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Math - Stats Basics Recap

Basic Statistics Concepts, Regression, Distributions, Covariance & Correlation, Bessel Correction

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 on are...

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

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, if y...

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

Math - Interpoloation

Linear, Cubic, Bicubic Splines

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 - Distance Metrics

KL Divergence , Chi-Squared Similarity

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 denotes h...

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 - 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 20, 2017

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

Covariance Matrix, PCA

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 square ma...

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 - 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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