Math

Stability Theory When a system has control, then comes the questions of whether we could make it stable under the control. Problem Setup A control affine system is this $$ \dot x =f(x)+\sum_i g_i(x)\bar u_i=f(x)+g(x)\bar u $$Interpretation:

May 10, 2021

Basic Notions Def Topological Equivalence: 2 dynamic systems are topological equivalence when there is a homeomorphism between their solutions. Def Conjugate: Two maps are connected by $$g=h^{-1}\circ f \circ h$$.

May 9, 2021

Bifurcation Normal Form Invariance and Stable Manifold Lyapnov Stability Theory Feedback Stablization

Apr 9, 2021

Stability Theory Motivation We want to know for a dynamic system, in this note majorly autonomous system, when it is stable? The meaning of stability? If it’s stable how to prove so. Majorly we are going to use Lyapnov functions and spectral properties of linearized system to prove.

Apr 9, 2021

Invariance Properties of an Invariance Set Stable and Unstable Manifold Theorem

Apr 9, 2021

Problem Statement Given a bunch of noisy data, you want a smooth curve going through the cloud. As the points are noisy, there is no need to going through each point.

Oct 13, 2020

Note on Gaussian Process Gaussian Process can be thought of as a Gaussian distribution in function space (or infinite dimension vector). One of its major usage is to tackle nonlinear regression problem and provide mean estimate and errorbar around it.

Jul 1, 2020

Note on Bayesian Optimization Related to Gaussain Process model Philosophy Bayesian Optimization applies to black box functions and it employs the active learning philosophy. Use Case and Limitation BO is preferred in such cases

Jul 1, 2020

Note on Laplacian-Beltrami (Diffusion) Operator Motivation Laplacian on graph and on discrete geometry (mesh) are very useful tools. One core intuition, just like Laplacian in $\R^n$ space, it’s related to diffusion and heat equation. Recall the diffusion equation is

May 8, 2020

Spectral Graph Theory and Segmentation Motivation Spectral Graph Theory is a powerful tool as it sits at the center of multiple representation. Connects to Graph and manifold, and linear algrbra. It’s related to dynamics on graph, related to Markov chain, random walk (diffusion.) Could be applied to any point cloud: images, meshes are suited. Could be used to perform clustering, segmentation etc. Linear Algebra Review There are several ways to see a eigenvalue problem

Apr 22, 2020