## Chart: Eigenvalue and Effective Dimension Plots ### Overview The image presents two plots side-by-side. The left plot shows the logarithm base 10 of the eigenvalue versus the logarithm base 10 of the index for different values of the parameter 2m+1. The right plot shows the logarithm base 10 of the effective dimension versus the logarithm base 10 of n for a specific value of 2m+1. ### Components/Axes **Left Plot:** * **X-axis:** log₁₀(index), with tick marks at 0, 1, 2, 3, and 4. * **Y-axis:** log₁₀(Eigenvalue), with tick marks at 0, -5, -10, -15, and -20. * **Legend (located in the bottom-left):** * Light Blue: 2m + 1 = 10⁰.⁵ * Light Blue: 2m + 1 = 10¹.⁰ * Light Blue: 2m + 1 = 10¹.⁵ * Light Blue: 2m + 1 = 10².⁰ * Light Blue: 2m + 1 = 10².⁵ * Light Blue: 2m + 1 = 10³.⁰ * Light Blue: 2m + 1 = 10³.⁵ * Blue: 2m + 1 = 10⁴.⁰ * Orange: Theoretical UB (Upper Bound) **Right Plot:** * **X-axis:** log₁₀(n), with tick marks at 1, 2, 3, and 4. * **Y-axis:** log₁₀(Effective dimension), with tick marks at 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. * **Legend (located in the bottom-center):** * Teal: 2m + 1 = 10⁴.⁰ ### Detailed Analysis **Left Plot (Eigenvalue vs. Index):** * **Theoretical UB (Orange):** This line shows a linear downward trend. It starts at approximately -1 on the y-axis when x=0, and ends at approximately -4 on the y-axis when x=4. * **2m + 1 = 10⁰.⁵ (Light Blue):** This line starts at approximately -1 on the y-axis when x=0, and quickly drops to approximately -20 on the y-axis at x=1. * **2m + 1 = 10¹.⁰ (Light Blue):** This line starts at approximately -1 on the y-axis when x=0, and drops to approximately -20 on the y-axis at x=1.5. * **2m + 1 = 10¹.⁵ (Light Blue):** This line starts at approximately -1 on the y-axis when x=0, and drops to approximately -20 on the y-axis at x=2. * **2m + 1 = 10².⁰ (Light Blue):** This line starts at approximately -1 on the y-axis when x=0, and drops to approximately -20 on the y-axis at x=2.5. * **2m + 1 = 10².⁵ (Light Blue):** This line starts at approximately -1 on the y-axis when x=0, and drops to approximately -20 on the y-axis at x=3. * **2m + 1 = 10³.⁰ (Light Blue):** This line starts at approximately -1 on the y-axis when x=0, and drops to approximately -20 on the y-axis at x=3.5. * **2m + 1 = 10³.⁵ (Light Blue):** This line starts at approximately -1 on the y-axis when x=0, and drops to approximately -20 on the y-axis at x=4. * **2m + 1 = 10⁴.⁰ (Blue):** This line starts at approximately -1 on the y-axis when x=0, and drops to approximately -20 on the y-axis at x=4. **Right Plot (Effective Dimension vs. n):** * **2m + 1 = 10⁴.⁰ (Teal):** This line shows an upward trend. * At x=1, y ≈ 0.45 * At x=2, y ≈ 0.75 * At x=3, y ≈ 0.90 * At x=4, y ≈ 0.98 ### Key Observations * In the left plot, as the value of 2m+1 increases, the point at which the eigenvalue drops significantly shifts to the right. * The "Theoretical UB" line in the left plot provides an upper bound for the eigenvalues. * In the right plot, the effective dimension increases with n, but the rate of increase slows down as n increases. ### Interpretation The plots illustrate the relationship between eigenvalues, index, effective dimension, and the parameter 2m+1. The left plot shows how the eigenvalues decay as the index increases, and how this decay is affected by the value of 2m+1. The right plot shows how the effective dimension increases with n for a specific value of 2m+1. The data suggests that increasing 2m+1 delays the decay of eigenvalues, and that the effective dimension approaches a limit as n increases. The theoretical upper bound provides a benchmark for the eigenvalue decay.

\n ## Charts: Eigenvalue Spectrum and Effective Dimension ### Overview The image presents two charts. The left chart displays the log10 of the Eigenvalue versus the log10 of the index for different values of `2m+1`. The right chart shows the log10 of the Effective Dimension versus the log10 of n, for a single value of `2m+1`. ### Components/Axes **Left Chart:** * **X-axis:** `log10(Index)` ranging from approximately 0 to 4. * **Y-axis:** `log10(Eigenvalue)` ranging from approximately 0 to -20. * **Data Series:** Multiple lines representing different values of `2m+1`: `10^5`, `10^4`, `10^3`, `10^2`, `10^1`, `10^0`. * **Additional Line:** A line labeled "Theoretical UB" (Upper Bound). * **Legend:** Located in the top-left corner, listing the values of `2m+1` and "Theoretical UB" with corresponding colors. **Right Chart:** * **X-axis:** `log10(n)` ranging from approximately 0 to 4. * **Y-axis:** `log10(Effective dimension)` ranging from approximately 0.5 to 1.0. * **Data Series:** A single line with markers representing data points for `2m+1 = 10^6`. * **Legend:** Located in the bottom-right corner, indicating `2m+1 = 10^6` with a teal color. ### Detailed Analysis or Content Details **Left Chart:** * **`2m+1 = 10^5` (Dark Blue):** The line starts at approximately `log10(Eigenvalue) = 0` at `log10(Index) = 0` and decreases steadily to approximately `log10(Eigenvalue) = -18` at `log10(Index) = 4`. * **`2m+1 = 10^4` (Blue):** The line starts at approximately `log10(Eigenvalue) = 0` at `log10(Index) = 0` and decreases steadily to approximately `log10(Eigenvalue) = -16` at `log10(Index) = 4`. * **`2m+1 = 10^3` (Light Blue):** The line starts at approximately `log10(Eigenvalue) = 0` at `log10(Index) = 0` and decreases steadily to approximately `log10(Eigenvalue) = -14` at `log10(Index) = 4`. * **`2m+1 = 10^2` (Medium Blue):** The line starts at approximately `log10(Eigenvalue) = 0` at `log10(Index) = 0` and decreases steadily to approximately `log10(Eigenvalue) = -12` at `log10(Index) = 4`. * **`2m+1 = 10^1` (Blue):** The line starts at approximately `log10(Eigenvalue) = 0` at `log10(Index) = 0` and decreases steadily to approximately `log10(Eigenvalue) = -10` at `log10(Index) = 4`. * **`2m+1 = 10^0` (Orange):** The line starts at approximately `log10(Eigenvalue) = 0` at `log10(Index) = 0` and decreases steadily to approximately `log10(Eigenvalue) = -8` at `log10(Index) = 4`. * **Theoretical UB (Gray):** The line starts at approximately `log10(Eigenvalue) = 0` at `log10(Index) = 0` and decreases steadily to approximately `log10(Eigenvalue) = -6` at `log10(Index) = 4`. **Right Chart:** * **`2m+1 = 10^6` (Teal):** The line starts at approximately `log10(Effective dimension) = 0.5` at `log10(n) = 0` and increases to approximately `log10(Effective dimension) = 0.98` at `log10(n) = 4`. The line exhibits a curved, increasing trend. Specific data points (approximate): * `log10(n) = 1`: `log10(Effective dimension) = 0.65` * `log10(n) = 2`: `log10(Effective dimension) = 0.78` * `log10(n) = 3`: `log10(Effective dimension) = 0.90` ### Key Observations * In the left chart, as `2m+1` increases, the rate of decrease in `log10(Eigenvalue)` slows down. * The "Theoretical UB" line consistently lies above all the other lines in the left chart. * In the right chart, the effective dimension increases with `n`, approaching 1 as `n` increases. ### Interpretation The left chart illustrates the eigenvalue spectrum for different values of `2m+1`. The decreasing trend of the eigenvalues suggests a diminishing contribution from higher-order components. The "Theoretical UB" provides an upper bound on the eigenvalue distribution. The fact that the rate of decrease slows down with increasing `2m+1` suggests that larger values of `2m+1` lead to a more persistent contribution from higher-order components. The right chart shows how the effective dimension grows with `n`. The effective dimension represents the number of independent components needed to represent the data. As `n` increases, the effective dimension approaches 1, indicating that the data becomes increasingly concentrated in a lower-dimensional space. This suggests that the system is becoming more predictable or less complex as `n` increases. The relationship between the two charts is that the eigenvalue spectrum (left) influences the effective dimension (right). A slower decay of eigenvalues implies a higher effective dimension for a given `n`. The charts together provide insight into the dimensionality and complexity of the system being analyzed.

## [Dual Plot Analysis]: Eigenvalue Decay and Effective Dimension Scaling ### Overview The image contains two separate but related scientific plots presented side-by-side. Both plots use logarithmic scales on both axes (log-log plots). The left plot analyzes the decay of eigenvalues across an index, while the right plot examines the scaling of an "Effective dimension" with respect to a parameter `n`. The plots appear to be from a technical paper, likely in the fields of machine learning, statistics, or numerical analysis, investigating properties of a model or system parameterized by `m` and `n`. ### Components/Axes **Left Plot:** * **Chart Type:** Line plot with multiple data series. * **X-axis:** Label: `log10(index)`. Scale: Linear from 0 to 4. * **Y-axis:** Label: `log10(Eigenvalue)`. Scale: Linear from approximately -22 to 2. * **Legend:** Positioned in the top-left corner. Contains 8 entries: 1. `2m + 1 = 10^0.5` (Lightest blue, almost white) 2. `2m + 1 = 10^1.0` 3. `2m + 1 = 10^1.5` 4. `2m + 1 = 10^2.0` 5. `2m + 1 = 10^2.5` 6. `2m + 1 = 10^3.0` 7. `2m + 1 = 10^3.5` 8. `2m + 1 = 10^4.0` (Darkest blue) 9. `Theoretical UB` (Orange line) * **Data Series:** 8 blue lines of varying shades (darker for higher `2m+1` values) and one orange line. **Right Plot:** * **Chart Type:** Line plot with a single data series. * **X-axis:** Label: `log10(n)`. Scale: Linear from approximately 0.5 to 4. * **Y-axis:** Label: `log10(Effective dimension)`. Scale: Linear from 0.45 to 1.0. * **Legend:** Positioned in the bottom-right corner. Contains 1 entry: 1. `2m + 1 = 10^4.0` (Teal/green diamond marker) * **Data Series:** A single teal/green line with diamond markers. ### Detailed Analysis **Left Plot - Eigenvalue Spectrum:** * **Trend Verification:** All blue data series show a general downward trend as `log10(index)` increases. The orange "Theoretical UB" (Upper Bound) line is a straight line with a constant negative slope, serving as an envelope. * **Data Series Behavior:** Each blue line follows the theoretical bound closely for low indices, then exhibits a sharp, near-vertical drop-off at a specific index. The index of this drop increases with the value of `2m+1`. * For `2m+1 = 10^0.5`, the drop occurs near `log10(index) ≈ 0.7` (index ≈ 5). * For `2m+1 = 10^4.0`, the drop occurs near `log10(index) ≈ 3.7` (index ≈ 5000). * **Approximate Values:** The eigenvalues start near `10^1` (log value ~1) for low indices. The theoretical bound line passes through approximately (0, 1) and (4, -7.5). The sharp drops plummet to values below `10^-20` (log value < -20). **Right Plot - Effective Dimension Scaling:** * **Trend Verification:** The single data series shows a clear upward, concave-down trend. The rate of increase slows as `log10(n)` increases. * **Data Points (Approximate):** * At `log10(n) ≈ 0.5` (n ≈ 3.16), `log10(Effective dimension) ≈ 0.46` (Effective dimension ≈ 2.88). * At `log10(n) ≈ 1.0` (n = 10), `log10(Effective dimension) ≈ 0.67` (Effective dimension ≈ 4.68). * At `log10(n) ≈ 2.0` (n = 100), `log10(Effective dimension) ≈ 0.85` (Effective dimension ≈ 7.08). * At `log10(n) ≈ 3.0` (n = 1000), `log10(Effective dimension) ≈ 0.94` (Effective dimension ≈ 8.71). * At `log10(n) ≈ 4.0` (n = 10000), `log10(Effective dimension) ≈ 0.99` (Effective dimension ≈ 9.77). * The curve appears to be approaching an asymptote near `log10(Effective dimension) = 1.0` (Effective dimension = 10). ### Key Observations 1. **Eigenvalue Decay Regime:** The system exhibits a clear "phase transition." For a given `2m+1`, eigenvalues decay according to a power law (linear on log-log plot) until a critical index, after which they vanish almost completely. This critical index scales with `2m+1`. 2. **Theoretical Bound:** The orange "Theoretical UB" line accurately predicts the pre-cutoff decay rate for all series, regardless of `2m+1`. 3. **Effective Dimension Saturation:** The effective dimension for the case `2m+1 = 10^4.0` grows with `n` but shows diminishing returns, suggesting a capacity limit or saturation point is being approached as `n` becomes large. 4. **Parameter Relationship:** The right plot uses the highest value of `2m+1` from the left plot (`10^4.0`), suggesting the two plots are analyzing different aspects of the same high-complexity model configuration. ### Interpretation These plots together tell a story about the **capacity and complexity** of a parameterized system (likely a neural network or kernel method). * **Left Plot (Eigenvalue Spectrum):** This reveals the **effective rank** or **information content** of the system's representation. The sharp cutoff indicates that only a finite number of components (eigenvectors) are significant for a given model size (`2m+1`). Larger models (higher `2m+1`) can sustain meaningful components across a wider spectrum (higher indices), implying they can capture more complex patterns or features. The theoretical bound suggests this decay is a fundamental property, not an artifact. * **Right Plot (Effective Dimension):** This quantifies the **functional complexity** or **degrees of freedom** of the model as the dataset size (`n`) grows. The sub-linear growth and saturation indicate that while adding more data increases the model's effective complexity, it does so at a decreasing rate. The model's capacity, set by `2m+1 = 10^4.0`, ultimately limits how complex its learned function can become, regardless of how much data (`n`) is provided. **In essence:** The left plot shows the *potential* complexity available in the model's parameter space (which grows with `2m+1`), while the right plot shows how much of that potential is *actually utilized* as a function of data availability (`n`). The system demonstrates a controlled, power-law-based complexity that is bounded both by its architecture (`2m+1`) and by the data (`n`). This is characteristic of analyses studying generalization, overfitting, or the "double descent" phenomenon in machine learning.

## Line Graphs: Eigenvalue Decay and Effective Dimension Trends ### Overview The image contains two side-by-side graphs. The left graph is a log-log plot showing eigenvalue decay across indices for different system sizes, with a theoretical upper bound (UB). The right graph is a line plot showing the relationship between effective dimension and system size (n). ### Components/Axes **Left Graph (Eigenvalue Decay):** - **X-axis**: `log₁₀(index)` ranging from 0 to 4 - **Y-axis**: `log₁₀(eigenvalue)` ranging from -20 to 0 - **Legend**: Located at bottom-left, with entries: - Blue dots: `2m + 1 = 10⁰.⁵, 10¹.⁰, 10¹.⁵, 10².⁰, 10².⁵, 10³.⁰, 10³.⁵, 10⁴.⁰` - Red line: `Theoretical UB` - **Data Points**: Blue dots with vertical error bars, connected by dashed lines to the x-axis. **Right Graph (Effective Dimension):** - **X-axis**: `log₁₀(n)` ranging from 1 to 4 - **Y-axis**: `log₁₀(effective dimension)` ranging from 0.5 to 1.0 - **Legend**: Located at bottom-right, with entry: - Green diamonds: `2m + 1 = 10⁴.⁰` - **Data Points**: Green diamonds connected by a solid line. ### Detailed Analysis **Left Graph Trends:** - All blue data series show a **downward slope** in log-log space, indicating exponential decay of eigenvalues with increasing index. - The `2m + 1 = 10⁴.⁰` series (darkest blue) has the slowest decay, with eigenvalues reaching ~-15 at index 4. - The `2m + 1 = 10⁰.⁵` series (lightest blue) decays most rapidly, reaching ~-20 at index 4. - The **Theoretical UB** (red line) is a straight diagonal line with a slope of -1, serving as an asymptotic boundary for all series. **Right Graph Trends:** - The green line shows a **steady upward trend** in log-log space, indicating sublinear growth of effective dimension with system size. - At `log₁₀(n) = 1` (n=10), effective dimension ≈ 0.5. - At `log₁₀(n) = 4` (n=10,000), effective dimension approaches 1.0. - The curve suggests a **saturation effect**, where effective dimension plateaus near 1 as n increases. ### Key Observations 1. **Eigenvalue Decay**: Larger system sizes (`2m + 1`) exhibit slower eigenvalue decay, consistent with the theoretical UB. 2. **Effective Dimension**: The logarithmic relationship implies the system's dimensionality grows sublinearly with size, approaching a fixed value. 3. **Convergence**: The right graph's saturation at effective dimension ≈1.0 may indicate a fundamental limit in the modeled system. ### Interpretation The left graph demonstrates that eigenvalues decay exponentially with index, with larger systems (`2m + 1`) approaching the theoretical UB more closely. This suggests the UB represents an optimal or maximum decay rate for the system. The right graph reveals that effective dimension grows logarithmically with system size but saturates near 1.0, implying the system's complexity or representational capacity stabilizes despite increasing size. These trends could reflect properties of matrix factorizations, neural network representations, or other high-dimensional systems where dimensionality and eigenvalue spectra are critical metrics.