## Chart: Cross Sections of J ### Overview The image contains two line charts displaying cross-sections of a function J in 8 dimensions. The left chart shows the cross-section with respect to x1, while the right chart shows the cross-section with respect to x2. Each chart plots two functions: "LPN 2" and "Ref J(x) = -1/4||x||_2". Both charts exhibit a parabolic shape, indicating a quadratic relationship. ### Components/Axes **Left Chart:** * **Title:** Cross sections of J(x1, 0, ...) Dim 8 * **X-axis:** x1, ranging from -4 to 4 in increments of 1. * **Y-axis:** Priorfunctions(x1, 0, ...), ranging from 0 to 10 in increments of 2. * **Legend:** Located in the top-right corner. * LPN 2 (solid blue line) * Ref J(x) = -1/4||x||_2 (dashed brown line) **Right Chart:** * **Title:** Cross sections of J(0, x2, 0, ...) Dim 8 * **X-axis:** x2, ranging from -4 to 4 in increments of 1. * **Y-axis:** Priorfunctions(0, x2, 0, ...), ranging from 0 to 10 in increments of 2. * **Legend:** Located in the top-right corner. * LPN 2 (solid blue line) * Ref J(x) = -1/4||x||_2 (dashed brown line) ### Detailed Analysis **Left Chart (x1):** * **LPN 2 (blue line):** The line forms a parabola. * At x1 = -4, Priorfunctions(x1, 0, ...) ≈ 10.5 * At x1 = 0, Priorfunctions(x1, 0, ...) ≈ 0.1 * At x1 = 4, Priorfunctions(x1, 0, ...) ≈ 5 * **Ref J(x) = -1/4||x||_2 (dashed brown line):** The line forms a parabola. * At x1 = -4, Priorfunctions(x1, 0, ...) ≈ 9.8 * At x1 = 0, Priorfunctions(x1, 0, ...) ≈ 0.1 * At x1 = 4, Priorfunctions(x1, 0, ...) ≈ 4.1 **Right Chart (x2):** * **LPN 2 (blue line):** The line forms a parabola. * At x2 = -4, Priorfunctions(0, x2, 0, ...) ≈ 9.5 * At x2 = 0, Priorfunctions(0, x2, 0, ...) ≈ 0.1 * At x2 = 4, Priorfunctions(0, x2, 0, ...) ≈ 8 * **Ref J(x) = -1/4||x||_2 (dashed brown line):** The line forms a parabola. * At x2 = -4, Priorfunctions(0, x2, 0, ...) ≈ 8.5 * At x2 = 0, Priorfunctions(0, x2, 0, ...) ≈ 0.1 * At x2 = 4, Priorfunctions(0, x2, 0, ...) ≈ 7.5 ### Key Observations * Both charts show similar parabolic trends for LPN 2 and Ref J(x). * The minimum value for both functions in both charts occurs near x1 = 0 and x2 = 0, respectively. * The LPN 2 function generally has slightly higher values than the Ref J(x) function, especially at the extremes of x1 and x2. ### Interpretation The charts compare the cross-sections of the LPN 2 function with a reference function Ref J(x). The similarity in shape and values suggests that LPN 2 approximates the reference function, particularly in the region around x1 = 0 and x2 = 0. The slight differences at the extremes indicate that LPN 2 might deviate from the reference function as the input values increase in magnitude. The plots demonstrate the behavior of these functions in a high-dimensional space by visualizing their cross-sections along specific axes.

## Chart: Cross Sections of Function J ### Overview The image presents two charts, side-by-side, visualizing cross-sections of a function J with dimensionality 8. Both charts depict the relationship between a variable (x1 in the left chart, x2 in the right chart) and the "Prior functions(x1, 0, ...)" or "Prior functions(0, x2, 0, ...)" respectively. Two curves are plotted on each chart: one representing "LPN 2" and the other representing "Ref f(x) = -1/4 ||x||₂". ### Components/Axes Each chart shares the following components: * **Title:** "Cross sections of J(x1, 0, ... ) Dim 8" (left chart) and "Cross sections of J(0, x2, 0, ... ) Dim 8" (right chart). * **X-axis:** Labeled "x1" (left chart) and "x2" (right chart), ranging from approximately -4 to 4. * **Y-axis:** Labeled "Prior functions(x1, 0, ... )" (left chart) and "Prior functions(0, x2, 0, ... )" (right chart), ranging from 0 to 10. * **Legend:** Located in the top-right corner of each chart. * "LPN 2" - represented by a solid purple line. * "Ref f(x) = -1/4 ||x||₂" - represented by a dashed orange line. ### Detailed Analysis **Left Chart (x1):** The solid purple line ("LPN 2") exhibits a U-shaped curve, reaching a minimum value around x1 = 0. The curve starts at approximately y = 9.5 at x1 = -4, decreases to a minimum of approximately y = 0.2 at x1 = 0, and then increases again to approximately y = 9.5 at x1 = 4. The dashed orange line ("Ref f(x) = -1/4 ||x||₂") also forms a U-shaped curve, but is more symmetrical and has a shallower slope. It starts at approximately y = 10 at x1 = -4, decreases to a minimum of approximately y = 0 at x1 = 0, and increases to approximately y = 10 at x1 = 4. **Right Chart (x2):** The solid purple line ("LPN 2") mirrors the shape of the left chart's "LPN 2" curve. It starts at approximately y = 9.5 at x2 = -4, decreases to a minimum of approximately y = 0.2 at x2 = 0, and increases to approximately y = 9.5 at x2 = 4. The dashed orange line ("Ref f(x) = -1/4 ||x||₂") also mirrors the shape of the left chart's "Ref f(x)" curve. It starts at approximately y = 10 at x2 = -4, decreases to a minimum of approximately y = 0 at x2 = 0, and increases to approximately y = 10 at x2 = 4. ### Key Observations * Both charts show that both "LPN 2" and "Ref f(x)" have a minimum value at x = 0. * The "LPN 2" curve is significantly higher than the "Ref f(x)" curve for values of x away from 0. * The "Ref f(x)" curve is more symmetrical around x = 0 than the "LPN 2" curve. * The shapes of the curves are very similar in both charts, suggesting a consistent relationship between the variable (x1 or x2) and the prior function. ### Interpretation The charts demonstrate the cross-sectional behavior of a function J in two dimensions (varying x1 and x2 while holding other dimensions constant at 0). The "Ref f(x) = -1/4 ||x||₂" function serves as a reference, likely representing a standard or baseline function. The "LPN 2" function represents a different approach or model. The U-shaped curves indicate that both functions have a minimum value at the origin (x=0). However, the "LPN 2" function exhibits larger values for non-zero x, suggesting a stronger prior or penalty for deviations from the origin. The difference in shape between the two curves suggests that "LPN 2" may be a more complex or regularized function compared to the reference function. The fact that the patterns are consistent across both charts (x1 and x2) suggests that the behavior of the function J is similar along these cross-sections. This could indicate that the function is relatively well-behaved or that the chosen dimensions (x1 and x2) are particularly important.

\n ## [Chart Type]: Dual Cross-Section Line Plots ### Overview The image displays two side-by-side line charts, each showing a cross-sectional view of a function `J` in an 8-dimensional space. The left chart plots the function against the first dimension (`x₁`), while the right chart plots it against the second dimension (`x₂`). Both charts compare a learned function ("LPN 2") against a theoretical reference function. ### Components/Axes **Titles:** * Left Chart: `Cross sections of J(x₁, 0, ...) Dim 8` * Right Chart: `Cross sections of J(0, x₂, 0, ...) Dim 8` **Axes:** * **X-Axis (Left Chart):** Label is `x₁`. Scale ranges from -4 to 4 with major tick marks at every integer (-4, -3, -2, -1, 0, 1, 2, 3, 4). * **X-Axis (Right Chart):** Label is `x₂`. Scale ranges from -4 to 4 with major tick marks at every integer (-4, -3, -2, -1, 0, 1, 2, 3, 4). * **Y-Axis (Both Charts):** Label is `Priorfunctions(x₁, 0, ...)` for the left chart and `Priorfunctions(0, x₂, 0, ...)` for the right chart. The scale ranges from 0 to 10 with major tick marks at 0, 2, 4, 6, 8, 10. **Legend (Both Charts, positioned in the top-right corner):** * **Solid Blue Line:** Label is `LPN 2`. * **Dashed Orange Line:** Label is `Ref J(x) = -1/4||x||₂²`. This denotes a reference function defined as negative one-quarter of the squared L2 norm of the input vector `x`. ### Detailed Analysis **Left Chart (Cross-section along x₁):** * **Trend Verification:** Both curves form a symmetric, upward-opening parabola with a minimum near `x₁ = 1`. * **Data Series - LPN 2 (Blue):** * At `x₁ = -4`, the value is approximately 10.2. * The curve descends to a minimum value of approximately 0.2 at `x₁ ≈ 1`. * It then ascends, reaching approximately 4.8 at `x₁ = 4`. * **Data Series - Reference (Orange Dashed):** * At `x₁ = -4`, the value is approximately 9.8. * The curve descends to a minimum value of approximately 0.0 at `x₁ = 1`. * It then ascends, reaching approximately 4.5 at `x₁ = 4`. * **Comparison:** The "LPN 2" curve closely follows the reference parabola but is consistently shifted slightly upward. The minimum of "LPN 2" is slightly above zero, while the reference function's minimum is exactly zero at `x₁=1`. **Right Chart (Cross-section along x₂):** * **Trend Verification:** Both curves form a symmetric, upward-opening parabola with a minimum near `x₂ = 1`. * **Data Series - LPN 2 (Blue):** * At `x₂ = -4`, the value is approximately 9.5. * The curve descends to a minimum value of approximately 0.5 at `x₂ ≈ 1`. * It then ascends, reaching approximately 8.0 at `x₂ = 4`. * **Data Series - Reference (Orange Dashed):** * At `x₂ = -4`, the value is approximately 8.5. * The curve descends to a minimum value of approximately 0.4 at `x₂ ≈ 1`. * It then ascends, reaching approximately 7.2 at `x₂ = 4`. * **Comparison:** Similar to the left chart, "LPN 2" tracks the reference function but sits slightly above it. The vertical offset appears more pronounced at the extremes (`x₂ = ±4`) compared to the left chart. ### Key Observations 1. **Parabolic Shape:** Both cross-sections reveal that the function `J` is quadratic (parabolic) along the `x₁` and `x₂` dimensions when other dimensions are held at zero. 2. **Minimum Location:** The minimum of the function occurs at `x₁ = 1` and `x₂ = 1` for both the learned and reference functions. 3. **Model Fidelity:** The "LPN 2" model successfully captures the overall parabolic shape and location of the minimum of the reference function `J(x) = -1/4||x||₂²`. 4. **Systematic Offset:** There is a consistent, small positive offset between the "LPN 2" curve and the reference curve across both plots. This offset is not uniform; it appears slightly larger at the boundaries of the plotted range (`x = ±4`) than near the minimum. 5. **Scale Difference:** The function values at the boundaries differ between the two plots. For example, at `x = -4`, the value is ~10.2 for `x₁` but ~9.5 for `x₂`. This indicates the function's behavior is not perfectly symmetric across all dimensions, despite the reference function being rotationally symmetric (depending only on the norm). ### Interpretation The charts demonstrate a validation or analysis of a learned model ("LPN 2") against a known theoretical prior distribution. The reference function `J(x) = -1/4||x||₂²` represents a simple, isotropic Gaussian-like prior (its negative log would be proportional to the squared norm). The fact that the cross-sections are parabolas confirms this. The "LPN 2" model appears to be a neural network or similar function approximator trained to represent this prior. The close match indicates successful learning of the core quadratic structure. The persistent positive offset suggests the model has learned a function that is everywhere slightly higher than the target. This could be due to: * A regularization effect during training. * An inherent bias in the model architecture. * The model approximating a slightly different function that includes a small constant term. The slight asymmetry between the `x₁` and `x₂` cross-sections (different values at `x=±4`) is notable. Since the reference function is perfectly symmetric, this asymmetry must originate from the "LPN 2" model itself, indicating its learned representation is not perfectly isotropic. This could be a result of the training data, optimization path, or model capacity. Overall, the visualization confirms the model has learned the essential geometric properties of the target prior, with minor, systematic deviations.

## Line Graphs: Cross Sections of J(x) in 8 Dimensions ### Overview The image contains two side-by-side line graphs comparing the behavior of two mathematical functions: **LPN 2** (solid blue line) and a reference function **J(x) = -1/4||x||²** (red dashed line). Both graphs represent cross-sections of an 8-dimensional function J, with one graph varying along the **x₁** axis and the other along the **x₂** axis. The y-axis represents "Priorfunctions" values. ### Components/Axes - **Left Graph (x₁-axis):** - **X-axis:** Labeled **x₁**, ranging from **-4 to 4** in increments of 1. - **Y-axis:** Labeled **Priorfunctions(x₁, 0, ..., )**, with values from **0 to 10**. - **Legend:** Top-right corner, with: - **Solid blue line:** "LPN 2" - **Red dashed line:** "Ref J(x) = -1/4||x||²" - **Right Graph (x₂-axis):** - **X-axis:** Labeled **x₂**, ranging from **-4 to 4** in increments of 1. - **Y-axis:** Labeled **Priorfunctions(0, x₂, 0, ..., )**, with values from **0 to 10**. - **Legend:** Top-right corner, identical to the left graph. ### Detailed Analysis #### Left Graph (x₁-axis): - **Trend Verification:** - Both lines form a **parabolic curve** opening upward, symmetric about **x₁ = 0**. - The **blue line (LPN 2)** and **red dashed line (Ref J(x))** overlap closely but diverge slightly near the bottom of the curve. - **Key Data Points (approximate):** - At **x₁ = -4**: Both lines ≈ **10**. - At **x₁ = -2**: Both lines ≈ **3**. - At **x₁ = 0**: Both lines ≈ **0.5** (minimum value). - At **x₁ = 2**: Both lines ≈ **3**. - At **x₁ = 4**: Both lines ≈ **10**. #### Right Graph (x₂-axis): - **Trend Verification:** - Identical parabolic shape to the left graph, symmetric about **x₂ = 0**. - The **blue line (LPN 2)** and **red dashed line (Ref J(x))** again overlap closely but show minor divergence near the minimum. - **Key Data Points (approximate):** - At **x₂ = -4**: Both lines ≈ **10**. - At **x₂ = -2**: Both lines ≈ **3**. - At **x₂ = 0**: Both lines ≈ **0.5** (minimum value). - At **x₂ = 2**: Both lines ≈ **3**. - At **x₂ = 4**: Both lines ≈ **10**. ### Key Observations 1. **Symmetry:** Both graphs exhibit perfect symmetry about their respective axes (x₁ = 0 and x₂ = 0). 2. **Function Similarity:** The LPN 2 and reference function J(x) are nearly identical, with deviations of **< 0.5 units** at all points. 3. **Minimum Values:** The minimum of both functions occurs at **x₁ = 0** and **x₂ = 0**, with a value of **~0.5**. 4. **Edge Behavior:** Both functions increase quadratically as |x₁| or |x₂| approaches ±4, reaching ~10. ### Interpretation - The graphs demonstrate that **LPN 2** closely approximates the reference function **J(x) = -1/4||x||²** in 8-dimensional space. The minor deviations suggest potential numerical or computational approximations in the LPN 2 model. - The symmetry implies that the function J(x) is **even** in its variables (i.e., J(x) = J(-x)), consistent with the quadratic form of the reference function. - The near-identical behavior of LPN 2 and J(x) indicates that LPN 2 is a robust approximation for this specific cross-section, though further analysis across other dimensions or variables would be needed to confirm generalizability. - The slight divergence near the minimum (x₁/x₂ = 0) could hint at differences in curvature or higher-order terms not captured by the LPN 2 model. ### Language Note All text in the image is in **English**. No non-English content is present.