Category Archives: Blog

CNNs_Part_1 $\textbf{Problem}$: Write functions that take an arbitrary grayscale image and convolve them with a Sobel edge detection kernel. Apply both functions to a grayscale image of your choice. $\textbf{Solution}$: In [1]: import numpy as np import matplotlib.pyplot as plt sobel_horizontal_kernel … Continue reading →

Problem: Deduce the Hamilton-Jacobi equation of classical mechanics. Solution: Instead of viewing the action \(S=S[\mathbf x(t)]\) as a functional of the particle’s trajectory \(\mathbf x(t)\), it can be viewed more simply as a scalar field \(S(\mathbf x,t)\) in which the … Continue reading →

Problem: Define the signature of a matrix. Hence, state and prove Sylvester’s law of inertia. Solution: The signature of an \(n\times n\) matrix \(A\) is a \(3\)-tuple \((n_+,n_-,n_0)\) where \(n_+\) is the number of positive eigenvalues of \(A\) (including multiplicity), … Continue reading →

Problem: How does the paradigm of reinforcement learning (RL) fit within the broader context of machine learning? Solution: It is instructive to compare/contrast reinforcement learning with supervised learning. In this way, it will be seen that RL can in fact … Continue reading →

Problem: In one sentence, what is the essence of DFT? Solution: To replace \(\Psi\mapsto n\), where the number density of a system of \(N\) identical quantum particles (usually electrons) \(n(\mathbf x)\) is: \[n(\mathbf x):=N\int d^3\mathbf x_2…d^3\mathbf x_N |\Psi(\mathbf x,\mathbf x_2,…,\mathbf … Continue reading →

Problem: Do an end-to-end walkthrough of the PyTorch machine learning workflow using the most basic univariate linear regression example. In particular, generate some linear data over a normalized feature space (whose slope \(w\) and intercept \(b\) would in practice be … Continue reading →

Problem: Illustrate some of the basic fundamentals involved in using the PyTorch deep learning library. In particular, discuss the attributes of PyTorch tensors (e.g. dtype, CPU/GPU devices, etc.), how to generate random PyTorch tensors with/without seeding, and operations that can … Continue reading →

Problem: Let \(|\mathcal V|,N_c\in\mathbf Z^+\) be positive integers (where \(|\mathcal V|\) is the cardinality of an arbitrary set \(\mathcal V\) called the vocabulary and \(N_c\) will come to be seen as the number of codebooks), and let \(\mathcal T\) be … Continue reading →

JAX_tutorial $\textbf{Problem}$: What is JAX? $\textbf{Solution}$: JAX = Autograd + XLA, where Autograd refers to automatic differentiation, and XLA refers to accelerated linear algebra (compiler developed by Google that optimizes code to run fast on GPUs/TPUs). At a high level, … Continue reading →

Problem: Distinguish between Las Vegas methods and Monte Carlo methods. Solution: Both are umbrella terms referring to broad classes of methods that draw (repeatedly) from (not necessarily i.i.d.) random variables to compute the value of some deterministic variable. Here, “compute” … Continue reading →