Neural Networks (Pytorch)

Adversarial Attacks on CIFAR-10

Adversarial attacks on neural networks were developed by Goodfellow et al. as techniques to fool models through perturbations applied to input data. As a result, the accuracy of the trained model can drop to almost zero. At the same time, perturbed inputs, e.g. images, are imperceptibly different to humans or AI solutions for self-driving cars, the military, or medicine. In this post, I set up a trained model on CIFAR-10 and demonstrate how applied perturbations shift classifications to the false negatives, which can be easily visualised through confusion matrices.

Option Pricing (Python)

Explicit Method of Lines for European Option

Method of Lines (MOL) is a semi-analytical approach to solve PDEs where space dimension is discretised but time stays continuous. As a result, there is a system of ODEs coupled by the matrix operator, which is solved along the time. MOL seems to be good to price options because it decreases the error of estimation. The method is accurate, relatively fast and in general reliable. Nevertheless, the stability of the explicit scheme still depends upon space-grid discretisation.

Option Pricing (Python)

Implicit Method of Lines for European Option

In this post, I present MOL for the EU option. This time it is the implicit method, which is unconditionally stable and allows for more accurate price approximation.

Universal Approximation (Pytorch)

Physics-Informed Neural Network

In Physic-Informed Learning a neural network is trained by means of the inherent physical laws ancoded a priori. As a result, we obtain data-efficient approximators able to reconstruct physical states. This post shows how Neural Network approximates the solution of the PDE Heat Equation, U_tt - c²U_xx = 0 , by minimising the loss function made up of the physist cost. i.e., residual of the PDE equation with some (not many) points in the interior and the boundary of the space-time domain. The physist cost works as a regulalizator for the suprvised boundary cost. Namely, we deal with the forward problem here.

Linear Algebra in Machine Learning (Python/Numpy)

Singualar Value Decomposition (SVD) in Machine Learning

SVD decomposition is the background for many ML algorithms such as PCA, Dimensionality Reduction (e.g. image compression), denoising data, linear autoencoders or matrix completion. Imagine, we want to compress a data matrix X from dimension s to 0 < k < s. Namely, we approximate matrix X ∈ ℝ^s × n by SVD matrix factorisation subject to rank(R) = k such that $$\underset{U_k,U_k^T}{\min } \left\| X-X U_k^T U_k \right\| _{\text{Fro}}^2=\underset{\left\{\sigma _j\right\}_{j=1}^{k < \min (s,n)}}{\min }\left\| X-\underbrace{\text{U$\Sigma $}_k V^T}_{R}\right\| _{\text{Fro}}^2$$ Understanding SVD will be necessary for the later post of robust PCA and matrix completion.

Linear Algebra in Machine Learning (Python/Numpy)

Robust PCA

Singular value decomposition (SVD) and principal component analysis (PCA) are powerful tools for finding low-rank representations of data matrices. However, many data matrices are only approximately low-rank, e.g., due to measurement errors and artefacts of other non-linear influences on the data acquisition process. Suppose we are given a large data matrix X ∈ ℝ^n × s and know it might be decomposed as X = L + S , where L ∈ ℝ^n × s has low-rank and S ∈ ℝ^n × s is sparse. The aim is to recover approximated matrix through recovering low-rank, L̂, and sparse components, Ŝ, by solving convex optimisation problem as follows $$\left(\hat{L},\hat{S}\right)=\,\,\, \underset{L\in \mathbb{R}^{n\times s},S\in \mathbb{R}^{n\times s}}{\arg \min } \left\{\| S\| _1+\alpha \| L\|_*\,\, \text{subject}\,\, \text{to}\,\, X=L+S\right\}$$ In this post, we will discuss a PCA-variant that is robust towards outliers in the data. Before I introduce the method, I want to reformulate the problem of finding low-rank approximations first.

Cryptocurrencies with Machine Learning

Market Capitalization of cryptocurrencies with Pandas and Plotly

Eigenmode Decomposition

Dynamic Mode Decomposition for El Niño Discovering

Eigenmode Decomposition

Forward and Inverse PINN for the heat equation.

Functional Analysis

Solutions from Introductory Functional Analysis by Kreyszig