A chef has the same ability to understand multidimensional concepts as a professor of mathematics. n-dimensional concepts are so fundamental to artificial intelligence that I believe before diving into any maths it is important to first understand them from an everyday perspective. After all our brains handle these concepts constantly when we are awake and in our sleep.
We are so used to being told that we live in a 3-dimensional space (or 4-dimensional universe if we consider time), that we become blind to the concept of all the other dimensions we constantly use to mentally describe the world we perceive. So lets start from the 3-D and move up.
Let’s conduct a “thought experiment” : imagine the house you live in, and use sat-nav to describe its location. With GPS we have longitude and latitude coordinates. With a military grade GPS system we can add in height above sea level. So instantly we have located your house in three dimensions (longitude, latitude and altitude). Now let’s assume we agree to meet there at 6pm on the 24th of December. We have described a point in the four dimensions of space time.
In our thought experiment, let’s go back to three dimensions, but add some complexity. We have the location, now how wide is your house, how deep is it, how tall is it, and which way does it face? At this point we are still in three dimensions, but we need more than three numbers to describe our model. We have migrated from an array of three numbers to a matrix to describe our model. Finally our maths professors are starting to take interest, but we really don’t need the maths, your brain and imagination have it all under control.
Now I want you to mentally enter your house and stand in the living room, face the window:
How bright is it?
Turn around, how bright is it now?
We have added an extra dimension brightness, and that is dependent on which way we are facing. A matrix is no-longer enough, we need a mathematical concept known as a tensor to hold the functions to connect our dimensions. Now lets add time back in:How bright is your living room at different times of the day?
Now lets add a few more functions:How does each room smell?
How do they feel on your bare feet?
How do they sound as you walk around?
The mathematics of describing your house has become n-dimensionally complex with functions that depend or where you are and which way you are facing to join the dots. Our senses, touch, smell and sound added extra dimensions, but nevertheless we can walk around this multidimensional construct in our imagination with relative ease.
The tensor based mathematics of describing your house are actually pretty complex, but our brains handle this all day and everyday without us giving it a second thought.
One of the core concepts and secrets to success in modern day Neural Networks is to create the multidimensional model and then map that into our neural network. For example if brightness is our desired output then we enter position and orientation. e.g. in the living room facing the window at midday. Hopefully your brain has given you an answer to the expected brightness, proving the point that you handle multi-dimensions without giving it too much thought.
Now lets bring the conversation back to our chef, combining ingredients and flavours at different times in the cooking process and at different temperatures. To describe that mathematically we might use the mathematical concept of tensors. To create an AI Robot chef we would transfer our test data via tensors into a neural network. Our Robot AI chef uses relatively complex mathematics to generate recipes and AI based training to learn what we like. Our chef does much the same just without the maths, and the added advantage of having their own taste buds!!!