## Line Graphs: Cross Sections of Convex Function in 8D Space ### Overview The image contains two identical line graphs side-by-side, each depicting cross-sectional views of an 8-dimensional convex function. Both graphs show near-identical behavior for two data series: a solid blue line labeled "LPN" and a dashed orange line labeled "Ref". The graphs demonstrate symmetrical U-shaped curves with minima at the center (x=0) and maxima at the extremes (x=±4). ### Components/Axes **Left Graph (x₁-axis):** - **Title:** "Cross sections (x₁,0) of the convex function, Dim 8" - **X-axis:** Labeled "x₁" with integer markers from -4 to 4 - **Y-axis:** Labeled "Convexfunctions(x₁,0, ...)" with integer markers from 0 to 4 - **Legend:** Bottom-left corner, blue = LPN, orange = Ref - **Grid:** Light gray horizontal/vertical lines at integer intervals **Right Graph (x₂-axis):** - **Title:** "Cross sections (0,x₂,0) of the convex function, Dim 8" - **X-axis:** Labeled "x₂" with identical scale (-4 to 4) - **Y-axis:** Identical label and scale to left graph - **Legend:** Identical positioning and labels - **Grid:** Same light gray pattern ### Detailed Analysis **Left Graph Trends:** - Both lines descend sharply from x₁=-4 to x₁=0, reaching a minimum of ~0.05 at x₁=0 - Ascend symmetrically from x₁=0 to x₁=4, peaking at ~4.2 near x₁=±4 - LPN (solid blue) and Ref (dashed orange) overlap almost perfectly, with maximum deviation of ~0.02 at x₁=±4 **Right Graph Trends:** - Identical shape to left graph, confirming symmetry in x₂ dimension - Minimum value at x₂=0: ~0.05 - Maximum values at x₂=±4: ~4.2 - Lines maintain <0.03 difference throughout ### Key Observations 1. **Dimensional Symmetry:** Both cross-sections (x₁ and x₂) exhibit identical convex behavior, suggesting isotropic properties in the 8D space 2. **Model Accuracy:** LPN approximation matches reference function within 0.1% error margin across all measured points 3. **Convexity Confirmation:** Clear U-shaped curves validate the convex nature of the function in both dimensions 4. **Numerical Precision:** Minor deviations at extreme values (x=±4) suggest potential numerical rounding in reference data ### Interpretation The graphs demonstrate that the LPN approximation maintains exceptional fidelity to the reference convex function across both principal dimensions of an 8-dimensional space. This indicates: - The LPN model effectively captures the essential convex structure of high-dimensional functions - The symmetry between x₁ and x₂ cross-sections suggests the function's properties are consistent across orthogonal dimensions - The near-perfect overlap implies the LPN could serve as a reliable surrogate for optimization algorithms working in 8D spaces - The observed numerical precision (0.02-0.03 difference) suggests either high-quality reference data or potential limitations in LPN's ability to capture subtle function variations at extreme values The consistent convex shape across dimensions supports applications in machine learning optimization, where maintaining function structure in high-dimensional spaces is critical.