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1890 learning items

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Quiz

MML-09-03: Marginal & Conditional Distributions of Multivariate Gaussians

Extract Gaussian marginals and compute conditional means and covariances using covariance or precision blocks. Analyze Schur complements, residual independence, conditional-dependence structure, and Gaussian observation updates. Normal-law notation uses variance or covariance as its second parameter. 40 questions combining single-select and multiselect formats, with worked explanations and feedback for every choice. Prerequisites: QS 9.2. Cognitive range: Tier 3–4. The local authored sequence progresses from accessible checks to computations, modeling diagnostics, and derivation recognition within that range. Reference: [Mathematics for Machine Learning, Chapter 6](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Oct 31, 2026

40 questions

Quiz

MML-09-02: The Multivariate Normal Distribution: PDF, Covariance Geometry & Contours

Evaluate multivariate Gaussian densities and log-densities, interpret Mahalanobis distance, and connect covariance eigenspaces to contour geometry. Diagnose dimension effects, singular support, unstable covariance estimates, and numerical scoring choices. Normal-law notation uses covariance as its second parameter. 45 questions combining single-select and multiselect formats, with worked explanations and feedback for every choice. Prerequisites: QS 9.1, Part 04. Cognitive range: Tier 1–3. The local authored sequence progresses from accessible checks to computations, modeling diagnostics, and derivation recognition within that range. Reference: [Mathematics for Machine Learning, Chapter 6](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Oct 31, 2026

45 questions

Quiz

MML-09-01: The Univariate Gaussian Distribution: Properties, Normalization & Integrals

Build fluency with Gaussian location, scale, PDF normalization, standardized probabilities, and moments. Work through Gaussian integrals, the error-function connection, tail bounds, and residual-model diagnostics. Normal-law notation uses variance as its second parameter. 30 questions combining single-select and multiselect formats, with worked explanations and feedback for every choice. Prerequisites: Part 08. Cognitive range: Tier 1–3. The local authored sequence progresses from accessible checks to computations, modeling diagnostics, and derivation recognition within that range. Reference: [Mathematics for Machine Learning, Chapter 6](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Oct 31, 2026

30 questions

Quiz

MML-08-08: Change of Variables: Univariate & Multivariable Jacobian Determinant Transformations

Transform scalar and multivariate densities while tracking inverse maps, support, branch contributions, and Jacobian volume factors. Apply the ideas to polar sampling, triangular coupling layers, coordinate permutations, normalizing-flow scores, and singular-map diagnostics. 35 questions combining single-select and multiselect formats, with worked explanations and feedback for every choice. Prerequisites: QS 8.7, Part 06. Cognitive range: Tier 3–4. The local authored sequence progresses from accessible checks to computations, modeling diagnostics, and derivation recognition within that range. Reference: [Mathematics for Machine Learning, Chapter 6](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Oct 31, 2026

35 questions

Quiz

MML-08-07: Covariance, Correlation Matrices & Linear Independence of RVs

Compute covariance and Pearson correlation from moments, expand variances of linear combinations, and interpret covariance and correlation matrices. Distinguish linear redundancy, uncorrelatedness, and statistical independence through counterexamples and feature diagnostics. 35 questions combining single-select and multiselect formats, with worked explanations and feedback for every choice. Prerequisites: QS 8.6. Cognitive range: Tier 2–3. The local authored sequence progresses from accessible checks to computations, modeling diagnostics, and derivation recognition within that range. Reference: [Mathematics for Machine Learning, Chapter 6](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Oct 31, 2026

35 questions

Quiz

MML-12-06: Stochastic Gradient Descent (SGD): Noise, Mini-Batch Variance & Learning Rate Schedules

Study Stochastic Gradient Descent (SGD): Noise, Mini-Batch Variance & Learning Rate Schedules. Master rigorous derivations, step-by-step numerical computations, multiselect conceptual checks, and diagnostic applications in machine learning models. Prerequisites and cognitive tiers follow the master syllabus.

mathematics

Oct 31, 2026

40 questions

Quiz

MML-12-05: Line Search Methods (Armijo, Wolfe Conditions) & Ill-Conditioned Ravines

Study Line Search Methods (Armijo, Wolfe Conditions) & Ill-Conditioned Ravines. Master rigorous derivations, step-by-step numerical computations, multiselect conceptual checks, and diagnostic applications in machine learning models. Prerequisites and cognitive tiers follow the master syllabus.

mathematics

Oct 31, 2026

30 questions

Quiz

MML-06-04: Vector-Valued Functions & The Jacobian Matrix

35 worked single-select and multiselect questions. Tier 2–3. Prerequisites: QS 6.3. Learning goals: Arrange Jacobian rows by outputs and columns by inputs; Compute and evaluate small vector-function Jacobians; Use Jacobian-vector products for local output predictions; Distinguish an affine local approximation from its linear derivative; Derive polar and spherical coordinate Jacobians with stated conventions; Interpret signed determinants and absolute volume distortion; Diagnose singular Jacobians, coordinate charts and local invertibility. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://mml-book.github.io/book/mml-book.pdf), [Source 2](https://mml-book.github.io/book/mml-book.pdf), [Source 3](https://openstax.org/books/calculus-volume-3/pages/5-7-change-of-variables-in-multiple-integrals).

mathematics

Oct 31, 2026

35 questions

Quiz

MML-06-03: Directional Derivatives, Steepest Ascent & Level Set Geometry

30 worked single-select and multiselect questions. Tier 1–3. Prerequisites: QS 6.2, Part 03. Learning goals: Compute directional derivatives for differentiable scalar fields; Normalize directions and distinguish spatial rates from path speed; Find Euclidean steepest-ascent and descent directions; Use Cauchy–Schwarz to prove directional-rate bounds; Relate gradients to regular level curves and tangent planes; Diagnose zero gradients and zero first-order rates; Apply directional geometry to local ML loss diagnostics. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient).

mathematics

Oct 31, 2026

30 questions

Quiz

MML-06-02: Multivariable Functions, Partial Derivatives & The Gradient Vector

35 worked single-select and multiselect questions. Tier 1–3. Prerequisites: QS 6.1, Part 02. Learning goals: Distinguish scalar-valued functions from vector-valued mappings; Compute partial derivatives while holding other coordinates fixed; Assemble and evaluate a column gradient; Use gradient/Jacobian layout conventions consistently; Interpret gradient fields and critical points on standard surfaces; Diagnose coordinate-wise sensitivity in small ML loss examples; Recognize why partial derivatives alone do not guarantee differentiability. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives), [Source 2](https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives), [Source 3](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Oct 31, 2026

35 questions

Quiz

MML-06-01: Univariate Calculus Foundations: Limits, Derivatives & Taylor Approximations

25 worked single-select and multiselect questions. Tier 1–2. Prerequisites: High school calculus. Learning goals: Interpret limits and derivatives through difference quotients; Apply power, product and quotient rules; Differentiate nested scalar functions by the chain rule; Build tangent-line approximations and interpret derivative units; Construct second-order Taylor polynomials at specified centers; Bound local truncation error using a stated derivative bound; Recognize nondifferentiability and limits of finite differences. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://openstax.org/books/calculus-volume-1/pages/3-3-differentiation-rules), [Source 2](https://openstax.org/books/calculus-volume-1/pages/3-6-the-chain-rule), [Source 3](https://openstax.org/books/calculus-volume-1/pages/4-2-linear-approximations-and-differentials), [Source 4](https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series).

mathematics

Oct 31, 2026

25 questions

Quiz

MML-05-07: Non-negative Matrix Factorization (NMF) & QR Factorization

30 worked single-select and multiselect questions. Tier 2–3. Prerequisites: QS 5.6. Learning goals: Interpret reduced QR dimensions and triangular solves; Compute small QR factorizations and least-squares coefficients; Diagnose rank and numerical-stability issues in QR; Formulate NMF with nonnegative data and factors; Distinguish joint nonconvexity from fixed-factor convexity; Compute Euclidean multiplicative updates with stated assumptions; Interpret additive components and factor nonuniqueness. Column gradients and output-by-input Jacobians are used where relevant. Detailed solutions explain every choice. References: [Source 1](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-02-23.html), [Source 2](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-02-23.html), [Source 3](https://proceedings.neurips.cc/paper_files/paper/2000/file/f9d1152547c0bde01830b7e8bd60024c-Paper.pdf).

mathematics

Oct 31, 2026

30 questions