COSMO · STUDY + TEACHING DECK

Computer Science & AI

July 16, 2026

A question worth sharing

Embeddings and Representations

Turn objects into vectors so relationships can be measured, compared, and tested.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use feature, representation, embedding to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

A model maps cat to (1.0, 0.9), dog to (0.9, 1.0), and bicycle to (-0.8, -0.7). What relationship is the representation expressing?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

feature

measurable input used to describe an object

representation

data form chosen for a task

embedding

vector representation whose geometry can encode relationships between objects

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

\phi:\mathcal X\longrightarrow\mathbb R^d

The idea

Turn objects into vectors so relationships can be measured, compared, and tested.

What to remember

An embedding is a hypothesis about which relationships deserve to become geometry.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Compare the vectors a=(1,2), b=(2,3), and c=(-3,-2) without naming the objects.

Predict each next step

Compute or estimate the pairwise differences.
Vectors a and b differ by only (1,1).
Vector c points into a different region.
The representation therefore treats a and b as the closest pair.

Which step needs a reason, not just a calculation?

See the structure

The structure behind embeddings and representations

\phi:\mathcal X\longrightarrow\mathbb R^d

A representation maps objects to coordinates; the task determines which relationships those coordinates should preserve.

Read the representation

Turn objects into vectors so relationships can be measured, compared, and tested.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate embeddings and representations

Your shared task

Groups invent vector features for familiar objects, then swap representations and infer what relationship each group tried to preserve.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

An embedding is a hypothesis about which relationships deserve to become geometry.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Design two features for representing short text messages.

Predict each next step

Choose a task first, such as separating questions from statements.
One feature could measure question-mark presence.
Another could measure interrogative-word frequency.
The representation is useful only relative to the task.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A useful embedding preserves relationships that matter for the intended task.

Build the argument

Coordinates have no value by themselves.
A metric turns coordinate differences into comparisons.
Evaluation checks whether those comparisons match the relationships the task needs.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Two embeddings preserve different neighbor relationships. How would you decide which is better?

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Embeddings support search, recommendation, clustering, retrieval, visualization, and language-model systems.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

A model maps cat to (1.0, 0.9), dog to (0.9, 1.0), and bicycle to (-0.8, -0.7). What relationship is the representation expressing?

Question 2

Explain why this claim is true: A useful embedding preserves relationships that matter for the intended task.

Question 3

Two embeddings preserve different neighbor relationships. How would you decide which is better?

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

An embedding is a hypothesis about which relationships deserve to become geometry.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.