## Histograms: TurboQuant Performance by Bitwidth ### Overview The image contains eight histograms arranged in a 2x4 grid, comparing TurboQuant performance metrics (TurboQuant_prod and TurboQuant_mse) across four bitwidths (1-4). Each bitwidth has two subplots: one for product distortion frequency and one for mean squared error (MSE). The histograms show distributions of inner product distortion values with statistical summaries. ### Components/Axes - **X-axis**: "Inner Product Distortion" (range: -0.15 to 0.15) - **Y-axis (Top Row)**: Frequency (1e6 scale) - **Y-axis (Bottom Row)**: TurboQuant_mse values (1e7 scale) - **Legends**: Positioned in top-right corner of each subplot, showing: - Mean (μ) - Standard deviation (σ) - Maximum value - **Colors**: - Bitwidth 1: Blue - Bitwidth 2: Dark blue - Bitwidth 3: Brown - Bitwidth 4: Dark green ### Detailed Analysis #### Bitwidth 1 - **TurboQuant_prod (a)**: - Mean: -0.0002 - Std: 0.0192 - Max: 0.1056 - Distribution: Symmetric around -0.0002 with right-skewed tail - **TurboQuant_mse (b)**: - Mean: 0.0312 - Std: 0.0263 - Max: 0.3526 - Distribution: Peaks near 0.03 with gradual decline #### Bitwidth 2 - **TurboQuant_prod (c)**: - Mean: -0.0000 - Std: 0.0116 - Max: 0.0660 - Distribution: Narrower than Bitwidth 1, centered at 0 - **TurboQuant_mse (d)**: - Mean: 0.0103 - Std: 0.0112 - Max: 0.1185 - Distribution: Tighter clustering around 0.01 #### Bitwidth 3 - **TurboQuant_prod (e)**: - Mean: 0.0000 - Std: 0.0066 - Max: 0.0354 - Distribution: Very narrow peak at 0 - **TurboQuant_mse (f)**: - Mean: 0.0030 - Std: 0.0052 - Max: 0.0515 - Distribution: Sharp peak at 0.003 #### Bitwidth 4 - **TurboQuant_prod (g)**: - Mean: 0.0000 - Std: 0.0036 - Max: 0.0212 - Distribution: Extremely narrow peak at 0 - **TurboQuant_mse (h)**: - Mean: 0.0008 - Std: 0.0026 - Max: 0.0180 - Distribution: Tightest distribution, centered at 0.0008 ### Key Observations 1. **Bitwidth Correlation**: Higher bitwidths show progressively tighter distributions: - Bitwidth 1: Broadest spread (σ=0.0192 for prod) - Bitwidth 4: Narrowest spread (σ=0.0036 for prod) 2. **Mean Trends**: - TurboQuant_prod means approach 0 as bitwidth increases - TurboQuant_mse means decrease from 0.0312 (Bitwidth 1) to 0.0008 (Bitwidth 4) 3. **Y-axis Scaling**: Bottom row uses 1e7 scale to accommodate larger MSE values 4. **Symmetry**: All distributions are approximately symmetric around their means ### Interpretation The data demonstrates that increasing bitwidth improves measurement precision: - **Product Distortion**: Higher bitwidths reduce both mean error and variability (σ decreases by 80% from Bitwidth 1 to 4) - **MSE**: Shows similar improvement, with Bitwidth 4 achieving 38x lower mean MSE than Bitwidth 1 - **Precision Tradeoff**: The sharpest peaks at Bitwidth 4 suggest near-perfect alignment but limited dynamic range - **Practical Implication**: Bitwidth 3-4 achieves sub-0.01 distortion levels, suitable for high-precision applications All histograms follow a bell-curve pattern, confirming normal-like distributions despite slight skews in lower bitwidths. The consistent legend placement and color coding enable direct comparison across bitwidths.