Softmax Function Overview

Overview

A concise animation that introduces the softmax function, shows how it converts a vector of raw scores into a probability distribution, and visualizes its effect on a simple 2‑D and 3‑D example. The key takeaway is that softmax emphasizes larger inputs while ensuring all outputs sum to 1.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title "Softmax Function" fades in at the top, while a brief outline of the problem (raw scores) appears as points on a number line.
2	Definition	~5s	The formula $\sigma(\mathbf{z})_i = \frac{e^{z_i}}{\sum_{j} e^{z_j}}$ is written in the bottom area and highlighted term‑by‑term as the narration explains numerator and denominator.
3	2‑D Example	~8s	Left area shows a 2‑D bar chart of raw scores $[2, 1]$. Right area animates the exponentiation, normalization, and resulting probability bars, with the total bar height constrained to 1.
4	3‑D Example	~8s	Left area displays three points on a 3‑D axis representing a vector $[1, 3, 0]$. The points rise to heights proportional to $e^{z_i}$, then shrink proportionally so that their heights sum to 1, illustrating the softmax mapping.
5	Comparison & Intuition	~4s	A side‑by‑side split shows the original scores versus softmax probabilities, with arrows and color coding to emphasize how larger scores dominate.
6	Outro	~2s	The title fades out, and a concise caption "Softmax turns scores into a probability distribution (sum = 1)" appears at the bottom.

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA                    │
├──────────────────────┬──────────────────────┤
│        LEFT AREA      │      RIGHT AREA      │
│  (main visual:       │  (supporting labels, │
│   bar/3‑D plot)       │   step‑by‑step math) │
├──────────────────────┴──────────────────────┤
│                BOTTOM AREA                 │
│   Formula, captions, brief notes           │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Softmax Function" (fades in/out)	Appears in Intro, disappears in Outro
Left	Main visual: bar chart for 2‑D case, 3‑D points for 3‑D case	Central focus, animated transformations
Right	Supporting step‑by‑step annotations, exponentiation arrows, normalization arrows	Appears from Definition onward, fades between examples
Bottom	Full softmax formula, final caption	Formula appears in Definition, caption appears in Outro

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Notes

• Keep total runtime under 30 seconds; the durations above sum to ~30 s.
• Use smooth fade and scale transitions; avoid abrupt jumps.
• No textual narration is required in the spec, but visual cues (arrows, color changes) should guide the viewer through each computational step.
• All numbers are kept small for clarity; exponentiation is shown with a simple animation (e.g., a growing bar) rather than calculating exact values.
• The scene must be implemented as a single Manim Scene class.