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MML-15-08: t-SNE Mathematics: High-Dimensional Affinities, Student-t Kernels & Gradient Flow

Deconstruct the complete mathematical machinery of Stochastic Neighbor Embedding (SNE) and t-Distributed Stochastic Neighbor Embedding (t-SNE): asymmetric Gaussian conditional probabilities $$p_{j|i}$$, Shannon entropy and Perplexity-driven binary search for local bandwidths $$\sigma_i$$, symmetrized joint affinities $$p_{ij}$$, the high-dimensional crowding problem, heavy-tailed Cauchy / Student-$$t$$ kernels $$q_{ij}$$, and the analytical KL divergence gradient flow. Aligns with Prerequisites: QS 15.7, Part 11; Cognitive Level: Tier 2–4.

mathematics

Nov 1, 2026

35 questions

Quiz

MML-15-07: Non-linear Manifolds: Multidimensional Scaling (MDS), Isomap & Graph Laplacians

Master the mathematical foundations of non-linear manifold learning and spectral embedding: the manifold hypothesis, Classical Multidimensional Scaling (MDS) via double-centering of squared Euclidean distance matrices, Isomap shortest-path geodesic graph distances, and Spectral Graph Theory using the unnormalized and normalized Graph Laplacians ($$L = D - W$$, $$L_{\text{sym}}$$, $$L_{\text{rw}}$$) and Laplacian Eigenmaps. Aligns with Prerequisites: QS 15.6, Part 03; Cognitive Level: Tier 2–4.

mathematics

Nov 1, 2026

35 questions

Quiz

MML-16-11: Variational Inference & VAEs: Evidence Lower Bound (ELBO) & The Reparameterization Trick

Master the mathematical theory of Variational Inference (VI) and Variational Autoencoders (VAEs): intractable marginal likelihoods, reverse KL minimization, exact decomposition and Jensen derivation of the Evidence Lower Bound (ELBO), closed-form Gaussian KL divergence and gradients, score-function (REINFORCE) vs. pathwise Reparameterization Trick gradients, amortized inference, posterior collapse, beta-VAEs, and Importance Weighted Autoencoders (IWAE).

mathematics

Nov 1, 2026

45 questions

Quiz

MML-16-10: Monte Carlo Methods, Rejection Sampling, Importance Sampling & MCMC

Master the mathematical theory of Monte Carlo integration and Markov Chain Monte Carlo: dimension-free O(1/sqrt(S)) standard error vs. deterministic quadrature, Rejection Sampling envelopes and high-dimensional acceptance collapse, unnormalized and self-normalized Importance Sampling with optimal proposals and ESS, Markov chain ergodicity, Detailed Balance, Metropolis-Hastings acceptance ratios, and Gibbs Sampling.

mathematics

Nov 1, 2026

40 questions

Quiz

MML-16-09: PAC Learning, VC Dimension & Rademacher Complexity Bounds

Master the mathematical theory of PAC learning and capacity measures: realizable and agnostic finite-class sample complexity bounds, dichotomies, growth functions, shattering and exact VC dimensions of geometric classifiers, the Sauer-Shelah polynomial bound, and empirical/population Rademacher complexity with Massart and Ledoux-Talagrand contraction inequalities.

mathematics

Nov 1, 2026

30 questions

Quiz

MML-16-08: Generalization Theory: Empirical Risk Minimization, Bias-Variance Decomposition

Master the mathematical foundations of statistical learning and generalization: population vs. empirical risk minimization, optimism of training error, approximation-estimation excess risk decomposition, exact derivation of the Bias-Variance-Noise decomposition under squared loss, spectral variance blow-up at the interpolation threshold, and the modern Double Descent curve.

mathematics

Nov 1, 2026

35 questions

Quiz

MML-16-07: Loss Functions & Regularization: Weight Decay, Dropout Math & Label Smoothing

Master the mathematical theory and derivations of deep learning regularization: L2 weight decay and Hessian eigenspace shrinkage, Bernoulli and Inverted Dropout expectations/variances, adaptive Ridge equivalence, Dropout-BatchNorm variance shifts, and Label Smoothing KL decompositions and finite optimal logit bounds.

mathematics

Nov 1, 2026

35 questions

Quiz

MML-16-06: Self-Attention Mathematics: Scaled Dot-Product, Multi-Head Projections & Softmax Geometry

Master the linear algebra, probability simplex geometry, variance-scaling derivations, Softmax Jacobians, Multi-Head subspace decompositions, permutation equivariance, causal masking, and computational complexity of Self-Attention in Transformer architectures.

mathematics

Nov 1, 2026

45 questions

Quiz

MML-16-05: Normalization Mathematics: BatchNorm, LayerNorm & RMSNorm Formulas and Gradients

Master the mathematical foundations, tensor reductions, invariance theorems, and exact backward gradients of Batch Normalization (BatchNorm), Layer Normalization (LayerNorm), Group/Instance Normalization, and Root Mean Square Normalization (RMSNorm) in deep neural networks.

mathematics

Nov 1, 2026

40 questions

Quiz

MML-14-06: Bayesian Linear Regression: Weight Posterior & Predictive Distribution

40 worked single-select and multiselect questions on Bayesian Linear Regression: Weight Posterior & Predictive Distribution. Cognitive level: Tier 2–4. Prerequisites: QS 14.5, Part 09. Develop conceptual understanding, step-by-step computations, ML diagnostics and the assigned theoretical derivations. Each explanation analyzes every choice by its content. Learning objectives: Complete the square to obtain Gaussian weight posteriors. Combine prior and data precision in scalar and matrix examples. Calculate posterior predictive mean and observation variance. Separate epistemic uncertainty from future observation noise. Analyze sequential updating, information gains and rank deficiency. Relate MAP and posterior means to ridge under Gaussian assumptions. Derive evidence and correlated predictions. Diagnose conditioning assumptions and limits of plug-in prediction. Study references: [Deisenroth, Faisal & Ong, Mathematics for Machine Learning, Section 9.3, equations 9.43–9.61](https://mml-book.github.io/book/mml-book.pdf).

mathematics

Nov 1, 2026

40 questions

Quiz

MML-14-05: Regression Diagnostics: Residual Analysis, Coefficient of Determination, Condition Numbers and Multicollinearity

35 worked single-select and multiselect questions on Regression Diagnostics: Residual Analysis, $$R^2$$, Condition Numbers & Multicollinearity. Cognitive level: Tier 2–3. Prerequisites: QS 14.4. Develop conceptual understanding, step-by-step computations, ML diagnostics and the assigned theoretical derivations. Each explanation analyzes every choice by its content. Learning objectives: Calculate TSS, RSS, ESS and ordinary coefficient of determination. Apply adjusted model-fit measures with correct degrees of freedom. Interpret residual patterns and avoid unjustified diagnostic conclusions. Calculate and interpret VIF from auxiliary regressions. Relate singular values and condition numbers to coefficient sensitivity. Distinguish leverage, residual magnitude and influence. Separate training fit, prediction quality and causal interpretation. Study references: [James et al., An Introduction to Statistical Learning, second edition, Sections 3.1–3.3 and 6.2](https://www.stat.cmu.edu/~brian/valerie/617-2022/0%20-%20books/ISLRv2_website.pdf).

mathematics

Nov 1, 2026

35 questions

Quiz

MML-14-04: Lasso Regression with Absolute-Value Regularization: Sparsity Geometry, Laplace Prior and Subgradient Calculus

45 worked single-select and multiselect questions on Lasso Regression ($$L_1$$): Sparsity Geometry, Laplace Prior & Subgradient Calculus. Cognitive level: Tier 2–4. Prerequisites: QS 14.3, Part 03, Part 13. Develop conceptual understanding, step-by-step computations, ML diagnostics and the assigned theoretical derivations. Each explanation analyzes every choice by its content. Learning objectives: Use absolute-value subdifferentials in lasso optimality. Compute soft-thresholds and scaled coordinate updates. Explain sparsity geometry and constrained-penalized equivalence. Derive Laplace-prior MAP with correct normalization factors. Analyze correlated predictors, uniqueness and support selection. Calculate proximal updates and stopping checks. Connect standardization, intercept treatment and regularization paths. Derive dual correlations and shrinkage bias. Study references: [Tibshirani, Proximal Gradient Descent and Acceleration, Soft-thresholding and lasso examples](https://www.stat.cmu.edu/~ryantibs/convexopt-F15/lectures/08-prox-grad.pdf); [James et al., An Introduction to Statistical Learning, second edition, Sections 3.1–3.3 and 6.2](https://www.stat.cmu.edu/~brian/valerie/617-2022/0%20-%20books/ISLRv2_website.pdf).

mathematics

Nov 1, 2026

45 questions