M5 · Lesson 1 — Mathematical Reasoning

Definitions —
Being Precise

Informal language lets ideas slide past each other.
A definition pins an idea down so it can't escape.

01
M5 · L1 — Why Precision Matters

The cost of vagueness

Informal language
breaks proofs

"A cold-start user is a user we don't know much about."

This sounds fine in a conversation. But in a paper — what is "not much"? 1 interaction? 5? 0? Two readers will interpret this differently and draw different conclusions.

Mathematical definitions remove ambiguity by specifying:

  • The name of the concept
  • The type of object being defined
  • The condition that must hold exactly
  • The notation to be used consistently
02
M5 · L1 — Structure of a Definition

Anatomy

What a definition
is made of

Part 1

The Term

The name being defined. Always bold or italicised on first use. e.g., cold-start user

Part 2

The Domain

What type of object is this? A user? A set? A function? e.g., a user u ∈ U such that...

Part 3

The Condition

The exact criterion that defines membership. Quantified, unambiguous. e.g., |𝒩_u| < k

Definition · Cold-Start User
A user u ∈ 𝒰 is called a cold-start user if the number of their observed interactions satisfies |𝒩_u| < k, where k ∈ ℕ is a pre-defined threshold.
Three parts present: term (cold-start user), domain (u ∈ 𝒰), condition (|𝒩_u| < k). No ambiguity.
03
M5 · L1 — RS Examples

Applied to recommender systems

Three definitions you'll
write in RS papers

Definition · Top-K Recommendation
Given a user u and a ranking function f: 𝒰 × ℐ → ℝ, the top-K recommendation list is the ordered set \( \mathcal{R}_u^K = \text{argtop-}K_{i \in \mathcal{I} \setminus \mathcal{I}_u^+} f(u, i) \).
Definition · Implicit Feedback
An implicit feedback signal is an observation r_{ui} ∈ {0,1} where r_{ui} = 1 indicates user u has interacted with item i (e.g., clicked, viewed, purchased), without expressing an explicit preference rating.
Definition · Fairness (Exposure)
A recommender system satisfies exposure fairness across groups G₁, G₂ ⊆ ℐ if \( \left|\mathbb{E}[\text{rank}(i)] - \mathbb{E}[\text{rank}(j)]\right| \leq \epsilon \) for all i ∈ G₁, j ∈ G₂.
04
M5 · L1 — Informal vs Formal

The transformation

Informal → formal

❌ Informal (vague)

"We consider items to be popular if many users interact with them, and unpopular otherwise."

How many? Relative to what? Where is the cutoff? This means nothing precisely.

✅ Formal (precise)

"Item i ∈ ℐ is called popular if its interaction count satisfies \(c_i = |\{u \in \mathcal{U} : r_{ui} > 0\}| \geq \tau\), where τ is the p-th percentile of the interaction count distribution."

Now τ is a parameter, the criterion is unambiguous, and reproducibility is possible.

Key test: After reading your definition, could two different researchers implement the same thing independently and get identical results? If yes — your definition is good.

05
M5 · L1 — Common Mistakes

What to avoid

Four definition
anti-patterns

  • Circular definition — defining X in terms of X. "A fair system is one that treats users fairly." Adds no information.
  • Missing domain — saying "let k be the number of interactions" without saying k ∈ ℕ or what it refers to.
  • Undefined symbols — using r_{ui} before defining R. Every symbol must be introduced before use.
  • Imprecise quantifiers — "for most users" or "in many cases". Formal definitions use ∀ and ∃, not prose approximations.

Rule of thumb: If your definition contains the words "many", "most", "few", "large", "small", "some", or "often" — it needs to be formalised further.

06
M5 · L1 — Key Takeaways

What to remember

01

Term + Domain + Condition

Every definition has three parts. If one is missing, the definition is incomplete.

02

Precision enables proof

You cannot prove something about a vague concept. Formalising a definition is the first step toward any theorem.

03

The reproducibility test

Two researchers, same definition, same implementation — if this fails, the definition is broken.

Next: M5 · L2 — Theorems, Lemmas, Propositions, Corollaries

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