# Learning Representations ## Linear Classifiers and Limitations A `Linear Classifier` can be defined as an ***hyperplane** with the following expression: $$ \sum_{i=1}^{N} w_{i}x_{i} + b = 0 $$ So, the output is $o = sign \left( \sum_{i=1}^{N} w_{i}x_{i} + b = 0 \right)$ ### Limitations The question is, how probable is to divide $P$ `points` in $N$ `dimensions`? >*The probability that a dichotomy over $P$ points in $N$ dimensions is* ***linearly separable*** >*goes to zero as $P$ gets larger than $N$* [Cover’s theorem 1966] So, once you have the number of features $d$, the number of dimensions is **exponentially higher** $N^{d}$. ## Deep vs Shallow Networks While it is **theoretically possible** to have only `shallow-networks` as **universal predictors**, this is **not technically possible** as it would require more **hardware**. Instead `deep-networks` trade `time` for `space`, taking longer, but requiring **less hardware** ## Invariant Feature Learning When we need to learn something that may have varying features (**backgrounds**, **colors**, **shapes**, etc...), it is useful to do these steps: 1. Embed data into **high-dimensional** spaces 2. Bring **closer** data that are **similar** and **reduce-dimensions** ## Sparse Non-Linear Expansion In this case, we break our datas, and then we aggreate things together ## Manifold Hypothesis We can assume that whatever we are trying to represent **doesn't need that much features** but rather **is a point of a latent space with a lower count of dimensions**. Thus, our goal is to find this **low dimensional latent-space** and **disentangle** features, so that each `direction` of our **latent-space** is a `feature`. Essentially, what we want to achieve with `Deep-Learning` is a **system** capable of *learning* this **latent-space** on ***its own***.