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In 1940, one of the most brilliant mathematicians, G.H. Hardy, wrote in his essay, ‘A Mathematician’s Apology’: “I have never done anything ‘useful’. […] Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.”
Hardy was deeply disturbed by the notion that mathematics should have real life value and took great pride in the “uselessness” of his work. Since then, though, a lot has changed due to the rise of the digital computer, mathematics being one of them.
The modern mathematics used today is very different than the mathematics Hardy studied. Many modern mathematicians have shifted their focus from classic subjects, such as Number Theory and Complex Analysis, to more applicable fields such as Graph Theory, Algorithm Analysis, Machine Learning, and Deep learning.
The Craze That is Deep Learning
Deep Learning, the latest buzz in the applied mathematics world, is a branch of Machine Learning, which works on inputs and outputs of data. It has endless applications ranging from stock 
market prediction to medical treatment and makes the wonders of Facebook’s face recognition and Amazon’s recommendation system possible.
When a Deep Learning model is used, a “neural network” is built. A digital brain, so to speak, that we train to perform various tasks by using knowledge from previously solved problems.
For example: Think of a baby learning the difference between an apple and a watermelon or between a cat and a dog. The baby wasn’t born with pre-existing knowledge of the two, rather its parents told the baby, “this is a cat” or “this is a dog,” over and over again. This repetitive “training” is so effective that after a few times, the child will be able to recognize the cat or dog even without its parents present.
Deep Learning is, in essence, the creation and training of a brain using mathematical models.
From Deep Learning to Deep Mirroring
Using advanced Deep Learning techniques, we are able to learn the structural design of sample data for the purpose of generating more data. This makes it possible for the platform to look at rows and rows of data and generate a similar set. Not a replica, and not a copy, but rather, a mirroring. Hence the term, Deep Mirroring. Deep Mirroring has the ability to take small, independent sample data $latex X$, and create more data similar to $latex X$ in logic and rules.
One of the ways we achieve this is by following the assumption that the data we’re given is generated from some complicated unknown probability distribution $latex \Phi$.
Furthermore, we assume the independence of our data : $latex \Phi(x) = \Phi(x_1)\Phi(x_2)…\Phi(x_m) = \prod_{i=1}^{m}\Phi(x_i)$ This is called the Likelihood function.
We then model this unknown distribution by a weighted sum of k simpler, known distributions: $latex \Phi(x_i) = w_1 P_1(x_i) + … + w_k P_k(x_i) = \sum_{j=1}^{k}w_jP_j(x_i)$
So overall we have: $latex \Phi(X) = \prod_{i=1}^{m}\sum_{j=1}^{k}w_jP_j(x_i)$
If we find $latex w_j$, $latex P_j$ that maximizes $latex \Phi (X)$, we have effectively found our initial unknown distribution and from it can generate our own samples. How do we find such parameters that maximize the Likelihood? Well, that’s in the magic of Deep Learning.
The Power of a Nearly Exact Replica of Data
These data rich environments are one of the key aspects breaking down the barriers withholding enterprises from innovating. They’re what allow enterprises to test innovations with multiple startups at once without having to worry about security threats or data breaches.
Here’s to solving PoC pains and advancing startup-enterprise collaborations, one mathematical equation at a time.
* To read more of Oria’s writing on mathematical concepts, read his blog!
