# Big Idea - Representation Below I am linking all examples of different ways of representing things: * [Parametric-Curves](Parametric-Curves.md) * [Change of variables](Integrals.md#Change%20of%20Variables) * [Polar coordinates for certain integrals](Integration-Difficulty.md#Answer%201). * [Using different basis to represent a vector](Abstract%20Vector%20Spaces.md) * [Representing a function in frequency space via the fourier transform](Fourier-Series.md) * [Algorithms and Representation](Algorithms%20and%20Representation.md) #### Consider adding... * Adjacency matrix vs graph * pattern space with a metric? Derivative? (you probably need to figure this out, then you can do this) * eigen-something? * Boolean one hot vs continuous/categorical counterpart? * Different types of means * complex numbers * [Data Matrices and Linear Transformation](Data%20Matrices%20and%20Linear%20Transformation.md) * [Does Linearity provide Information](Does%20Linearity%20provide%20Information.md) * [representing data as a DAG](Semantic%20Fabric%20Theory%20Questions.md) * probably many other unsupervised things... * [Univarate-model-vs-Linear-transformation](Univarate-model-vs-Linear-transformation.md) * exponential functions & $e$ * Taylor series (power series), and the role in integration. See newton representing circle via Taylor series, in infinite powers by strogatz, pg 191 * What does a language have to do with this, what does it allow? Deep down what does structure mean? What “objects” are we working with…? * Derivatives and calculus, how unbelievably linked to xy plane are they? Analytic geometry, algebra were essential to creation of calculus. See strogatz page 194 * Newton’s approach vs liebnez approach, chapter 7,8 infinite powers * Fourier, series, and heat equation - pg 251, infinite powers * His problem was incredibly hard to solve, he imagined a situation/world/model in which it would be easier (if he could represent temperature as a bunch of sine waves), then his problem was could he represent temperature as a bunch of sine waves? * Data structures are specifically a form of representation - see 5.4 war story algorithm design manual * roman numerals vs decimal * SVM Dual (optimization theory primal vs. dual) * Greek numbering system on allowed recording not calculation (footnote I against the gods) * Linear maps and matrices are the same object, just different representations (see pg 155 Jkun text) #### Points to consider * You are frequently exploiting information that was not captured in one description * You may be making additional assumptions * Look at convo with JW and Matt G in information in linear transform * post on things to think about * Think about how this also relates to algorithms [Algorithms and Representation](Algorithms%20and%20Representation.md). For instance, algorithms seem to be a way in which we use a heuristic and some how exploit the most valuable information about the process that we are interested in. Probability plays into this in an even more interesting way. * You think a lot about representation, finding mechanical processes that some how exploit structure of reality to find an answer, etc * Strogatz has some interesting points here. See chapter 1 of infinity book * One way to think of it is that mathematics is allowing us to build a scaffolding/framework to mechanically transform representation in an algorithmic, repeatable process. It gives us *tools* that can be used to more easily transform our representation. We fundamentally are operating in a high (infinite?) dimensional information space. In order to reduce the complexity confronting us, mathematics provides a way to almost "bootstrap" our way in. * ### Interesting thoughts from Carlos Perez > There is of course a question whether we should explore different kinds of Representations. I mean this at a more general level. In Deep Learning, there are all kinds of different neural embeddings. These embeddings are vector representations of semantics. These vector spaces are learned over the course of training. These vectors are supposed to represent an invariant form of the actual concept. One major difficulty of Deep Learning systems is that these representations are extremely entangled. The fact that they are entangled can explain why Deep Learning systems have zero conceptual understanding of what they predict. Understanding requires the creation of concepts, if concepts cannot be factored out, then what does that imply for understanding? It is important to realize, that there are many cases where understanding is not needed for competence. See more [here](https://medium.com/intuitionmachine/exploration-exploitation-and-imperfect-representation-in-deep-learning-9472b67fdecd). --- Date: 20210524 Links to: [Mathematics MOC](Mathematics%20MOC.md) References: * []()