## Line Graphs: Comparison of Learned vs. True Function J(x) in 8D Space ### Overview The image contains two identical line graphs side-by-side, comparing a learned function (LPN) and a true mathematical function J(x) = -1/4||x||² in an 8-dimensional space. Both graphs plot the function along a single axis (x₁ for the left graph, x₂ for the right graph) while holding other dimensions constant at zero. The y-axis represents the function value J(x), ranging from -4 to 0. --- ### Components/Axes - **Left Graph**: - **X-axis**: Labeled "x₁" with ticks from -4 to 4. - **Y-axis**: Labeled "J(x₁, 0, ..., 0)" with ticks from -4 to 0. - **Legend**: Located at the bottom-left corner, with: - **Blue solid line**: "LPN (Learned J)" - **Red dashed line**: "True J(x) = -1/4||x||²" - **Right Graph**: - **X-axis**: Labeled "x₂" with ticks from -4 to 4. - **Y-axis**: Labeled "J(0, x₂, 0, ..., 0)" with ticks from -4 to 0. - **Legend**: Identical to the left graph, positioned at the bottom-left. --- ### Detailed Analysis 1. **True Function (Red Dashed Line)**: - Both graphs show a symmetric downward-opening parabola. - Peak at (0, 0) with J(x) = 0. - At x₁ = ±4 (left graph) or x₂ = ±4 (right graph), J(x) ≈ -4. - Intermediate points: - At x₁ = ±2 (left graph): J(x) ≈ -1. - At x₁ = ±1 (left graph): J(x) ≈ -0.25. 2. **Learned Function (Blue Solid Line)**: - Nearly identical to the true function in shape and position. - Minor deviations observed: - Slightly flatter near the peak (x₁ = 0 or x₂ = 0). - Slightly sharper curvature at x₁ = ±4 or x₂ = ±4. - At x₁ = ±2 (left graph): J(x) ≈ -1.02 (vs. true -1). - At x₁ = ±1 (left graph): J(x) ≈ -0.27 (vs. true -0.25). --- ### Key Observations - **Symmetry**: Both functions are symmetric about their respective axes (x₁ = 0 and x₂ = 0). - **Alignment**: The LPN closely approximates the true function, with deviations < 5% in magnitude. - **Consistency**: Identical behavior in both graphs despite differing axes (x₁ vs. x₂), suggesting the learned model generalizes well across dimensions. --- ### Interpretation The graphs demonstrate that the LPN (Learned J) effectively approximates the true quadratic function J(x) = -1/4||x||² in 8D space. The near-perfect alignment suggests the model successfully captures the underlying mathematical relationship, even when projected onto lower-dimensional subspaces (x₁ or x₂). The minor deviations may indicate limitations in the learning process or numerical precision, but the overall performance is robust. This implies the LPN could reliably model similar high-dimensional functions in practical applications.