Statistical Learning

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.

Oct 13, 2020

TOC {:toc} Philosophy The spirit of Variational Inference is to solve Bayesian inference problem with optimization. In the scenario of latent factor It’s not trying to use Bayes rule directly, but to fit this distribution within a class of distributions $q(z;\nu)$, by minimizing the KL-divergence between the 2 models.

Jul 27, 2020

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.

Jul 1, 2020

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

Jul 1, 2020

TOC {:toc} Note on Online Regression Algorithm Least Square Problem Classical least square linear regression is $$ \hat \beta_{ls}=\arg\min_\beta\|y-X\beta\|^2_2 $$ With regularizations it becomes a ridge or lasso regression problem

Dec 15, 2019

TOC {:toc} L-BFGS algorithm Motivation L-BFGS is one of the not so simple optimization algorithm that we may encounter in large scale optimization problems. Not so simple means it’s not simply a first order algorithm, and the deviation from that is well motivated by theoretical arguments. So this note target to understand this algorithm

Nov 11, 2019

TOC {:toc} Problem Setting The original problem of non-negative matrix factorization is simple, if the dissimarity $D(A\|HW)$ between original matrix and reconstructed one is L2 distance than, $$ argmin_{H,W} \|A-HW\|_F^2, \\ s.t.\ W\succeq0, H\succeq0 $$The non-negative constraint applies element-wise.

Jun 25, 2019

TOC {:toc} Constrained CMA-ES Algorithm Target CMA-ES is originally used in unconstrained optimization. To adapt it into constrained optimization and we have to handle the boundary in some way. So how could it handle this geometric boundary?

Jun 20, 2019

TOC {:toc} 最近在阅读1，是以为记。 Objective of Algorithm 目标 Belief Propagation算法想解决的是Markov随机场，Bayes网络等图模型的边缘概率估计，以及求解最可能的状态的问题。 有许多名字称呼这一General的算法，如sum-product, max-product, min-sum, Message Passing等，属于更general的Message Passing算法范畴。 同时这一算法可以说是一种通用框架或者philosophy，因此在不同结构的模型中有许多著名的特例，这些具体算法也有各自的名字(如前向后向算法，Kalman Filter等等) 对于统计学习问题，通常会区分模型与算法，模型设定一些假设，抽象现实的某个方面，建立问题的结构；而算法求解问题(很多时候是转化为优化问题来求解)。在这个post中将要介绍的Belief Propagation算法，属于后者，但为了理解他，我们首先需要理解他对应的模型，即概率图模型。 Graphical Models： What relates graph to probability? 第一次接触概率图模型的人（像我）都会问，概率和图这两者有什么关系呢？ 我们知道图是一种直观的表征事物之间二元关系的方法通常由$(\mathcal V, \mathcal E)$定点和边组成。在概率图模型中，顶点通常代表随机变量，而边代表随机变量之间的关系。

May 18, 2019