Completeness contract
This article follows all fifteen pages of 4_CH4-Backgrounds for AI(1).pdf. Ledgers cover numbered and unnumbered displayed equations, meaningful inline mathematics, figure equations and numbers, ?, ??, blanks, every table cell, captions, footnotes, references, and legible handwriting. Source blanks and notation issues remain visible; completions and corrections receive separate statuses.

Keep three layers distinct.
Separate the 2025 PDF reconstruction, editorial and mathematical verification, and the research update dated August 18, 2026. Later research does not retroactively rewrite the lecture’s claims.

#Ledger status

• PDF mapping: 15 of 15 pages
• Total equation records: 163 (MAI-P3-001–MAI-P3-163)
• Source equations: 140
• Displayed-equation explanation cards: 99
• Equation statuses: source-exact, source-suspect, editorially-completed, corrected-variant
• Source PDF SHA-256: 2a3855c52d688a79d38e714ab7ef7223dd7a5bc5b46f8385f55da777aa1c78af
• New SVG/HTML diagrams preserve figure structures, numbers, and relationships without reproducing original images.

#Reconstructing the PDF source

#3. Backgrounds for AI and deep learning

The chapter begins with a single perceptron, then develops multilayer perceptrons (MLPs), convolutional neural networks (CNNs), global/local matrix representations of convolution, and core PyTorch modules.

#3.1 A single perceptron model

A perceptron mathematically simplifies a biological neuron. The lecture relates dendrites, synapses, and firing thresholds to numerical inputs, weights, weighted sums, and activation functions.

#3.1.1 Linear discriminant

One perceptron learns a linear boundary between two linearly separable classes. A 2D line is $y=ax+b$, where coefficient b, not multiplying a coordinate, is the bias.

An N-dimensional point has coordinates $x_1,x_2,\ldots,x_N$, and boundary coefficients are $w_1,w_2,\ldots,w_N,b$.

Rearranging $y=ax+c$ gives $y-ax-c=0$. A more general two-variable linear boundary takes this form.

The source asks how to rewrite y=ax+c as w₁x₁+w₂x₂+b=0 and leaves ?. The question is preserved, with its completed answer in the editorial layer.

For N-dimensional features, classification uses this hyperplane.

Vector notation is

The classifier applies a sign decision rule to linear score g.

Its activation is the threshold-type sign function.

• The boundary is perpendicular to the weight vector. The PDF writes distance from the origin as $\frac{b}{\lVert\mathbf{w}\rVert_2}$.

• The PDF writes distance from point x to the boundary as $\frac{g(\mathbf{x})}{\lVert\mathbf{w}\rVert_2}$.

• On the boundary, $g(\mathbf{x})=0$; on the positive side indicated by w, $g(\mathbf{x})>0$; on the opposite side, $g(\mathbf{x})<0$.

Let the linearly separable training set be $\mathcal{D}=\{(\mathbf{x}_1,y_1),\ldots,(\mathbf{x}_M,y_M)\}$. Correct classification of every sample expands classwise as follows.

This compresses into a zero-empirical-risk condition.

#3.1.2 Perceptron learning by gradient descent

The initial learning model starts from a necessary and sufficient condition for zero empirical risk.

The lecture first constructs reward maximization by increasing the sum of all signed scores.

It then writes the equivalent loss-minimization problem.

Lecture annotation — PDF p. 2
The sketch shows w perpendicular to the boundary, with perpendicular distance normalized by ‖w‖. The notation No training error appears beside the signed-margin condition. Other small notes lack sufficient context and remain uncertain to read.

Applying gradient descent to equation (1) requires derivatives with respect to w and b. The source leaves both as ??.

It then gives these two updates.

The iteration index is $i\ge 0$; initial values $\mathbf{w}^{(0)},\ b^{(0)}$ are chosen arbitrarily.

• Linear separability ensures at least one pair (w,b) satisfying $\mathcal{L}(\mathbf{w},b)<0$.

• The lecture explains that any solution attaining an equal or smaller value suffices for this gradient-descent objective.

But equation (1) is unbounded below. Once loss is negative, positive scaling of w or b can yield $\mathcal{L}(\mathbf{w},b)\to-\infty$, making the objective numerically unsuitable.

The modified model focuses only on misclassified points, defining an error set each iteration.

Summing loss only over that set gives equation (2).

• Signed scores in E are negative, so $\mathcal{L}_{\mathcal{E}}(\mathbf{w},b)\ge0$.

• Since $\mathcal{L}_{\mathcal{E}}(\mathbf{w},b)=0\Longleftrightarrow\mathcal{E}=\varnothing$, zero loss means every sample is correctly classified.

Gradient updates likewise use only samples in the error set.

Lecture annotation — PDF p. 3
Handwritten -Σ yₘxₘ and -Σ yₘ appear beside the two ?? placeholders. A lower note connects loss and objective functions as descriptions of the same optimization goal. Some Korean handwriting is blurred and remains uncertain to read.

An alternative learns by minimizing MSE between desired and actual linear responses.

Here, $y_m\in\{+1,-1\}$, the augmented parameters are $\widetilde{\mathbf{w}}\triangleq[\mathbf{w}^{\mathsf T},b]^{\mathsf T}$, and the augmented input is $\widetilde{\mathbf{x}}_m\triangleq[\mathbf{x}_m^{\mathsf T},1]^{\mathsf T}$. The source states $\mathcal{L}_{\mathrm{MSE}}(\widetilde{\mathbf{w}})>0$.

The source leaves this MSE’s gradient as ??.

Its LMS-style update accumulates residuals multiplied by inputs.

The lecture states $0<\alpha<2$ is necessary but insufficient for convergence and gives typical range $0.1<\alpha<1$. The exact data-dependent upper bound is checked separately in the editorial layer.

#3.2 Nonlinear discriminants: FCNs and MLPs

#3.2.1 Limits of a single perceptron

XOR cannot be solved by one linear boundary. The source places false-class points at $S_1=\{(0,0),(1,1)\}$ and true-class points at $S_2=\{(1,0),(0,1)\}$.

Geometry shows no single separating line. The source proposes two linear boundaries or a nonlinear boundary such as a quadratic.

Lecture annotation — PDF p. 4
A sketch stacks w with b, and x with constant 1, into augmented vectors. Beside the MSE blank is derive the gradient directly; beside the learning rate are lower bound / upper bound. Remaining small writing is uncertain to read.

The source asks readers to sketch the geometry of both proposed solutions and leaves the answer blank.

It then asks whether multiple perceptrons can solve XOR and, if so, the minimum number of units.

#3.2.2 MLP overview

Figure 2 depicts a general multilayer network. The lecture defines:

• A column of nodes forms a layer.

• Every node is an artificial neuron.

• A network is fully connected when each node’s output connects to every node in the next layer.

• A network without loops is feedforward.

• Neuron k’s output or activation in layer l is $a_k(l)$.

The goal is classifying unknown input-image patterns. A network with N_L outputs selects the class with greatest output activation.

#3.2.3 Forward pass

A forward pass maps input x to output-layer values used for classification. Equation (4) gives neuron i’s scalar net input and activation.

Output nodes are $a_i(L)=h(z_i(L)),\quad i=1,\ldots,N_L$ and the input layer is initialized with $a_j(1)=x_j,\quad j=1,\ldots,N_1,\qquad N_1=N$.

A footnote allows softmax instead of sigmoid in the final layer. Its activations sum to one and permit probabilistic interpretation.

Lecture annotation — PDF p. 5
Near XOR and forward-pass equations are MLP and arrows indicating activations. Most short symbols and English notes remain uncertain to read because of resolution and overlapping handwriting.

Collecting equation (4) into vector/matrix operations gives the activation/net-input relationship in equation (5).

Equation (6) defines layer l’s weight matrix, but every source entry remains ??.