Quizmaster

Learning objectives

What you will be able to explain

Linear factor-model assumption
Probabilistic PCA and factor analysis
PCA as a latent-variable model
Why ICA needs non-Gaussianity
What ICA can and cannot identify
Slow feature analysis

Section 01

Linear factor-model assumption to PCA as a latent-variable model

Linear factor-model assumption

The models differ mainly in assumptions on latent factors and noise, which make different structure identifiable or useful.

Guided checkpoint

Which generative form captures the common structure of the chapter's linear factor models?

Source: Chapter 13, pp. 489-490

Probabilistic PCA and factor analysis

Probabilistic PCA constrains residual noise to equal variance in every direction; factor analysis allows a diagonal residual covariance.

Guided checkpoint

Match each covariance component to its meaning in $x=Wh+\mu+\epsilon$ .

Source: Chapter 13, section 13.1, pp. 490-491

PCA as a latent-variable model

Only the latent subspace is identified under isotropic Gaussian assumptions; rotating both factors and loadings leaves the observed distribution unchanged.

Guided checkpoint

Evaluate each statement.

Source: Chapter 13, section 13.1, pp. 490-491

Section 02

Why ICA needs non-Gaussianity to Slow feature analysis

Why ICA needs non-Gaussianity

A Gaussian joint distribution is rotationally symmetric after whitening. Non-Gaussian independent sources break that symmetry, up to scale and permutation ambiguities.

Guided checkpoint

What enables independent component analysis to identify latent directions beyond a Gaussian subspace?

Source: Chapter 13, section 13.2, pp. 491-493

What ICA can and cannot identify

ICA separates independent directions more strongly than PCA but still cannot assign canonical order or scale without extra conventions.

Guided checkpoint

Match each ambiguity or operation to its role.

Source: Chapter 13, section 13.2, pp. 491-493

Slow feature analysis

Many semantic causes vary more slowly than raw measurements. Constraints prevent the constant function from winning the slowness objective.

Guided checkpoint

What representation does slow feature analysis seek?

Source: Chapter 13, section 13.3, pp. 493-496

Section 03

Sparse-coding objective to Compare linear factor objectives

Sparse-coding objective

2+0.5(3)=3.5

Guided checkpoint

For one example, reconstruction error is 2, $||h||_1=3$ , and $\lambda=0.5$ . Compute $||x-Wh||_2^2+\lambda||h||_1$ .

Source: Chapter 13, section 13.4, pp. 496-499

Sparse coding mechanics

Sparse coding trades reconstruction fidelity for using few active atoms, and code inference is not necessarily a single feedforward pass.

Guided checkpoint

Judge each statement.

Source: Chapter 13, section 13.4, pp. 496-499

Compare linear factor objectives

Each method adds a different prior on useful latent causes; there is no universally privileged linear representation.

Guided checkpoint

Match each method to the structure it prioritizes.

Source: Chapter 13, sections 13.1-13.4, pp. 490-499

Section 04

PCA through the manifold lens

PCA's affine subspace is the simplest manifold model. Curved or topologically complex data require nonlinear or multiple-chart descriptions.

Guided checkpoint

Evaluate each claim.

Source: Chapter 13, section 13.5, pp. 499-501

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

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