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Study Support Vector Regression (SVR) & $$\epsilon$$-Insensitive Loss. 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.
Nov 1, 2026
Study Popular Kernels: Linear, Polynomial, Gaussian RBF & Sigmoid Kernels. 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.
Nov 1, 2026
Study Mercer's Theorem, Positive Definite Kernels & Reproducing Kernel Hilbert Spaces (RKHS). 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.
Nov 1, 2026
Study Soft-Margin SVM: Slack Variables ($$\xi$$), Penalty Parameter $$C$$ & Hinge Loss. 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.
Nov 1, 2026
Study Dual Formulation of Hard-Margin SVM & Support Vector Geometry. 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.
Nov 1, 2026
Study Separating Hyperplanes, Geometric Margins & Hard-Margin Primal SVM. 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.
Nov 1, 2026
Study The Expectation-Maximization (EM) Algorithm: ELBO Derivation & Monotonic Convergence. 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.
Nov 1, 2026
Master the probability theory and variance propagation derivations behind deep neural network weight initialization: symmetry breaking failure of constant/zero weights, forward activation variance $$\operatorname{Var}(z) = n_{\text{in}}\operatorname{Var}(w)\operatorname{Var}(x)$$, backward gradient variance scaling, Glorot/Xavier normal and uniform bounds for symmetric activations, why Xavier collapses by $$2^{-L}$$ on ReLU, and He/Kaiming initialization derivations for ReLU and Leaky ReLU. Aligns with Prerequisites: QS 16.3, Part 08; Cognitive Level: Tier 2–4.
Nov 1, 2026
Explore the theoretical foundations of neural network expressive capacity: Cybenko (1989) Universal Approximation Theorem for sigmoidal networks, Hornik (1991) and Leshno-Lin-Pinkus-Schocken (1993) non-polynomial characterization, constructive bump/hat function approximations, Barron spectral bounds, width-bounded deep ReLU universality, and Telgarsky depth separation proofs demonstrating exponential width requirements for shallow networks. Aligns with Prerequisites: QS 16.2; Cognitive Level: Tier 3–4.
Nov 1, 2026
Analyze the calculus, geometry, and optimization diagnostics of neural network activation functions (Sigmoid, Tanh, ReLU, Leaky ReLU, PReLU, ELU, Softplus, Swish/SiLU, and GELU): exact analytical derivatives, second derivatives, saturation bounds, vanishing gradients, non-zero-centered gradient zig-zagging, subdifferentials at derivative singularities, Dying ReLU mechanics, and non-monotonic smooth gating in modern Transformers. Aligns with Prerequisites: QS 16.1; Cognitive Level: Tier 2–3.
Nov 1, 2026
Develop rigorous mathematical and geometric mastery of feedforward neural network architectures: multi-layer functional composition $$f(\mathbf{x}) = (f_L \circ \dots \circ f_1)(\mathbf{x})$$, matrix-vector and batched affine transformations, exact parameter counting, algebraic proofs of linear collapse in the absence of non-linearities, rank bottlenecks, coordinate warping for the non-separable XOR problem, and hyperplane arrangements partitioning input space into convex polytopes. Aligns with Prerequisites: Parts 02, 06; Cognitive Level: Tier 1–3.
Nov 1, 2026
Master Uniform Manifold Approximation and Projection (UMAP) from first principles: local Riemannian metric normalization, nearest-neighbor distance offset $$\rho_i$$ for local connectivity, fuzzy simplicial set membership $$p_{j|i}$$, probabilistic t-conorm (fuzzy union) symmetrization $$p_{ij} = p_{i|j} + p_{j|i} - p_{i|j}p_{j|i}$$, smooth low-dimensional curve fitting $$q_{ij} = (1 + a\|\mathbf{y}_i - \mathbf{y}_j\|^{2b})^{-1}$$, and the Fuzzy Set Cross-Entropy objective. Aligns with Prerequisites: QS 15.8; Cognitive Level: Tier 2–4.
Nov 1, 2026
