crossword-app/backend-py/docs/softmin_visualization_ideas.md

Soft Minimum Visualization Ideas

This document outlines visualization concepts to showcase how the soft minimum method works for multi-topic word intersection in the crossword generator.

Overview

The soft minimum method uses the formula -log(sum(exp(-beta * similarities))) / beta to find words that are genuinely relevant to ALL topics simultaneously. Unlike simple averaging, which can promote words that are highly relevant to just one topic, soft minimum penalizes words that score poorly on any individual topic.

Visualizations would help users understand:

• How soft minimum differs from averaging
• Why it produces better semantic intersections
• How the beta parameter affects results
• How adaptive beta mechanism works

Visualization Concepts

1. Heat Map Comparison (🌟 Most Impactful)

Concept: Side-by-side heat maps showing individual topic similarities vs soft minimum scores.

Layout:

• Left Heat Map: Individual Similarities

  • Rows: Top 50-100 words
  • Columns: Individual topics (e.g., "universe", "movies", "languages")
  • Color intensity: Similarity score (0.0 = white, 1.0 = dark blue)

• Right Heat Map: Soft Minimum Results

  • Same rows (words)
  • Single column: Soft minimum score
  • Color intensity: Final soft minimum score

Key Insights:

• Words like "anime" would show moderate blue across all topics → high soft minimum score
• Words like "astronomy" would show dark blue for "universe", white for others → low soft minimum score
• Visually demonstrates how soft minimum penalizes topic-specific words

Implementation:

• Frontend: Use libraries like D3.js or Plotly for interactive heat maps
• Backend: Return individual topic similarities alongside soft minimum scores

2. 3D Scatter Plot (For 3-Topic Cases)

Concept: 3D space where each axis represents similarity to one topic.

Layout:

• X-axis: Similarity to topic 1
• Y-axis: Similarity to topic 2
• Z-axis: Similarity to topic 3
• Point size/color: Soft minimum score
• Point labels: Word names (on hover)

Key Insights:

• Words near the center (similar to all topics) = large, bright points
• Words near axes (similar to only one topic) = small, dim points
• Shows the "volume" of intersection vs union

Implementation:

• Use Three.js or Plotly 3D
• Interactive rotation and zoom
• Filter points by soft minimum threshold

3. Interactive Beta Slider

Concept: Real-time visualization of how beta parameter affects word selection.

Layout:

• Horizontal slider: Beta value (1.0 to 20.0)
• Bar chart: Word scores (sorted descending)
• Threshold line: Current similarity threshold
• Counter: Number of words above threshold

Key Insights:

• High beta (strict): Only a few words pass, distribution is peaked
• Low beta (permissive): More words pass, distribution flattens
• Shows adaptive beta mechanism in action

Implementation:

• React component with range slider
• Real-time recalculation of soft minimum scores
• Animated transitions as beta changes

4. Venn Diagram with Words

Concept: Position words in Venn diagram based on topic similarities.

Layout (for 2-3 topics):

• Circles represent individual topics
• Words positioned based on similarity combinations
• Words in intersections = high soft minimum scores
• Words in single circles = low soft minimum scores
• Word opacity/size based on final soft minimum score

Key Insights:

• Visual representation of "true intersections"
• Words in overlap regions are what soft minimum promotes
• Empty intersection regions explain why some topic combinations yield few words

Implementation:

• SVG-based Venn diagrams
• Dynamic positioning algorithm
• Interactive word tooltips

5. Before/After Word Clouds

Concept: Compare averaging vs soft minimum results using word clouds.

Layout:

• Left Cloud: "Averaging Method"

  • Word size based on average similarity
  • May prominently feature problematic words like "ethology" for Art+Books

• Right Cloud: "Soft Minimum Method"

  • Word size based on soft minimum score
  • Should prominently feature true intersections like "literature"

Key Insights:

• Dramatic visual difference in word prominence
• Shows quality improvement at a glance
• Easy to understand for non-technical users

Implementation:

• Use word cloud libraries (wordcloud2.js, D3-cloud)
• Color coding by topic affinity
• Interactive word selection

6. Mathematical Formula Animation

Concept: Step-by-step visualization of soft minimum calculation.

Layout:

• Example word with similarities: [0.8, 0.2, 0.1] (universe, movies, languages)
• Animated steps:
  1. Show individual similarities as bars
  2. Apply exponential transformation: exp(-beta * sim)
  3. Sum the exponentials
  4. Apply logarithm and normalization
  5. Compare result to simple average (0.37)

Key Insights:

• How the minimum similarity dominates the calculation
• Why soft minimum ≈ minimum similarity for high beta
• Mathematical intuition behind the formula

Implementation:

• Animated SVG or Canvas
• Step-by-step button progression
• Mathematical notation display

7. Adaptive Beta Journey

Concept: Show the adaptive beta retry process as a timeline.

Layout:

• Horizontal timeline showing beta decay: 10.0 → 7.0 → 4.9 → 3.4...
• For each beta value:
  • Histogram of soft minimum scores
  • Threshold line (adjusted)
  • Count of valid words
  • Decision: "Continue" or "Stop"

Key Insights:

• How threshold adjustment makes lower beta more permissive
• Why word count increases with each retry
• When the algorithm decides to stop

Implementation:

• Timeline component with expandable sections
• Small multiples showing score distributions
• Real-time data from debug logs

Implementation Priorities

Phase 1: Essential (MVP)

1. Heat Map Comparison - Most educational value
2. Interactive Beta Slider - Shows parameter effects clearly

Phase 2: Enhanced Understanding

3. Before/After Word Clouds - Easy to understand impact
4. Mathematical Formula Animation - Educational for technical users

Phase 3: Advanced Analysis

5. 3D Scatter Plot - For deep analysis of 3-topic cases
6. Venn Diagram - Complex positioning algorithms
7. Adaptive Beta Journey - Comprehensive debugging tool

Technical Implementation Notes

Backend Changes Needed

• Return individual topic similarities alongside soft minimum scores
• Add debug endpoint for visualization data
• Include beta parameter and threshold information in responses

Frontend Integration

• Add to existing debug tab
• Use React components for interactivity
• Responsive design for different screen sizes
• Export/save visualization capabilities

Data Format

{
  "visualization_data": {
    "individual_similarities": {
      "word1": [0.8, 0.2, 0.1],
      "word2": [0.3, 0.9, 0.4]
    },
    "soft_minimum_scores": {
      "word1": 0.15,
      "word2": 0.32
    },
    "beta_used": 7.0,
    "threshold_adjusted": 0.175,
    "topics": ["universe", "movies", "languages"]
  }
}

Expected Impact

These visualizations would:

1. Educate users about the soft minimum method
2. Build confidence in the algorithm's choices
3. Enable debugging of problematic topic combinations
4. Facilitate research into parameter optimization
5. Demonstrate value of the multi-topic intersection approach

The heat map comparison alone would be worth implementing, as it clearly shows why soft minimum produces higher-quality word intersections than simple averaging.