Examples of linear algebra on machine learning

• Linear regression 
• Regularization 
• Principal component analysis (PCA)
• Singular-value decomposition (SVD)
• Deep learning. 

• Linear Regression - It is an old method form statistics for describing the relationships between variables. It is often used in machine learning for predicting numerical values in simpler regression problems. There are many ways to describe and solve the linear regression problem, i. e. finding a set of coefficients that when multiplied by each of the input variables and added together results in the best prediction of the output variables. If you have used a machine learning tool or library, the most common way of solving linear regression is via a least-squares optimization that is solved using matrix factorization methods from linear regression, such as an LU decomposition or a singular-value decomposition or SVD. Even the common way of summarizing the linear regression equation uses linear algebra notation

                                                                  x=B.

         where x is the output variable B is the dataset and a is the model coefficients.

• Regularization - In applied machine learning, we often seek that achieve the best skill in our problem. Simpler models are often better at generalizing from specific examples to unseen data. In many methods that involve coefficients, such as regression methods and artificial neural networks, simpler models are often characterized by models that have smaller coefficient values. A technique that is often used to encourage a model to minimize the size of coefficients while if is being fit on data is called regularization. Common implementations include the L2 and L1 forms of regularization. Both of these forms of regularization are in fact a measure of the magnitude of length of the coefficients as a vector and are methods lifted directly from linear algebra called the vector norm.

•  Principal Component Analysis (PCA) - Often a dataset has many columns, perhaps tens, hundreds, thousands of more. Modeling data with many features is challenging, and models built from data that include irrelevant features are often less skillful than models trained from the most relevant data. It is hard to know which features of the data are relevant and which are not. Methods for automatically reducing the number of columns of a dataset are called dimensionality reduction, and perhaps the most popular method is called the principal component analysis or  PCA for short. This method is used in machine learning to create projections of high-dimensional data for both visualizations and for training models. The core of the PCA method is a matrix factorization method from linear algebra. The eigendecomposition can be used and more robust implementations may use the singular- value decomposition or SVD.

• Singular-value Decomposition (SVD) -Another popular dimensionality reduction method is the singular-value decomposition method or SVD for short. As mentioned and as the name of the method suggests, it is a matrix factorization method from the field of linear algebra. It has wide use in linear algebra and can be used directly in applications such as feature selection, visualization, noise reduction, and more.

• Deep Learning -  Artificial neural networks are nonlinear machine learning algorithms that are inspired by elements of the information processing in the brain and have proven effective at a range of problems not least predictive modeling. Deep learning is the recent resurged use of artificial neural networks training of larger and deeper (more layers) networks on very large datasets. Deep learning methods routinely achieve state-of-the-art results on a range of challenging problems such as machine translation, photo captioning, speech recognition, and much more.  

               At their core, the execution of neural networks involves linear algebra data structure multiplied and added together. Scaled up to multiple dimensions. deep learning methods work with vectors, matrices, and even tensors of inputs and coefficients, where a tensor is a matrix with more than two dimensions. Linear algebra is central to the description of deep learning methods via matrix notation to the implementation of deep learning methods such as Google's TensorFlow Python library that has the word "tensor " in its name.