A Map of the Mathematical Chassis
| Concept | Common roles |
|---|---|
| Vectors, matrices, and tensors | Represent data, parameters, activations, and linear transformations |
| Dot products | Similarity scores, attention logits, and linear heads |
| Eigenvalues and singular values | PCA, spectral methods, conditioning, and low-rank analysis |
| Logarithms | Likelihoods, information measures, and stable probability computation |
| Probability | Uncertainty models, objectives, calibration, and decoding |
| Normal distributions | Selected noise models, initialization analysis, and statistical approximations |
| Losses and evaluation | Define optimization targets and measure behavior on relevant data |
| Derivatives and the chain rule | Compute local sensitivity through a graph |
| Backpropagation and optimizers | Obtain gradients efficiently and update parameters |
This is a foundation, not the entire mathematics of modern AI. Information theory, numerical linear algebra, optimization, discrete algorithms, dynamical systems, causal inference, and domain-specific mathematics may all become important.
What the Foundation Lets You Ask
- Do the tensor shapes and axes match the intended operation?
- Is a loss numerically stable and aligned with the desired behavior?
- Does an evaluation set represent deployment, or has the process adapted to it?
- Are gradients absent because of saturation, detachment, scale, or architecture?
- Is a normal or independence assumption supported by the data-generating process?
These questions guide diagnosis; they do not make any one remedy automatic. ReLU, normalization, regularization, and residual connections solve different failure modes under different conditions.
Recognizing the chassis is the beginning of technical judgment. Name the assumption, operation, and failure mode before selecting a fix.
Track Reward
A forward pass computes the current model, a loss expresses an objective, backpropagation computes sensitivities, and an optimizer proposes an update. That loop is central to gradient-trained models, while the architecture, data, distributed system, and evaluation protocol determine what the loop actually produces.
인공지능은 마법이 아니라 수학이라는 레고 블럭을 쌓고 틀릴때마다 고쳐가는 끈기 있는 연습벌레이다. 리스펙!