Math Foundations
Math foundation note는 AI paper와 implementation note를 읽는 데 필요한 최소한의 재사용 가능한 정의를 모읍니다. 목표는 완전한 수학 교과서를 만드는 것이 아니라 architecture, learning, generation, evaluation note의 수식을 읽기 쉽게 만드는 것입니다.
Most machine learning methods combine:
이 페이지들은 수식 뒤의 object, distribution, parameter, estimator를 식별하기 위한 입구입니다. 증명 자체보다 AI, Computational Biology, evaluation note를 읽는 데 필요한 정의와 해석을 우선합니다.
Route Map
| Area | Use For | Start |
|---|---|---|
| Linear algebra | vectors, matrices, projections, embeddings, rank, similarity | Linear algebra, Tensor shape notation, Vector norm and similarity |
| Spectral structure | representation analysis, PCA/SVD, low-rank approximations, stability | Eigenvalue and eigenvector, Singular value decomposition |
| Calculus | gradients, chain rule, Jacobians, Hessians, backpropagation notation | Calculus, Chain rule, Matrix calculus, Jacobian and Hessian |
| Constrained optimization | feasibility, penalties, Lagrangian views, projected updates | Constrained optimization |
| Geometry and symmetry | coordinate sets, rigid motions, equivariance, invariant targets | Geometry, Symmetry group |
| Probability | random variables, distributions, expectations, normal assumptions | Random variable, Probability distribution, Expectation, Normal distribution |
| Estimation and statistics | finite-sample estimates, uncertainty, hypothesis checks, bias/variance | Estimator vs metric, Statistical estimator, Monte Carlo estimation, Central limit theorem, Hypothesis testing, Bias-variance tradeoff |
| Dependence | covariance, correlation, representation diagnostics, label relationships | Covariance and correlation |
| Likelihood and information | MLE, entropy, KL, cross-entropy, Bayesian updates | Maximum likelihood, Entropy and KL divergence, Bayes rule, Bayesian inference |
Checks
- What object is the formula about: scalar, vector, matrix, graph, distribution, or function?
- What does each tensor axis mean, and which axis is reduced, mixed, normalized, or broadcast?
- Is a derivative with respect to inputs, parameters, coordinates, or time?
- Is the quantity optimized, estimated, sampled, normalized, or measured?
- Is a probability statement conditional or marginal?
- Is a loss a negative log-likelihood, distance, ranking loss, or surrogate?
- Are expectations taken under data, model, sampling, or deployment distributions?
- If the object has coordinates, what transformations should preserve meaning?
Formula Reading Template
Concept page를 추가하거나 확장할 때는 quantity를 직접 정의하는 수식을 우선합니다.
- Define symbols before using them.
- Separate population quantities from finite-sample estimates.
- State the distribution behind each expectation.
- State whether a matrix/vector shape matters.
- State tensor shape and axis semantics when the operation depends on batch, token, graph, coordinate, head, or candidate axes.
- Name the variable being optimized or differentiated.
- Link the formula to the modeling decision it supports.
Example:
이 값은 test risk의 empirical estimate입니다. Population risk 자체와는 다른 object입니다.