## Line Charts: Prior J(x1, 0, ...) and Prior J(0, x2, 0, ...) ### Overview The image contains two line charts side-by-side, comparing the performance of "LPN 2" against a "True J(x)" function. Both charts depict a similar downward-sloping curve, with the x-axis representing either x1 or x2, and the y-axis representing the function J(x1, 0, ...) or J(0, x2, 0, ...). The "True J(x)" curve is slightly above the "LPN 2" curve in both charts. Both charts are titled "Prior J(...) Dim 8". ### Components/Axes **Left Chart:** * **Title:** Prior J(x1, 0,...) Dim 8 * **X-axis:** x1, ranging from -4 to 4 in increments of 1. * **Y-axis:** J(x1, 0,...), ranging from -4 to 0 in increments of 1. * **Legend (bottom-left):** * Blue line: LPN 2 * Dashed orange line: True J(x) = -1/4||x||² **Right Chart:** * **Title:** Prior J(0, x2, 0,...) Dim 8 * **X-axis:** x2, ranging from -4 to 4 in increments of 1. * **Y-axis:** J(0, x2, 0,...), ranging from -4.0 to 0.0 in increments of 0.5. * **Legend (bottom-left):** * Blue line: LPN 2 * Dashed orange line: True J(x) = -1/4||x||² ### Detailed Analysis **Left Chart (Prior J(x1, 0,...)):** * **LPN 2 (Blue):** The curve starts at approximately -4 when x1 is -4, rises to a peak of approximately 0 when x1 is 0, and then decreases back to approximately -4 when x1 is 4. * x1 = -4, J(x1, 0, ...) ≈ -4 * x1 = -2, J(x1, 0, ...) ≈ -1 * x1 = 0, J(x1, 0, ...) ≈ 0 * x1 = 2, J(x1, 0, ...) ≈ -1 * x1 = 4, J(x1, 0, ...) ≈ -4 * **True J(x) = -1/4||x||² (Dashed Orange):** The curve starts at approximately -4 when x1 is -4, rises to a peak of approximately 0 when x1 is 0, and then decreases back to approximately -4 when x1 is 4. The orange line is consistently slightly above the blue line. * x1 = -4, J(x1, 0, ...) ≈ -4 * x1 = -2, J(x1, 0, ...) ≈ -1 * x1 = 0, J(x1, 0, ...) ≈ 0 * x1 = 2, J(x1, 0, ...) ≈ -1 * x1 = 4, J(x1, 0, ...) ≈ -4 **Right Chart (Prior J(0, x2, 0,...)):** * **LPN 2 (Blue):** The curve starts at approximately -4 when x2 is -4, rises to a peak of approximately 0 when x2 is 0, and then decreases back to approximately -4 when x2 is 4. * x2 = -4, J(0, x2, 0, ...) ≈ -4 * x2 = -2, J(0, x2, 0, ...) ≈ -1 * x2 = 0, J(0, x2, 0, ...) ≈ 0 * x2 = 2, J(0, x2, 0, ...) ≈ -1 * x2 = 4, J(0, x2, 0, ...) ≈ -4 * **True J(x) = -1/4||x||² (Dashed Orange):** The curve starts at approximately -4 when x2 is -4, rises to a peak of approximately 0 when x2 is 0, and then decreases back to approximately -4 when x2 is 4. The orange line is consistently slightly above the blue line. * x2 = -4, J(0, x2, 0, ...) ≈ -4 * x2 = -2, J(0, x2, 0, ...) ≈ -1 * x2 = 0, J(0, x2, 0, ...) ≈ -0.2 * x2 = 2, J(0, x2, 0, ...) ≈ -1 * x2 = 4, J(0, x2, 0, ...) ≈ -4 ### Key Observations * Both charts show a symmetrical, downward-sloping curve, indicating a similar relationship between x1/x2 and the function J. * The "True J(x)" curve consistently outperforms "LPN 2", as it has slightly higher values across the range of x1 and x2. * The minimum value for both curves in both charts is approximately -4, occurring at x1/x2 = -4 and x1/x2 = 4. * The maximum value for both curves in both charts is approximately 0, occurring at x1/x2 = 0. ### Interpretation The charts compare the performance of "LPN 2" to a "True J(x)" function, which is defined as -1/4||x||². The charts suggest that "LPN 2" approximates the "True J(x)" function, but with a slight underestimation. The fact that both charts are titled "Prior J(...) Dim 8" suggests that these are prior distributions in an 8-dimensional space, with the charts showing the marginal distributions along the x1 and x2 axes, while holding the other dimensions at 0. The similarity between the two charts indicates that the function behaves similarly with respect to both x1 and x2. The difference between the "LPN 2" and "True J(x)" curves could be due to approximations or simplifications made in the "LPN 2" model.

\n ## Charts: Prior Distributions in 8 Dimensions ### Overview The image presents two separate charts, both depicting prior distributions in an 8-dimensional space. Each chart visualizes a function of two variables (x1, 0, ...) and (0, x2, 0, ...), respectively, comparing the approximation provided by an LPN 2 model to the "true" function. Both charts share a similar structure and scale. ### Components/Axes Both charts have the following components: * **Title:** "Prior J(x1, 0, ...)" Dim 8" (left chart) and "Prior 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 "J(x1, 0, ...)" (left chart) and "J(0, x2, 0, ...)" (right chart), ranging from approximately -4 to 0. * **Legend:** Located in the bottom-left corner of each chart. * "LPN 2" - represented by a solid blue line. * "True J(x) = -1/4||x||₂" - represented by a dashed orange line. ### Detailed Analysis or Content Details **Left Chart (x1):** * **LPN 2 (Blue Line):** The line starts at approximately -4 with a value of -3.8, increases to a peak around x1 = 0 with a value of approximately 0, and then decreases symmetrically to approximately -3.8 at x1 = 4. The curve appears parabolic. * **True J(x) (Orange Dashed Line):** The dashed line starts at approximately -4 with a value of -4, increases to a peak around x1 = 0 with a value of approximately 0, and then decreases symmetrically to approximately -4 at x1 = 4. This curve also appears parabolic, and closely follows the LPN 2 line. **Right Chart (x2):** * **LPN 2 (Blue Line):** The line starts at approximately -4 with a value of -3.5, increases to a peak around x2 = 0 with a value of approximately -0.5, and then decreases symmetrically to approximately -3.5 at x2 = 4. The curve appears parabolic. * **True J(x) (Orange Dashed Line):** The dashed line starts at approximately -4 with a value of -4, increases to a peak around x2 = 0 with a value of approximately -0.5, and then decreases symmetrically to approximately -4 at x2 = 4. This curve also appears parabolic, and closely follows the LPN 2 line. ### Key Observations * Both charts show a similar parabolic shape for both the LPN 2 approximation and the true function. * The LPN 2 approximation consistently underestimates the true function's value, particularly at the extremes of the x-axis. * The difference between the LPN 2 approximation and the true function appears to be relatively small, suggesting a good approximation. * The y-axis scales are slightly different between the two charts. ### Interpretation These charts demonstrate the behavior of a Locally Private Neural Network (LPN) 2 model when approximating a prior distribution in an 8-dimensional space. The "true" function, J(x) = -1/4||x||₂, represents the actual prior distribution. The LPN 2 model provides an approximation of this distribution. The fact that both curves are parabolic suggests that the LPN 2 model is capturing the general shape of the prior distribution. However, the consistent underestimation of the true function indicates that the LPN 2 model is introducing some bias. This bias is likely a consequence of the privacy mechanism employed by the LPN, which adds noise to the data to protect individual privacy. The slight differences in the y-axis scales between the two charts might indicate that the sensitivity of the function to changes in x1 and x2 is different. The charts provide a visual comparison of the approximation quality of the LPN 2 model, highlighting the trade-off between privacy and accuracy. The closeness of the lines suggests a reasonable balance between these two factors.

## [Line Graphs]: Prior Function Approximation Comparison ### Overview The image displays two side-by-side line graphs comparing an approximated function ("LPN 2") against a true mathematical function ("True J(x)") in an 8-dimensional space. The graphs plot the value of a function `J` against a single variable (`x₁` or `x₂`) while holding other dimensions at zero. ### Components/Axes * **Chart Type:** Two 2D line plots. * **Titles:** * Left Plot: `Prior J(x₁, 0, ...) Dim 8` * Right Plot: `Prior J(0, x₂, 0, ...) Dim 8` * **Axes:** * **X-Axis (Left Plot):** Labeled `x₁`. Scale ranges from -4 to 4 with major ticks at -4, -3, -2, -1, 0, 1, 2, 3, 4. * **X-Axis (Right Plot):** Labeled `x₂`. Scale ranges from -4 to 4 with major ticks at -4, -3, -2, -1, 0, 1, 2, 3, 4. * **Y-Axis (Left Plot):** Labeled `J(x₁, 0, ...)`. Scale ranges from -4 to 0 with major ticks at -4, -3, -2, -1, 0. * **Y-Axis (Right Plot):** Labeled `J(0, x₂, 0, ...)`. Scale ranges from -4.0 to 0.0 with major ticks at -4.0, -3.5, -3.0, -2.5, -2.0, -1.5, -1.0, -0.5, 0.0. * **Legend (Present in both plots, located at bottom-left):** * `LPN 2` (Solid blue line) * `True J(x) = -1/4||x||₂²` (Dashed orange line) * **Grid:** Both plots have a light gray grid. ### Detailed Analysis **Left Plot (`x₁`):** * **Trend Verification:** Both lines form downward-opening parabolas symmetric around `x₁ = 0`. * **Data Points & Comparison:** * **True Function (Orange Dashed):** Peaks at `(x₁=0, J=0)`. At `x₁ = ±4`, `J ≈ -4`. * **LPN 2 (Blue Solid):** Peaks at approximately `(x₁=0, J≈-0.2)`. At `x₁ = ±4`, `J ≈ -4.1`. The blue line is consistently below the orange line, with the greatest deviation at the vertex (`x₁=0`). **Right Plot (`x₂`):** * **Trend Verification:** Identical parabolic shape and relationship as the left plot. * **Data Points & Comparison:** * **True Function (Orange Dashed):** Peaks at `(x₂=0, J=0)`. At `x₂ = ±4`, `J ≈ -4`. * **LPN 2 (Blue Solid):** Peaks at approximately `(x₂=0, J≈-0.2)`. At `x₂ = ±4`, `J ≈ -4.1`. The deviation pattern is identical to the left plot. ### Key Observations 1. **Symmetry:** The function `J` is symmetric with respect to both `x₁` and `x₂` when other variables are zero. 2. **Approximation Error:** The "LPN 2" model consistently underestimates the true function value across the entire domain shown. The error is most pronounced at the maximum point (`x=0`), where the model predicts a value ~0.2 units lower than the true value of 0. 3. **Consistency:** The approximation error is visually identical for both the `x₁` and `x₂` dimensions, suggesting the model's behavior is consistent across these input dimensions. 4. **Function Shape:** The true function is a negative scaled squared L2 norm (`-1/4||x||₂²`), which is a downward-opening paraboloid in high-dimensional space. The 2D slices shown confirm this parabolic profile. ### Interpretation This visualization demonstrates the performance of a model (likely a neural network or similar approximator labeled "LPN 2") in learning a simple quadratic prior function in an 8-dimensional space. The plots show 1D cross-sections of this high-dimensional function. * **What the data suggests:** The model has successfully learned the general parabolic shape and symmetry of the target function. However, it exhibits a systematic bias, failing to reach the true maximum value at the origin. This could indicate issues with the model's capacity, training, or the specific prior being imposed. * **How elements relate:** The side-by-side comparison for `x₁` and `x₂` serves to validate that the model's approximation is consistent across different input dimensions, which is a desirable property. The legend directly links the visual representation (line style/color) to the mathematical entities being compared. * **Notable anomaly:** The consistent negative bias at the vertex is the primary anomaly. In an optimization or Bayesian inference context, where such a prior might be used, this bias could lead to systematically shifted estimates or suboptimal solutions. The perfect match at the tails (`x=±4`) suggests the model captures the curvature well away from the origin but struggles with the peak.

## 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.