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Mathematical foundations

The derivations behind the methods. This is where most candidates stop at the name of the technique, and it is where a hard interview loop — Google, DeepMind, Palantir, a research lab — spends its second half.

Each entry states the question the way an interviewer asks it, works the derivation, and names what a strong answer contains beyond the algebra. Where the result is measurable in this repository, the notebook cell is named.

Retrieval and scoring

Question Status
BM25 from first principles Derive the term-frequency component from the probabilistic relevance model. Why does it saturate, and what does k₁ control? ✅
Cosine, dot product and MIPS Why does L2 normalisation make them rank-equivalent, and when is that a bug? ✅
Reciprocal rank fusion Why 1/(k+rank)? Why not 1/rank or the nDCG discount? ✅
nDCG Why a log₂ discount, and what breaks when relevant-set sizes differ across queries? ◻ planned
Multi-hop recall Per-passage recall is 0.938 and you need three passages. What is full-chain recall? ✅
Question Status
Eckart–Young–Mirsky Prove that truncating the SVD at rank k is the optimal rank-k approximation. In which norm? ✅
Johnson–Lindenstrauss How many dimensions preserve pairwise distances among n points to within ε? ✅
Navigable small-world graphs Why is greedy search O(log n), and what breaks the guarantee? ✅
Product quantization Derive the memory saving and the distortion cost. What is asymmetric distance computation? ◻ planned

Evaluation and statistics

Question Status
Cohen's κ Derive it. Your judge and human agree 90% of the time — is that good? ✅
The paired bootstrap and power Why paired? How many eval questions to detect a 2-point gain? ✅
Multiple comparisons You evaluate 12 slices and one is significant. What is wrong? ◻ planned
Calibration and abstention How do you know a score is calibrated? ◻ planned

Contributing one: open a thread in Q&A with the derivation before writing the file. Getting the argument reviewed is worth more than getting the file merged, and the thread becomes the commentary the document does not have room for.