14 items tagged
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:
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 $$g=h^{-1}\circ f \circ h$$.
Bifurcation Normal Form Invariance and Stable Manifold Lyapnov Stability Theory Feedback Stablization
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.
Invariance Properties of an Invariance Set Stable and Unstable Manifold Theorem
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.
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.
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
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
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
Note on Hyperbolic Geometry Reference Notes 2018 Lec Note 2015 Lecture note Ch5-3 Measurement in Hyperbolic Geometry [Cheatsheet / Note](http://home.iiserb.ac.in/~kashyap/MTH 520/lp.pdf) Motivation Hyperbolic geometry is a great source of inspiration for math art. Besides it is used to model some hierarchical data structure. Here I collected a few models
Motivation This is a brief analytical note about how physical self movement of eye / camera will induce optic flow in a static environment. And then discuss how a system can separate these two components instantaneously.
Krylov Subspace, Lancosz Iteration, QR and Conjugate Gradient Motivation In practise, many numerical algorithms include iteratively multiply a matrix, like power method and QR algorithm. All these algorithms have their core connected to a single construct, Krylov subspace and a operation, Lancosz Iteration. So this note motivates to understand this core.
Some Computation on Sphere (Updating) Motivation Recently, in research, we encounter quite a few statistical problems on sphere. For example, Head direction tuning 3d direction of object 3d direction of body parts Some 3d tuning There are many standard statistical operations on Euclidean space, like getting mean, standard deviation and generate uniform distribution, fitting a model etc. We can perform these operation without thinking.