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Math - Self-Adjoint Matrices

Robust Information Matrix

Definition A complex matrix is self adjoint if \(A = \bar{A}^T\), reading “equal to its own conjugate transpose”. It’s also called “Hermitian”. For real values, this means \(A = A^T\) In Eigen, ...

Posted by Rico's Nerd Cluster on January 26, 2017

Math - Interpoloation

Linear, Cubic, Bicubic Splines, Slerp

Interpolation “Interpolation” is essentially “how to get through lines and predict the value of a point in between?”. The first step is to connect points. For that, we need a spline. A spline is a...

Posted by Rico's Nerd Cluster on January 26, 2017

Math - 3D Point Distribution Metrics

Median Absolute Deviation (mad), Auto-Correlated Function (ACF)

std/MAD Two ways to measure “spread”: std \(=\sqrt{\mathbb E[X^2]}\) — squares everything, so one big outlier dominates. Measures the extremes. MAD \(=\text{median}\vert X\vert\) — half the ...

Posted by Rico's Nerd Cluster on January 23, 2017

Math - Distance Metrics

KL Divergence , Chi-Squared Similarity, Mahalanobis Distance, Geodesic Distance

Statistical Distance & Divergence Metrics Kullback-Leibler (KL) Divergence Given two distributions, \(p(x)\), and \(q(x)\), denotes how different \(p(x)\) is from \(q(x)\), hence it further d...

Posted by Rico's Nerd Cluster on January 23, 2017

Math - Positive Definiteness and Cholesky Decomposition

A matrix is positive definite, if it always sees the glass as half full. But why does the matrix still go to therapy? To break down its issues with Cholesky decomposition. Just Joking.

Positive Definiteness A matrix is Positive definite when for any x: \[\begin{gather*} x^T A x > 0 \end{gather*}\] For an nxn real symmetric matrix. It’s equivalent to say “it’s positive defin...

Posted by Rico's Nerd Cluster on January 20, 2017

Math - Eigen Value, Eigen Vector, and Eigen Value Decomposition

Covariance Matrix, PCA, Condition Number

Eigen Values and Eigen Vectors Basic Definitions \[\begin{gather*} Ax = \lambda x \end{gather*}\] \(x\) is an eigen vector, \(\lambda\) is an eigen value Important properties Only squar...

Posted by Rico's Nerd Cluster on January 15, 2017

Math Trigonometry Review

t a— layout: post title: Math - Trigonometry Review date: ‘2017-01-13 13:19’ subtitle: comments: true tags: - Math — Half Angle Formulas \[\begin{gather*} cos \theta = 2 cos^2 \frac{\theta}{2...

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

Math - Null Space Pseudo Inverse

Column, Row Spaces, Projection Matrix, Full Rank Factorization

A very nice reference article in Mandarin Column, Row, Left Null and Right Null Spaces Given the linear system \(A\mathbf{x}=\mathbf{b}\) with \(A\in\mathbb{R}^{m\times n}\): \(\textbf{Row sp...

Posted by Rico's Nerd Cluster on January 11, 2017

Math - Matrix Derivatives

Jacobian and Hessian, Vector-Vector Derivative

Jacobian and Hessian Jacobian Suppose we have a vector-valued function \[\begin{gather*} \begin{aligned} & f = [f_0(x), ... f_m(x)] \\ & x = [x_0, ... x_n] \end{aligned} \end{gather*}\] ...

Posted by Rico's Nerd Cluster on January 10, 2017

Math - Different Types Of Convolutions

Transpose Convolution (U-Net), Dilated Convolution (DeepLab)

Transpose Convolution Definition Transpose Convolution (a.k.a upsampling convolution) was designed specifically for upsampling, which is then used in the decoder part of an encoder-decoder networ...

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

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