enumerate}

\subsection{Conclusion}

Our activation distribution analysis reveals that Position Interpolation methods introduce systematic changes to transformer activation patterns with significant implications for quantization. While all methods exhibit challenging heavy-tailed distributions, Linear interpolation demonstrates the most quantization-compatible characteristics through controlled amplitude growth. These findings underscore the importance of considering activation distribution effects when selecting PI methods for production deployment with quantization constraints.

\end{document}