MathsXDC-优快云博客

原创 MATHEMATICAL PRINCIPLE OF VAE

IntroductionWithin just three years, along with GAN, Variational Auto-encoder (VAE) became the most popular method of unsupervised learning of complicate probability density. This is because VAE is bui

2017-06-13 21:31:26 623

原创 MATHEMATICAL PRINCIPLE OF GAN

IntroductionThe potential advantage of deep learning lies in representing complicate probability densities of data with models of large scale hierarchical structure. Till now, the great success of deep

2017-06-13 18:51:41 597

原创 INEQUALITY PROOF

INEQUALITY PROOFEthanIn Loving Memory of Mamba Day, 4.13, 2016I. Common Inequalities(a). Lagrangian Formula for Summation EstimationSuppose f(x) is a continuous function on x >= 1 and F(x)

2016-05-14 16:02:38 961

原创 STATISTICS DISTRIBUTION OF NORMAL POPULATION

STATISTICS DISTRIBUTION OF NORMAL POPULATIONEthanFOR MUM AND MUM'S MUMIn probability and statistics, a statistic is a function of samples. Theoretically, it is also a random variable with a prob

2016-05-05 11:18:08 1266

原创 SHAH FUNCTION

Sha FunctionEtharaMathematically, Sha function is actually an extension of Delta function. This work will shed light on it in the context of its celebrated application --- Crystal Gazing --- for i

2016-03-06 20:43:52 1368

原创 MULTIVARIATE GAUSSIAN DISTRIBUTION

Multivariate Gaussian DistributionEtharaThis work will present a linear transformation interpretation on the multivariate Gaussian distribution. The entire derivation is originated from univariate

2015-05-24 19:13:20 1296

原创 LINEAR SYSTEM

Linear SystemEtharaI would rather consider this work as an appreciation towards linear systems that govern the entire 20th century. We are going to see how Fourier transform is applied to linear s

2015-05-08 11:17:44 1777 1

原创 GENERALIZED FUNCTION

Generalized FunctionEtharaThis work will extend the classic function to the generalized one typified by Dirac function in order to deal with the convergence issue of Fourier transform.DilemmaO

2015-02-27 15:49:05 1010

原创 FOURIER SERIES

Fourier SeriesEtharaThis work will present the derivation on Fourier series and give an intuitive understanding of it.Derivation on Fourier SeriesGiven a periodic functionf(t) with period T

2015-02-10 11:03:37 762

原创 DISCRETE FOURIER TRANSFORM

Discrete Fourier TransformEtharaParticularly in the modern world, it is the discrete Fourier transform (DFT) that can be used to analyze the spectrum of a signal in the frequency domain. Rather th

2015-01-06 12:58:04 1082

原创 SVD APPLICATIONS: PCA AND PSEUDO-INVERSE

SVD Applications: PCA and Pseudo-inverseEtharaAs far as I am concerned, there are two typical applications of Singular Value Decomposition, Principal Component Analysis (PCA) and matrix pseudo-inv

2014-12-31 10:27:09 1020

原创 SINGULAR VALUE DECOMPOSITION

Meditation on Singular Value DecompositionEtharaSingular Value Decomposition (SVD) is considered as the final and best factorization of a matrix, as described by Professor Gilbert Strang from Mass

2014-12-26 13:23:57 865

原创 POLYNOMIAL CURVE FITTING

Polynomial Curve FittingEtharaPolynomial curve fitting problem is such that, in a two dimensional plane, given a set ofm sample points,Figure 1you are asked to fit them into ann-or

2014-12-23 18:46:44 935

原创 CENTRAL LIMIT THEOREM

Derivation on Central Limit TheoremEtharaHere comes something really spooky.Suppose a random signal is created by generating a bunch of random numbers in between 0 and 1 and joining them by

2014-12-20 20:25:24 777

原创 PROBABILISTIC INTERPRETATION ON LEAST SQUARES

Probabilistic Interpretation on Least SquaresEtharaSuppose you are asked to make a regression on a training set,Youmight want to find a line as in the figure blow.Formally,to find an analy

2014-12-05 14:26:18 1037

原创 DISCRETE RANDOM DISTRIBUTION

Discrete Random DistributionEtharaMost common discrete random distributions are Bernoulli, Binomial, Multinomial, Geometric and Poisson distributions. Let’s elaborate on them one by one.Bernou

2014-12-02 18:33:54 691

原创 CONTOUR OF GAUSSIAN DISTRIBUTION

Contour of Gaussian DistributionEtharaHow much do you know about ellipse especially when I present you with an equation like this,Chances are that you fail to obtain an intuitive interpret

2014-11-29 21:15:00 1314

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