## Line Graphs: Cross Sections of Prior Functions in 32-Dimensional Space ### Overview The image contains two side-by-side line graphs comparing "LPN" (blue) and "Ref" (orange) prior functions across two cross-sectional views of a 32-dimensional space. Both graphs share identical axes ranges (-4 to 4 on the x-axis) but differ in their y-axis labels, reflecting different dimensional perspectives. ### Components/Axes - **Left Graph**: - Title: "Cross sections (x₁,0) of the prior function, Dim 32" - X-axis: Labeled "x₁" with ticks at -4, -3, -2, -1, 0, 1, 2, 3, 4 - Y-axis: Labeled "Prior functions (x₁,0)" - Legend: Top-right corner, blue = "LPN", orange = "Ref" - **Right Graph**: - Title: "Cross sections (0, x₂, 0) of the prior function, Dim 32" - X-axis: Labeled "x₁" with identical tick marks as the left graph - Y-axis: Labeled "Prior functions (0, x₂, 0)" - Legend: Top-right corner, same color coding as left graph ### Detailed Analysis #### Left Graph (x₁,0 Cross-Section) - **LPN (Blue)**: - Starts at ~14.5 at x₁ = -4 - Dips to ~0.5 at x₁ = 0 - Rises to ~10.5 at x₁ = 4 - **Ref (Orange)**: - Starts at ~9 at x₁ = -4 - Dips to ~0.5 at x₁ = 0 - Rises to ~4.5 at x₁ = 4 #### Right Graph (0, x₂, 0 Cross-Section) - **LPN (Blue)**: - Starts at ~14 at x₁ = -4 - Dips to ~0.5 at x₁ = 0 - Rises sharply to ~13.5 at x₁ = 4 - **Ref (Orange)**: - Starts at ~9 at x₁ = -4 - Dips to ~0.5 at x₁ = 0 - Rises to ~8 at x₁ = 4 ### Key Observations 1. **Symmetry**: Both graphs exhibit U-shaped curves, suggesting quadratic-like behavior in the prior functions. 2. **Divergence at Extremes**: LPN values are consistently higher than Ref values at x₁ = ±4 in both graphs. 3. **Convergence at Origin**: Both LPN and Ref lines intersect at their minimum point (~0.5) at x₁ = 0. 4. **Asymmetry in Right Graph**: The LPN line in the right graph rises more sharply than in the left graph, while the Ref line shows a more gradual increase. ### Interpretation The graphs compare two prior functions (LPN and Ref) across two orthogonal cross-sections of a high-dimensional space. The LPN function demonstrates: - **Higher sensitivity** to input values at the extremes (x₁ = ±4), with values exceeding Ref by ~50% in the right graph. - **Stronger curvature** in the right graph, suggesting dimensional dependencies in the prior function's behavior. - A **shared minimum** at x₁ = 0, indicating a common optimal point or constraint in both functions. The Ref function appears to represent a baseline or reference model, while LPN may incorporate additional constraints or regularization terms that amplify its response at the boundaries of the input space. The dimensional perspective (x₁,0 vs. 0,x₂,0) reveals how the prior functions behave differently when varying specific dimensions in isolation.