## Line Graphs: Prior J(x1, 0, ...) Dim 8 and Prior J(0, x2, 0, ...) Dim 8 ### Overview Two side-by-side line graphs compare the performance of an LPN 2 approximation (solid blue line) against the true quadratic prior function $ J(x) = -\frac{1}{4}||x||_2^2 $ (dashed orange line). Both graphs share identical axes ranges (-4 to 4) and y-axis scales (-4 to 0), with mirrored x-axis labels ($ x_1 $ for the left graph, $ x_2 $ for the right). ### Components/Axes - **Left Graph**: - **X-axis**: $ x_1 $ (labeled "x₁"), range: -4 to 4. - **Y-axis**: $ J(x_1, 0, \dots) $ (labeled "J(x₁, 0, ...)", range: -4 to 0). - **Legend**: Bottom-left corner, labels: - Solid blue: "LPN 2" - Dashed orange: "True $ J(x) = -\frac{1}{4}||x||_2^2 $". - **Right Graph**: - **X-axis**: $ x_2 $ (labeled "x₂"), range: -4 to 4. - **Y-axis**: $ J(0, x_2, 0, \dots) $ (labeled "J(0, x₂, 0, ...)", range: -4 to 0). - **Legend**: Identical to the left graph, positioned bottom-left. ### Detailed Analysis - **Left Graph Trends**: - The dashed orange line (True $ J(x) $) forms a symmetric downward-opening parabola, peaking at $ x_1 = 0 $ with $ J(0) = 0 $. - The solid blue line (LPN 2) closely follows the dashed line but dips slightly below it near the peak, with a maximum value of approximately -0.05 at $ x_1 = 0 $. - Both lines converge at $ x_1 = \pm4 $, where $ J(x_1) \approx -4 $. - **Right Graph Trends**: - Identical to the left graph, with the dashed orange line peaking at $ x_2 = 0 $ and the solid blue line mirroring its shape. - The LPN 2 approximation again underestimates the true function by ~0.05 at the peak. ### Key Observations 1. **Symmetry**: Both graphs exhibit perfect symmetry about their respective x-axis origins ($ x_1 = 0 $, $ x_2 = 0 $). 2. **Approximation Accuracy**: The LPN 2 model (solid blue) closely matches the true quadratic prior (dashed orange) across the entire domain, with minor deviations only at the peak. 3. **Consistency**: The behavior of LPN 2 is identical in both graphs, suggesting the approximation is dimensionally consistent for $ x_1 $ and $ x_2 $. ### Interpretation The graphs demonstrate that the LPN 2 approximation effectively captures the quadratic prior $ J(x) = -\frac{1}{4}||x||_2^2 $ in two dimensions. The slight underestimation at the peak ($ x_1 = 0 $ or $ x_2 = 0 $) may indicate a limitation in the LPN 2 model’s ability to represent the exact curvature of the true function. However, the near-perfect alignment elsewhere suggests the approximation is robust for practical purposes, particularly in regions away from the origin. The symmetry and consistency across dimensions imply the prior is isotropic in $ x_1 $ and $ x_2 $ when other variables are fixed at zero.