00.07 Notation and Symbols

00-START-HERE · 01-READING-ROUTES · 00_06_for_practicing_engineers · 00_08_acknowledgments This unit standardizes the symbols, abbreviations, overloads, and measurement conventions used throughout the retrieval-augmented generation (RAG) book.

TL;DR

• The semi-parametric frame separates knowledge in generator parameters $\theta$ from knowledge in the retrievable datastore $D$ .
• The main flow is $x$ to $q$ to a retrieval gate, a ranked retrieval result, $k$ contexts, and generated answer $y$ .
• Retrieval and ranking notation covers relevance, rank, term statistics, encoders, token lengths, loss, and temperature.
• Index notation names vector count, dimension, graph degree, search breadth, product quantization, and recall targets.
• Evaluation and training notation distinguishes ranking quality, chunking quality, retrieval distributions, generator supervision, and low-rank factors.
• Multimodal, graph, latency, and cost symbols connect retrieval choices to modality thresholds, graph size, stage timing, and query price.
• Several conventional symbols are overloaded. The source keeps them and requires context to identify the intended meaning.

The story

Learned generator parameters are $\theta$; datastore $D$ remains externally addressable. Component stores are $D_1,\ldots,D_m$. Transform input $x$ into retriever query $q=\Phi(x)$; the gate $g(x)\in\{0,1\}$ selects whether retrieval runs. These are local definitions, not a claim that weights and stores have the same update operations. When retrieval runs, write $\mathop{\mathrm{RET}}\nolimits(q,D)$ for the ranked candidates. Candidate depth $k'$ precedes filtering/reranking and final depth $k\le k'$; these are requested upper limits, and fewer eligible contexts may remain. The generator conditions on the request and retained $c_1,\ldots,c_k$ to generate $y$ token by token. The library has several control panels. Retrieval scores and ranks items. The vector index controls how broadly the shelves are searched. Evaluation checks ranked results and chunk boundaries. Training aligns the retriever’s distribution with the generator’s preference. Multimodal admission rules and graph grounding decide which rooms and relationships can join the search. The checkout counter records each stage latency $t_i$ and query cost $C_q$ . Some labels repeat across panels. The label $d$ can mean a document at the search desk or a dimension at the index panel. The label only becomes safe when its panel and first-use definition are visible.

Decoder table

Core mechanics

The Semi-Parametric Frame

• What: $\theta$ names the generator’s parametric knowledge. $D$ names test-time retrievable knowledge. The remaining symbols trace input, query transformation, gating, retrieval, context selection, and generation.
• Why: The frame keeps weight-based knowledge separate from datastore knowledge and makes each pipeline decision visible.
• Failure without it: A reader could confuse what the user typed with the retriever query, or confuse pre-reranking depth $k'$ with the final top-k cutoff.
• Stated cost: No explicit cost formula appears in this section.

Retrieval and Ranking

• What: This notation defines documents, relevance scores, rank positions, BM25 statistics, encoders, token lengths, late interaction, training loss, and softmax temperature.
• Why: It gives one vocabulary for lexical ranking, encoded retrieval, training, and score calibration.
• Failure without it: Rank could lose its 1-indexed convention, $d$ could be mistaken for dimension, or $\tau$ could be mistaken for a score instead of a temperature.
• Stated cost: Query length $L_q$ and passage length $L_p$ drive encoder floating-point operations.

Indexing and Vector Search

• $N_{\mathrm{vec}}$ counts indexed vectors, not necessarily documents, while $d$ is their dimension.
• HNSW $M$ controls link allocation; $M_0=2M$ is a common base-layer capacity. With four-byte slots, $8M\,\mathrm B$ is the base-layer-only link subtotal, not complete memory. Expected upper-layer slots add $4M/(M-1)\,\mathrm B$ under the stated geometric model.
• For PQ, $m$ equal subspaces have width $d/m$. If each sub-code has $b_{\mathrm{sub}}$ bits, $k=2^{b_{\mathrm{sub}}}$, total code length is $m b_{\mathrm{sub}}$ bits and fp32 centroid payload is $4kd\,\mathrm B$.
• $\mathrm{efSearch}$ and $n_{\mathrm{probe}}$ are search-work controls, not exact hop counts or guaranteed recall values.

Evaluation

• MRR uses the first relevant hit and permits several relevant items; it does not assume exactly one exists.
• Precision and recall at a fixed cutoff ignore order within that prefix, but changing membership changes them. Distinguish ANN-neighbor recall, relevance recall and complete supporting-evidence coverage.
• nDCG needs a gain function, positive IDCG or an explicit zero-IDCG policy, cutoff and judgment universe.
• $F_1$ requires declared averaging and zero-denominator policies; macro-$F_1$ is not accuracy or an overall error rate. Segmentation metrics and span IoU measure different objectives.

Training the RAG System

• A retrieved training support $D'$ must be explicitly defined. A top-candidate normalized distribution is conditional on that support; full-corpus objectives use a different normalizer.
• If $Q_{\mathrm{LM}}$ is the fixed teacher target, a common cross-entropy-equivalent direction is $\mathop{\mathrm{KL}}\nolimits(Q_{\mathrm{LM}}\Vert P_R)$. Reverse KL is different; writing generic $\mathop{\mathrm{KL}}\nolimits(P\Vert Q)$ does not specify which system is the target.
• Require positive softmax temperatures. Define mixture weights, support and whether gradients flow through teacher scores.
• For $W\in\mathbb R^{d_{\mathrm{out}}\times d_{\mathrm{in}}}$, use $A\in\mathbb R^{r\times d_{\mathrm{in}}}$ and $B\in\mathbb R^{d_{\mathrm{out}}\times r}$; a common LoRA update is $W+(\alpha/r)BA$. Other orientations require a consistent factor order.

Multimodal and Graph Retrieval

• Shared embeddings require compatible training/normalization; “multimodal” does not make arbitrary encoder spaces interchangeable.
• $\theta_m$ denotes a chosen modality-specific gate. Calibration may justify another policy, so “never global” is not a universal theorem.
• A CLIP-style batch has diagonal designated pairs under its pairing convention; duplicates and multiple valid matches can require a richer target.
• Distinguish full graph $\mathcal G=(V,E)$ from $\mathop{\mathrm{schema}}\nolimits(\mathcal G)$. Graph grounding is not proof that the extracted assertions or generated answer are true.

Latency and Cost

• Stage latency $t_i$ and attributable device service time $s_j$ are different quantities. A serial critical path can add stage wall times; parallel fan-out follows its dependency graph, not the sum of all branches.
• With resource price $\rho_j$ in dollars/hour and attributable service time $s_j$ in milliseconds, $C_q^{\mathrm{resource}}=\sum_j\rho_j s_j/(3.6\times10^6)$. This excludes fixed reservations, utilization policy and API billing unless explicitly included.
• A one-resource serial approximation can use $\rho\sum_i t_i/(3.6\times10^6)$ only when those times really represent attributable, nonoverlapped reservation/service. Batching, concurrency and different devices invalidate casual wall-time billing.
• Use $N_{\mathrm{param}}$ for parameters and $\rho$ for price to avoid the overloaded $P$ in calculations.

Symbols That Carry More Than One Meaning

• What: The source explicitly overloads $d$, $k$, $m$, $M$, $b$, $N$, $P$, and $\mathcal G$.
• Why: These symbols remain consistent with their separate retrieval, representation-learning, and systems literatures.
• Failure without it: A familiar symbol could silently acquire the wrong meaning and corrupt a formula or design explanation.
• Stated cost: No cost is assigned to the convention. Each section states the active meaning at first use.

Conventions

• State the base explicitly: natural logs for nats, log loss and the displayed BM25 IDF; base $2$ for DCG and bit counts.
• Use decimal $\mathrm{GB}$ and binary $\mathrm{GiB}$ precisely, not interchangeably. The storage convention does not require binary values to be quotations.
• Vectors are columns by default; stated row-stacked representations override that default. Check every product’s dimensions.
• The nominal $d=768$, $110\times10^6$-parameter encoder is an illustrative recurring setup. Parameter count does not uniquely determine output dimension or achieved throughput.

Diagrams

The manifest lists zero figures and zero captioned tables for physical PDF pages 28-31. No diagram redraw is required. The source contains seven unnumbered notation tables and one unnumbered overload table. The exhaustive Decoder table preserves every row from those layouts. The prose conventions block is preserved in both the Decoder table and Core mechanics.

Whiteboard pack

What to draw

1. Draw $x\to q=\Phi(x)$ and a retrieval gate.
2. Draw $\mathop{\mathrm{RET}}\nolimits(q,D)$, candidate depth $k'$, eligibility/reranking and final contexts $c_{1:k}$.
3. Condition $\mathop{\mathrm{GEN}}\nolimits_\theta$ on the request and contexts to generate $y$.
4. Separate vector counts from model parameters, PQ sub-code bits from total code bits, and configured HNSW slots from realized degree.
5. Add evaluation definitions and explicit teacher/retriever KL direction.
6. Finish with dimension checks, log bases, storage units, resource service time and query cost.

Spoken script

Start with input $x$ and transform it into retriever query $q=\Phi(x)$. A gate decides whether retrieval runs. Draw candidate contexts from datastore $D$, then eligibility checks, reranking and final contexts for the generator. Under that path, separate score, index, evaluation and training panels. Define every overloaded letter locally. PQ bits per sub-code differ from total code bits, and HNSW capacity parameters differ from realized degree. Finally distinguish stage latency from attributable resource time, state the direction of any KL loss, and write explicit logarithm bases and storage units.

Interview traps

1. Why does the notation keep overloaded symbols instead of inventing new ones?

Each symbol is conventional in its own literature. The source preserves those conventions and requires the active meaning to be stated at first use.

2. Are $k'$ and $k$ interchangeable?

No. Candidate depth $k'$ and final depth $k$ refer to different stages. In the simple truncation pipeline, $k\le k'$, but filters can leave fewer eligible items and other retrieval rounds need their own candidate counts.

3. What is the difference between $D$ and $D'$?

$D$ denotes the full store and $D'$ a declared retrieved training support. Top-candidate normalized distributions use $D'$. That is a modeling choice, not proof that every retrieval-training objective normalizes only over a retrieved subset.

4. Can $M$ denote product quantization?

This reference reserves $m$ for PQ and $M$ for HNSW, but external APIs can differ. Faiss PQ explicitly uses uppercase $M$ for its number of subquantizers. Map meaning and dimensions rather than assuming a global case-based prohibition.[1]

5. What does $P$ mean in cost arithmetic?

Use an explicit price symbol such as $\rho$ in cost arithmetic and $N_{\mathrm{param}}$ for model parameters. The millisecond-to-hour divisor is $3.6\times10^6$. Multiplying price by summed stage wall time is valid only under a stated attribution model; it is not a universal per-query cost formula.

Key numbers

00-START-HERE · 01-READING-ROUTES · 00_06_for_practicing_engineers · 00_08_acknowledgments