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+# Learning Representations
+
+## Linear Classifiers and Limitations
+
+A `Linear Classifier` can be defined as an ***hyperplane** with the following expression:
+
+$$
+\sum_{i=1}^{N} w_{i}x_{i} + b = 0
+$$
+<!-- TODO: Add images -->
+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***.