COSMO · STUDY + TEACHING DECK

Computer Science & AI

August 3, 2026

A question worth sharing

PCA: Reducing Dimensions

Compress correlated coordinates by projecting data onto directions that retain the most variance.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use centering, principal component, explained variance 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

Points lie close to the line y=x. Which one-dimensional direction should preserve more spread: horizontal or diagonal?

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.

centering

subtracting the mean from each feature

principal component

unit direction with maximal remaining projected variance

explained variance

share of total variance retained by selected components

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

The core idea

Keep the idea. Explain the conditions.

v_1=\underset{\|v\|_2=1}{\operatorname{argmax}}\;v^\top\Sigma v

The idea

Compress correlated coordinates by projecting data onto directions that retain the most variance.

What to remember

Dimensionality reduction preserves a chosen kind of structure; it never preserves everything.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Project centered points (-1,-1), (0,0), and (1,1) onto direction (1/ $\sqrt{2}$ ,1/ $\sqrt{2}$ ).

Predict each next step

Take a dot product with the unit direction.
The projections are - $\sqrt{2}$ , 0, and $\sqrt{2}$ .
All variation along the diagonal is retained.
The perpendicular coordinate becomes zero.

Which step needs a reason, not just a calculation?

See the structure

The structure behind pca: reducing dimensions

v_1=\underset{\|v\|_2=1}{\operatorname{argmax}}\;v^\top\Sigma v

For centered data with covariance Sigma, the first principal direction maximizes projected variance.

Read the representation

Compress correlated coordinates by projecting data onto directions that retain the most variance.

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 pca: reducing dimensions

Your shared task

Students rotate a two-dimensional point cloud, compare projection spreads, and choose a compression direction with a stated goal.

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

Dimensionality reduction preserves a chosen kind of structure; it never preserves everything.
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

A covariance matrix has eigenvalues 9, 3, and 1. What share does the first component explain?

Predict each next step

Total variance is 9+3+1=13.
The first component retains 9 units.
Compute $\frac{9}{13}$ .
The explained share is about 69.2 percent.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

The first principal component is the unit direction that maximizes projected variance after centering.

Build the argument

Every unit direction defines a one-dimensional projection.
Projection variance is a quadratic form of the covariance matrix.
Its maximum occurs at the eigenvector with the largest eigenvalue.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Why can PCA remove information needed for classification even when it retains most variance?

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

PCA supports visualization, compression, noise reduction, preprocessing, and inspection of embedding spaces.

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

Points lie close to the line y=x. Which one-dimensional direction should preserve more spread: horizontal or diagonal?

Question 2

Explain why this claim is true: The first principal component is the unit direction that maximizes projected variance after centering.

Question 3

Why can PCA remove information needed for classification even when it retains most variance?

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

Dimensionality reduction preserves a chosen kind of structure; it never preserves everything.

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

A good question belongs to everyone in the room.