dot product concept
| Definition | The sum of the coordinatewise products of two vectors. Equation 0.1. Also cosine similarity, cosine reader. |
|---|---|
| Example | (1, 2) · (3, 4) = 3 + 8 = 11. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id dot-product, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.1. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/DotProduct.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.1.
\[x\cdot y=\sum_{i=1}^{d}x_i y_i,\qquad \|x\|=\sqrt{x\cdot x},\qquad \cos\theta=\frac{x\cdot y}{\|x\|\,\|y\|}.\]
Book equation 11.1.
\[\cos\big(k,\hat k\big)=0.995\qquad\text{while}\qquad \mathrm{PPL}:\ 12.24\ \to\ 10643.\]
Conditions
- The sum of the coordinatewise products of two vectors, symmetric and bilinear. The cosine, the dot product over the two lengths, lies between minus one and one by Cauchy–Schwarz and is unchanged when either vector is scaled by a positive factor, while the dot product scales with the vector.
- A cosine reader has declared length a nuisance and a dot-product reader has not, which is the reader difference of chapter 3, and cosine 0.995 between keys and their reconstruction said nothing about the softmax that read them.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.1 and chapter 3 section 3.2 of Data Mining as
Observation, with the program’s cosine-versus-consumer case in
turboquant-pro/docs/KV_KEYS_FINDING.md:1-49.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/DotProduct.lean,
theorems self_nonneg, dot_comm,
dot_sq_le, cosine_mem, dot_smul,
cosine_smul, at observation-data-mining f3914f0; what the
check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation primer L, chapters 0, 1, 2, 3, 8, 11, 12.
Related
projection; Euclidean distance; quotient; nuisance.
See also
Ledger rows that cite the entry’s records without naming it: NEG-2.
Sources-table rows that share a record with the entry without naming it: chapter 1 section 1.4, chapter 2 section 2.5, chapter 3 section 3.2, chapter 8 section 8.2, chapter 11 section 11.1, chapter 11 section 11.2.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.