All Writings

High-dimensional feature selection and ill-posed inverse problems: The geometry of L1 diamond vs L2 spherical constraint manifolds, proximal gradient operators, and soft-thresholding.

Transcending linear separability: The mathematical foundations of Reproducing Kernel Hilbert Spaces (RKHS), Mercer’s positive semi-definite theorem, and implicit infinite-dimensional embeddings.

The definitive maximum-margin classifier: Vladimir Vapnik’s convex quadratic programming formulation, Karush-Kuhn-Tucker conditions, and the emergence of sparse support vector representations.

The bridge between classical statistics and linear classification: Ronald Fisher’s scatter ratio maximization, probabilistic generative vs discriminative models, and exponential family link...

From full-batch gradient descent to stochastic approximations: Polyak heavy-ball momentum, Nesterov acceleration, variance reduction, and the analytical derivation of Adam adaptive moments.

The algorithmic engine of machine learning: Convexity, Lipschitz smoothness, strong convexity, condition number geometry, and fundamental convergence rate proofs for first-order optimization.

The classic dichotomy of estimation error: Deriving the analytical decomposition of Expected Prediction Error into squared bias, variance, and irreducible noise, and...

Moving beyond finite spaces: Vladimir Vapnik and Alexey Chervonenkis’s dimension theory, combinatorial shattering, Sauer’s lemma, and distribution-free generalization bounds for continuous parameter...

The formal mathematical foundation of computational learnability: Leslie Valiant’s Probably Approximately Correct (PAC) framework, epsilon-delta guarantees, and sample complexity bounds under finite...

The foundational mathematical formulation of learning: how minimizing empirical loss over finite samples relates to true expected risk, why unconstrained hypothesis spaces...