Vector Calculus - Deepstash

Vector Calculus

Calculus is concerned with a perpetual change that consists of functions and limits. Vector calculus involves the differentiation and integration of the vector fields.

Useful terms:

  • Derivative is a function of real numbers that measure the change of the function (output) value concerning a change in its argument (input value)
  • Differentiation is the action of calculating a derivative.
  • Gradient is another word for derivative and is used for functions with several inputs and one output.

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Optimisation

Training a machine learning model is about finding a good set of parameters. The best value is the minimum value.

  • Global minima: The point where a function best values takes the minimum value. When the aim is to minimise the function and solve it using optimisation algorithms, the function could have a minimum value at various points. It is called local minima.
  • Unconstrained optimisation is where we see the minimum of a function.
  • Constrained optimisation introduces a set of constraints to limit possible value.

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  • Math enables you to select the right machine learning algorithm. It gives insight into how the model works, including selecting the right model parameter and validation strategies.
  • Maths helps with creating the right confidence interval and uncertainty measurements with the model.
  • Maths is needed to understand aspects such as metrics, training time, model complexity, number of parameters, and number of features.
  • By knowing the machine learning model's math, you could develop a customised model.

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Matrix decomposition is about how to reduce a matrix into its constituent parts. It tries to simplify complex matrix operations on the decomposed matrix instead of the original matrix.

There are many ways to decompose a matrix using a range of different techniques.

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Linear Algebra

Linear Algebra in machine learning is a systematic representation of data that computers can understand.

  • It is the part of mathematics that uses vector space and matrices to represent linear equations.
  • Vectors are special objects that can be added together and multiplied by scalars to produce another object of the same kind.
  • A matrix can be thought of as a group of column vectors or row vectors.

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Analytic Geometry (Coordinate Geometry)

Analytic geometry is concerned with defining and representing geometrical shapes numerically. It extracts numerical information from the shapes numerical definitions and representations.

Important terms that will help you start learning this subject:

  • Distance Function: It is a function that gives numerical information for the distance between the elements of a set.
  • Inner Product: This term introduces natural geometrical concepts, such as the length of a vector and the angle or distance between two vectors.

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Machine Learning is a tool that Data scientists use to obtain valuable patterns. Learning the math behind machine learning could give you an edge.

These six math subjects are the foundation for machine learning.

  1. Linear Algebra
  2. Analytic Geometry
  3. Matrix Decomposition
  4. Vector Calculus
  5. Probability & Distributions
  6. Optimisation

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Probability is the study of randomness. Probability distribution is a function that measures the probability of a specific outcome associated with the random variable.

Probability theory and statistics are about different aspects of uncertainty.

  • In statistics, we use probability to try and find the underlying process of something that has happened and strive to explain the observations.
  • Machine learning is similar to statistics. It makes a model that represents the process that generates the data.

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Statistics is using math to do technical analysis of data. Instead of guesstimating, data helps us get concrete and factual information.

The most widely used statistical concept in data science is called Statistical Features. It includes important measurements like bias, variance, mean, median and percentiles. It’s all code-friendly too.

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Matrices simplified

Imagine yourself walking into a pizzeria to buy a delicious pizza.

A pizzeria sells many kinds of pizzas.

and serves for each kind three different sizes (small-medium-large)

To remember all these data , the owner arranged kinds , sizes and prices in a respective way in a table (or The Menu) .

In this case, the menu is our matrix.

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The biggest problem, thoug h, is that models like this one are performed only on a single task. Future tasks require a new set of data points as well as equal or more amount of resources.

Transfer learning is an approach in deep learning (and machine learning) where knowledge is transferred from one model to another.

Deep learning models require a LOT of data for solving a task effectively. However, it is not often the case that so much data is available. For example, a company may wish to build a very specific spam filter to its internal communication system but does not possess lots of labelled data.

What you can do is using a pre-trained image classifier on dog photos to predict cat photos.

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