Understanding Transformers Through the Lens of Kernel Methods
Attention mechanisms are kernel smoothers—transformers learn in reproducing kernel Hilbert spaces with adaptive, hierarchical kernels.
Why Adam Works: Adaptive Learning Rates Explained
How momentum and adaptive scaling combine to navigate the diverse optimization landscapes of deep learning
Neural Tangent Kernels: When Networks Behave Like Linear Models
Understanding the exact conditions where neural networks become kernel methods reveals both their power and their deeper mysteries
Why Residual Connections Enable Deep Networks
How skip connections transform gradient dynamics, optimization geometry, and the very meaning of network depth
The Mathematics of Dropout Regularization
How random masking performs approximate Bayesian inference and adapts regularization to weight influence.
Rademacher Complexity: Measuring Model Capacity
Why your model's true capacity depends on the data it sees, not just its parameter count
Why Batch Normalization Accelerates Training
Unraveling why stabilizing distributions matters less than smoothing optimization landscapes for faster neural network convergence
The Bias-Variance Tradeoff in Modern Deep Learning
Why overparameterized neural networks generalize despite classical theory predicting catastrophic overfitting at the interpolation threshold.
PAC Learning: When Machine Learning Has Guarantees
The mathematical framework that proves when learning is possible and reveals the fundamental limits no algorithm can overcome.
Understanding Backpropagation Through Automatic Differentiation
Backpropagation revealed as reverse-mode automatic differentiation exploiting computational graph structure for linear-time exact gradients through billions of parameters.
Why Gradient Descent Works: The Hidden Geometry of Optimization
Understanding the geometric structure that makes the simplest optimization algorithm succeed in billion-dimensional spaces
Vapnik's Margin Theory: The Geometry Behind SVMs
How Vapnik proved that geometric margin width, not dimension count, determines whether classifiers generalize—revolutionizing machine learning theory.
The Mathematical Core of Attention Mechanisms
How softmax-weighted averaging transformed sequence modeling by creating differentiable retrieval with learnable geometric structure and provable stability guarantees.
Why Neural Networks Learn Hierarchical Features
Mathematical frameworks reveal why depth creates hierarchy: compositional efficiency, feature reuse, and information compression converge on inevitable abstraction.