点积范数偏差
The long vector may outrank a better-aligned short vector.
本地计算 / 模拟本地教学实验，不调用外部 API。合成示例不能替代生产环境测量。
学习指南与本课目录
LEARNING CONTRACT理解 → 实现 → 研究
目标：解释本课机制、复现一个例子、测试一个失败案例。读完不自动代表掌握。
- 理解层：写明假设、形状/单位和一个反例。
- 实现层：复算数值例子并检查边界。
- 研究层：在匹配预算下比较一个替代方法。
02 · 预测与配置
技术场景说明、参数名称与参考答案目前为英文。
03 · 检查证据
方法与参考资料
Five manually chosen 2D vectors, not embeddings generated by a language model. Zero-vector cosine is defined as 0 in this lab. Ties preserve fixture order.
Text / 原始公式
cos(q,d)=q·d/(||q|| ||d||); dot(q,d)=Σqᵢdᵢ
Geometry alone cannot establish semantic relevance; these vectors are hand-authored.
Introduction to Information Retrieval · vector space scoring ↗练习题
embedding-2 / 01What does “Dot-product norm bias” test?
The long vector may outrank a better-aligned short vector.
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 02Which computation produces the result in “Dot-product norm bias”?
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 03What is the baseline configuration for “Dot-product norm bias”?
{"x":1,"y":0.2,"metric":1}
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 04What numerical result should you check for “Dot-product norm bias”?
{"Top score":5.2,"Query norm":1.019803902718557}
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 05How would you run a controlled comparison for “Dot-product norm bias”?
Keep the supplied fixture and other settings fixed; change x. Compare ranking order, vector norms and scores on identical documents.
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 06What failure could invalidate a conclusion from “Dot-product norm bias”?
Geometry alone cannot establish semantic relevance; these vectors are hand-authored.
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 07Is “Dot-product norm bias” a benchmark of a deployed language model?
No. Five manually chosen 2D vectors, not embeddings generated by a language model. Zero-vector cosine is defined as 0 in this lab. Ties preserve fixture order.
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 08Which evidence should you report for “Dot-product norm bias”?
Compare ranking order, vector norms and scores on identical documents. Export inputs, engine, result and method; distinguish synthetic fixtures from observed production data.
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 09How can another researcher reproduce “Dot-product norm bias”?
Open embedding-2, reset the preset, retain the exact inputs and run. The calculation is deterministic for identical inputs.
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →embedding-2 / 10What is the next validation step after “Dot-product norm bias”?
Geometry alone cannot establish semantic relevance; these vectors are hand-authored. Test the hypothesis on a representative held-out workload and document a baseline, success metric and stopping rule before claiming deployment readiness.
Original teaching answer / local derivation. Method reference: Introduction to Information Retrieval · vector space scoring ↗
打开实验 →假设、参考实现与验收
零向量的数学余弦未定义。程序返回零作为哨兵值，不能解读为正交。点积与余弦的尺度敏感性不同。
已复核引擎公式、场景预设及数值保护。固定种子的合成输入不是生产基准。
运行前：预测一个边界情况
The long vector may outrank a better-aligned short vector.
提示：隔离假设
固定基线输入，只改变一个参数或边界；同时比较数值与声明单位。单例通过仅验证该样例。
最小可运行算例
将算例及所链接模块保存在同一目录。使用 Node.js 22 或更高版本，然后运行 node embedding-2.mjs. 无需第三方运行依赖。
import {calculate,network} from './academy-core.mjs';
const params={
"x": 1,
"y": 0.2,
"metric": 1
};
const result=calculate('embedding',params);
console.log(result.metrics);测试与迁移
The long vector may outrank a better-aligned short vector.
在下方笔记记录输入、代码版本、期望值和实测值；再构造违反上述假设的反例，解释原结论为何不再成立。
下载完整代码包（.tar.gz） · 完整参考代码与回归检查预测、检验与证据记录
展开答案前先预测一个结果；练习后记录差异和迁移测试。记录仅存本地，在不同语言版本间共用。
打开记录／导出／清除 →数学复核范围逐篇记录，不代表完成论文实证复现；可在笔记中记录疑似错误。 复核范围