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## Scatter Plot: Projection of Activations on t_G and t_P
### Overview
The image displays two side-by-side scatter plots under the main title "Projection of activations on t_G and t_P". The left plot is titled "Affirmative Statements" and the right plot is titled "Negated Statements". Each plot visualizes the relationship between two projected activation values, with data points colored according to a binary "True" or "False" label.
### Components/Axes
* **Main Title:** "Projection of activations on t_G and t_P"
* **Subplot Titles:**
* Left: "Affirmative Statements"
* Right: "Negated Statements"
* **X-Axis (Both Plots):** Label is `a_ij^T t_G`. The scale runs from -40 to 40, with major tick marks at intervals of 10 (-40, -30, -20, -10, 0, 10, 20, 30, 40).
* **Y-Axis (Both Plots):** Label is `a_ij^T t_P`. The scale runs from -50 to 20, with major tick marks at intervals of 10 (-50, -40, -30, -20, -10, 0, 10, 20).
* **Legend:** Located in the bottom-left corner of the "Negated Statements" plot.
* Red dot: "False"
* Blue dot: "True"
### Detailed Analysis
**1. Affirmative Statements (Left Plot):**
* **Data Series - "False" (Red):** The red points form a dense, elongated cluster that trends diagonally from the bottom-left quadrant (approximately x=-40, y=-50) to the center (approximately x=0, y=0). The trend is strongly positive and linear.
* **Data Series - "True" (Blue):** The blue points form a separate, dense cluster located almost exclusively in the top-right quadrant. This cluster is centered roughly around x=25, y=15. The points show a slight positive slope within their cluster but are distinctly separated from the red cluster.
* **Spatial Relationship:** There is a clear, wide separation between the two clusters along the diagonal. The "True" (blue) cluster is positioned entirely above and to the right of the "False" (red) cluster.
**2. Negated Statements (Right Plot):**
* **Data Series - "False" (Red):** The red points are widely scattered across the left and central portions of the plot, primarily in the upper half (y > -10). There is no strong linear trend; the distribution appears somewhat amorphous, centered roughly around x=-20, y=0.
* **Data Series - "True" (Blue):** The blue points form a distinct, vertically oriented cluster on the right side of the plot. This cluster spans a wide range of y-values (from approximately y=-50 to y=10) but is confined to a narrow band of x-values (approximately x=5 to x=25). The cluster has a slight negative slope.
* **Spatial Relationship:** The two clusters overlap significantly in the central region of the plot (around x=-10 to x=10). The "True" (blue) cluster is positioned to the right of the main mass of "False" (red) points.
### Key Observations
1. **Dichotomy in Affirmative Context:** For affirmative statements, the model's activations (as projected onto t_G and t_P) for "True" and "False" labels are linearly separable into two distinct, non-overlapping regions.
2. **Convergence in Negated Context:** For negated statements, the activations for "True" and "False" labels are not linearly separable. They show significant overlap, with the "True" cluster forming a vertical band that intersects the more scattered "False" distribution.
3. **Cluster Shape Difference:** The "True" cluster shape changes dramatically between plots: a tight, diagonal blob for affirmative statements versus a tall, narrow vertical band for negated statements.
4. **Axis Range Utilization:** Both plots use a similar range on the x-axis (-40 to 40). The y-axis range is also similar, but the data in the "Affirmative" plot utilizes the lower negative values more extensively for the "False" series.
### Interpretation
This visualization likely comes from an analysis of a neural network's internal representations, specifically examining how it processes affirmative versus negated statements. The projections `a_ij^T t_G` and `a_ij^T t_P` represent activations projected onto two specific directions (vectors t_G and t_P) in the model's activation space.
* **Affirmative Statements:** The clear separation suggests that for simple affirmative claims, the model's internal state (as captured by these projections) cleanly distinguishes between true and false propositions. The positive linear correlation for "False" statements indicates that as one projected component (`a_ij^T t_G`) increases, the other (`a_ij^T t_P`) also increases proportionally for false claims.
* **Negated Statements:** The overlapping clusters indicate that negation introduces complexity or ambiguity into the model's internal representation. The model does not map "True" and "Negated False" (or similar) to distinctly separable regions in this projected space. The vertical "True" cluster suggests that for negated statements labeled true, the `a_ij^T t_G` value is relatively consistent (positive), while the `a_ij^T t_P` value varies widely. This could imply that handling negation relies on a different or more distributed mechanism within the model compared to processing affirmative statements.
**In summary, the data suggests a fundamental difference in how the model represents truth value for affirmative versus negated linguistic constructs. Affirmative truth appears to be encoded in a simple, separable manner within this subspace, while negated truth results in a more entangled and complex representation.**
