Rico's Nerd Cluster

「离开世界之前 一切都是过程」

Math - SO(3) Perturbations

Left Perturbation, Right Perturbation, BCH Formula, Left Jacobian, Right Jacobian

Left and Right Perturbations Imagine we have a world frame, a car frame, and a pedestrian. Now the car has rotated from the car1 pose to the car2 pose. We assume the pedestrain has not moved: In ...

Posted by Rico's Nerd Cluster on February 22, 2017

Math - Lie Group

Group Definition, Lie Group and Lie Algebras, Skew Matrices, Manifold, SO(3) Derivatives

Group Definition Group (Group) is an algebraic structure consisting of a set and an operation. Denote the set as A, and the operation as ⋅. Then, the group can be written as G=(A,⋅). Closure (...

Posted by Rico's Nerd Cluster on February 22, 2017

Math - Multivariate Normal Distribution

PDF, Linear Transformation, Covariance Matrix

PDF “Variate” is basically a synonym to “random variable”. If we have a vector of random variables and we want to find their joint distributution, then we call this “multivariate” distribution. I...

Posted by Rico's Nerd Cluster on February 13, 2017

Math - MAP(Maximum A-Posteriori) and MLE (Maximum Likelihood Estimate)

Likelihood vs Probabilities

Likelihood VS Probabilities Probabilities describe the chances of discrete, mutually-exclusive possible states. These chances should sum up to 1. Likelihoods describe the chances or the plausibili...

Posted by Rico's Nerd Cluster on February 13, 2017

Math - t-SNE

Introduction van der Maaten, Geoff Hinton and Yoshua Bengio came up with “t-SNE” (pronounced as “tee-snee”) to reduce high dimensional data down to low dimensional data, so clusters between data i...

Posted by Rico's Nerd Cluster on February 7, 2017

Math - Approximation to SVD

How to find Null Space? How to implement that using Singular Value Decomposition (SVD)?

Introduction For a square full-rank matrix \(G\), the usual SVD is \[G = U\Sigma V^\top\] and the closest orthogonal matrix to \(G\) in Frobenius norm is the polar factor \[Q = UV^\top.\] But ...

Posted by Rico's Nerd Cluster on February 7, 2017

Math - Null Space and Singular Value Decomposition (SVD)

How to find Null Space? How to implement that using Singular Value Decomposition (SVD)?

Introduction The three most basic & important matrix factorizations in pure & applied linear algebra are: QR Decomposition (Gram-Schmidt Orthogonalization) LU Decomposition (Gaussian...

Posted by Rico's Nerd Cluster on February 7, 2017

Math - Welford Algo for Cov Calculation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 template <typename Container, typename VectorType, typename Getter> inline void compute_cov_and_mean(const Container &dat...

Posted by Rico's Nerd Cluster on February 5, 2017

Math - Stats Basics Recap

Basic Statistics Concepts, Regression, Distributions, Covariance & Correlation, Covariance Propagation, Variance Addition, Inverse-Variance Weighting, Bessel Correction, box filter

Basic Statistics Concepts Standard Deviation and Variance Variance of a distribution can be “biased” and “unbiased”. A biased variance is to always underestimate the real bias. \[\begin{gather*}...

Posted by Rico's Nerd Cluster on February 4, 2017

Math - Lagrange Multiplier

Pontryagin's Minimum Principle

Lagrange Multiplier Multivating example: maximize \(f(x,y)\), where \(xy+1\), given constraint \(g(x,y) = x^2+y^2-1 = 0\) Geometric Intuition: the value of f(x,y) and the constraint it must stay ...

Posted by Rico's Nerd Cluster on February 4, 2017

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