➩ Clustering: separates items into groups.

➩ Novelty (outlier) detection: finds items that are different (two groups).

➩ Dimensionality reduction: represents each item by a lower dimensional feature vector while maintaining key characteristics.

➩ Google news.

➩ Google photo.

➩ Image segmentation.

➩ Text processing.

➩ Data visualization.

➩ Efficient storage.

➩ Noise removal.

➩ The number of features of bag of words representation is the size of the vocabulary.

➩ The number of features of pixel intensity features is the number of pixels of the images.

➩ Find the direction of the greatest variability, \(u_{1}\).

➩ Find the direction of the greatest variability that is orthogonal (perpendicular) to \(u_{1}\), say \(u_{2}\).

➩ Repeat until there are \(K\) such directions \(u_{1}, u_{2}, ..., u_{K}\).

➩ If \(\mu \neq 0\), then \(X\) should be centered, that is, the mean of each column should be subtracted from each column.

➩ This constrained maximization problem has solution (local maxima) \(u_{k}\) that satisfies \(\Sigma u_{k} = \lambda u_{k}\), and by definition of eigenvalues, \(u_{k}\) is the eigenvector corresponding to the eigenvalue \(\lambda\) for the matrix \(\Sigma\): Wikipedia.

➩ At a solution, \(u^\top_{k} \Sigma u_{k} = u^\top_{k} \lambda u_{k} = \lambda u^\top_{k} u_{k} = \lambda\), which means, the larger the \(\lambda\), the larger the variability in the direction of \(u_{k}\).

➩ Therefore, if all eigenvalues of \(\Sigma\) are computed and sorted \(\lambda_{1} \geq \lambda_{2} \geq ... \geq \lambda_{m}\), then the corresponding eigenvectors are the principal components: \(u_{1}\) is the first principal component corresponding to the direction of the largest variability; \(u_{2}\) is the second principal component corresponding to the direction of the second largest variability orthogonal to \(u_{1}\), ...

➩ \(K\) can be selected based on prior knowledge or requirement (for example, \(K = 2, 3\) for visualization tasks).

➩ \(K\) can be the number of non-zero eigenvalues.

➩ \(K\) can be the number of eigenvalues that are larger than some threshold.

➩ Eigenfaces are eigenvectors of face images: every face can be written as a linear combination of eigenfaces. The first \(K\) eigenfaces and their coefficients can be used to determine and reconstruct specific faces: Link, Wikipedia.

➩ Dual optimization techniques can be used for kernels such as the RBF kernel which produces infinite number of new features.

➩ The hidden units (\(K < m\) units, fewer than the number of input features) can be considered a lower dimensional representation of the original features.

➩ The generator (deconvolutional neural network): input is noise, output is fake images (or text).

➩ The discriminator (convolutional neural network): input the fake or real image, output is binary class whether the image is real.

➩ The two networks are trained together, or competing in a zero-sum game: the generator tries to maximize the discriminator loss, and the discriminator tries to minimize the same loss: Wikipedia.

➩ The solution to the zero-sum game, called the Nash equilibrium, is the solution to \(\displaystyle\min_{w^{\left(G\right)}} \displaystyle\max_{w^{\left(D\right)}} C\left(w^{\left(G\right)}, w^{\left(D\right)}\right)\), where \(C\) is the loss of the discriminator.

➩ \(a^{\left(h\right)}_{t} = g\left(w^{\left(x\right)} \cdot x_{t} + b^{\left(x\right)}\right)\) is not recurrent.

➩ There are value units \(a^{\left(v\right)}_{t} = w^{\left(v\right)} \cdot a^{\left(h\right)}_{t}\), key units \(a^{\left(k\right)}_{t} = w^{\left(k\right)} \cdot a^{\left(h\right)}_{t}\), and query units \(a^{\left(q\right)}_{t} = w^{\left(q\right)} \cdot a^{\left(h\right)}_{t}\), and attention context can be computed as \(g\left(\dfrac{a^{\left(q\right)}_{s} \cdot a^{\left(k\right)}_{t}}{\sqrt{m}}\right) \cdot a^{\left(v\right)}_{t}\) where \(g\) is the softmax activation: here \(a^{\left(q\right)}_{s}\) represents the first word, \(a^{\left(k\right)}_{t}\) represents the second word, \(a^{\left(q\right)}_{s} \cdot a^{\left(k\right)}_{t}\) is the dot product (which represents the cosine of the angle between the two words, i.e. how similar or related the two words are), and \(a^{\left(v\right)}_{t}\) is the value of the second word to send to the next layer.

➩ The attention matrix is usually masked so that a unit \(a_{t}\) cannot pay attention to another unit in the future \(a_{t+1}, a_{t+2}, a_{t+3}, ...\) by making the \(a^{\left(q\right)}_{s} \cdot a^{\left(k\right)}_{t} = -\infty\) when \(s \geq t\) so that \(e^{a^{\left(q\right)}_{s} \cdot a^{\left(k\right)}_{t}} = 0\) when \(s \geq t\).

➩ There can be multiple parallel attention units called multi-head attention.

➩ Fine tuning for pre-trained models freezes a subset of the weights (usually in earlier layers) and updates the other weights (usually the later layers) based on new datasets: Wikipedia.

➩ Prompt engineering does not alter the weights, and only provides more context (in the form of examples): Link, Wikipedia.

➩ Reinforcement learning from human feedback (RLHF) uses reinforcement learning techniques to update the model based on human feedback (in the form of rewards, and the model optimizes the reward instead of some form of costs): Wikipedia.