## Line Graph: Estimate of tr(Hθ) vs. Number of Samples S ### Overview The graph depicts two time series trends (⟨zᵀHθz⟩ and ⟨κα⟩) plotted against a logarithmic scale of sample size (S). Both series exhibit initial volatility followed by stabilization, with convergence observed at larger sample sizes. ### Components/Axes - **X-axis**: "number of samples S" (logarithmic scale: 10⁰ to 10³) - **Y-axis**: "estimate of tr(Hθ)" (linear scale: -20 to 20) - **Legend**: - Solid black line: ⟨zᵀHθz⟩ - Dashed red line: ⟨κα⟩ - **Reference line**: Horizontal dashed gray line at y=0 ### Detailed Analysis 1. **⟨zᵀHθz⟩ (Black Line)**: - **S=10⁰**: Starts at ~15 - **S=10¹**: Drops sharply to ~-10 - **S=10²**: Fluctuates between -5 and 5 - **S=10³**: Stabilizes near 0 with minor oscillations (~±2) 2. **⟨κα⟩ (Red Dashed Line)**: - **S=10⁰**: Begins at ~5 - **S=10¹**: Plummets to ~-15 - **S=10²**: Rises to ~-2 - **S=10³**: Converges with ⟨zᵀHθz⟩ near 0 ### Key Observations - Both series show **initial divergence** (S=10⁰–10¹) followed by **gradual convergence** (S=10²–10³). - ⟨zᵀHθz⟩ exhibits **larger amplitude fluctuations** early on but recovers faster. - ⟨κα⟩ demonstrates a **deeper initial trough** but slower recovery. - Both lines **cross the zero reference line** between S=10¹ and S=10². - Final convergence at S=10³ suggests **asymptotic stability** toward tr(Hθ) = 0. ### Interpretation The graph illustrates how estimation error for two matrix traces (⟨zᵀHθz⟩ and ⟨κα⟩) evolves with increasing sample size. The logarithmic x-axis emphasizes performance across orders of magnitude in data collection. Key insights: 1. **Early Instability**: Both estimators show high variance in initial samples (S<10²), likely due to insufficient data for reliable trace estimation. 2. **Convergence Mechanism**: The shared asymptotic behavior (S≥10²) implies that both methods achieve similar accuracy with sufficient samples, despite differing initial performance. 3. **Zero-Centered Recovery**: The horizontal reference line at tr(Hθ)=0 suggests this is the true value being estimated, with both methods correcting toward it as S increases. 4. **Method Comparison**: ⟨zᵀHθz⟩ appears more robust to early-sample noise but requires more samples for final stabilization compared to ⟨κα⟩. *Note: All values are approximate due to the absence of gridlines. The convergence pattern suggests potential applications in statistical learning or signal processing where trace estimation stability is critical.*