Independence (probability theory) | Signal processing
Most real world data sets consist of data vectors whose individual components are not statistically independent. In other words, knowing the value of an element will provide information about the value of elements in the data vector. When this occurs, it can be desirable to create a factorial code of the data, i. e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent. Later supervised learning usually works much better when the raw input data is first translated into such a factorial code. For example, suppose the final goal is to classify images with highly redundant pixels. A naive Bayes classifier will assume the pixels are statistically independent random variables and therefore fail to produce good results. If the data are first encoded in a factorial way, however, then the naive Bayes classifier will achieve its optimal performance (compare Schmidhuber et al. 1996). To create factorial codes, Horace Barlow and co-workers suggested to minimize the sum of the bit entropies of the code components of binary codes (1989). Jürgen Schmidhuber (1992) re-formulated the problem in terms of predictors and binary feature detectors, each receiving the raw data as an input. For each detector there is a predictor that sees the other detectors and learns to predict the output of its own detector in response to the various input vectors or images. But each detector uses a machine learning algorithm to become as unpredictable as possible. The global optimum of this objective function corresponds to a factorial code represented in a distributed fashion across the outputs of the feature detectors. Painsky, Rosset and Feder (2016, 2017) further studied this problem in the context of independent component analysis over finite alphabet sizes. Through a series of theorems they show that the factorial coding problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a series of linear problems. In addition, they introduce a simple transformation (namely, order permutation) which provides a greedy yet very effective approximation of the optimal solution. Practically, they show that with a careful implementation, the favorable properties of the order permutation may be achieved in an asymptotically optimal computational complexity. Importantly, they provide theoretical guarantees, showing that while not every random vector can be efficiently decomposed into independent components, the majority of vectors do decompose very well (that is, with a small constant cost), as the dimension increases. In addition, they demonstrate the use of factorial codes to data compression in multiple setups (2017). (Wikipedia).
👉 Learn all about factorials. Factorials are the multiplication of a number in descending integer values back to one. Factorials are used often in sequences, series, permutations, and combinations. Factorial quotient expressions are simplified by canceling out common integer products or
From playlist Sequences
Simplifying a factorial divided by another factorial
👉 Learn all about factorials. Factorials are the multiplication of a number in descending integer values back to one. Factorials are used often in sequences, series, permutations, and combinations. Factorial quotient expressions are simplified by canceling out common integer products or
From playlist Sequences
How to simplify an expression with factorials
👉 Learn all about factorials. Factorials are the multiplication of a number in descending integer values back to one. Factorials are used often in sequences, series, permutations, and combinations. Factorial quotient expressions are simplified by canceling out common integer products or
From playlist Sequences
How to simplify an expression with factorials
👉 Learn all about factorials. Factorials are the multiplication of a number in descending integer values back to one. Factorials are used often in sequences, series, permutations, and combinations. Factorial quotient expressions are simplified by canceling out common integer products or
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Ex 2: Simplify Expressions with Factorials Containing Variables
This video provides three examples of how to evaluate factorials and how to simplify expression involving factorials. Site: http://mathispower4u.com
From playlist Using the Binomial Theorem / Combinations
Writing a Simple Factorial Program. (Python 2)
Writing a simple factorial program with a "for" loop
From playlist Computer Science
Evaluating a rational expression with factorials
👉 Learn all about factorials. Factorials are the multiplication of a number in descending integer values back to one. Factorials are used often in sequences, series, permutations, and combinations. Factorial quotient expressions are simplified by canceling out common integer products or
From playlist Sequences
Introduction into factoring quadratics
👉Learn how to factor quadratics. A quadratic is an algebraic expression having two as the highest power of its variable(s). To factor an algebraic expression means to break it up into expressions that can be multiplied together to get the original expression. To factor a quadratic with th
From playlist Factor Quadratic Expressions
RailsConf 2022 - Your test suite is making too many database calls! by Joël Quenneville
On a recent project, I sped up a test suite 15% by making a change to a single factory. This suite, like many others (including yours!), was making way too many database calls. It’s so easy to accidentally add extra queries to factories and test setup and these can compound to shockingly l
From playlist RailsConf 2022
A Beginner's Guide to Recursion
Recursion has an intimidating reputation for being the advanced skill of coding sorcerers. But in this tutorial we look behind the curtain of this formidable technique to discover the simple ideas under it. Through live coding demos in the interactive shell, we'll answer the following que
From playlist Software Development
Get the Code Here: http://goo.gl/Zyzvy In this tutorial I end my Code Refactoring Tutorial. It was a big one and between it and my design patterns video tutorial you should be able to do most anything. To end this tutorial, I'll cover the Abstract Factory Pattern again to make sure you u
From playlist Code Refactoring
Design Patterns Tutorials for Beginners | Design Patterns Tutorial | Edureka
Watch Sample Class recording: http://www.edureka.co/design-patterns?utm_source=youtube&utm_medium=referral&utm_campaign=design-patterns-tut-beginners Design patterns are solutions to software design problems you find again and again in real-world application development. Patterns are abo
From playlist Design Patterns Videos
Ruby Conf 12 - Y Not- Adventures in Functional Programming by Jim Weirich
One of the deepest mysteries in the functional programming world is the Y-Combinator. Many have heard of it, but few have mastered its mysteries. Although fairly useless in real world software, understanding how the Y-Combinator works and why it is important gives the student an important
From playlist Ruby Conference 2012
Spring Tutorial 35 - Understanding AOP Proxies
In this video, we go where no Spring AOP tutorial has gone before! We'll understand how Spring AOP works and how it uses proxy classes to apply advice. And we'll do this by implementing our own simple factory service. The way Spring AOP works is much more complicated, but this simple imple
From playlist Spring AOP
Design Patterns : The Ultimate Blueprint for Software | Edureka
Watch Sample Class recording: http://www.edureka.co/design-patterns?utm_source=youtube&utm_medium=webinar&utm_campaign=design-patterns- 25-3-15 Design patterns are solutions to software design problems you find again and again in real-world application development. Patterns are about reu
From playlist Webinars by Edureka!
Get everything here: http://goo.gl/kzBzL Best Design Patterns Book : http://goo.gl/W0wyie MY UDEMY COURSES ARE 87.5% OFF TIL July 16th ($9.99) https://www.udemy.com/ ➡️ Python Data Science Series for $9.99 : Highest Rated & Largest Python Udemy Course + 56 Hrs + 200 Videos + Data Scienc
From playlist Java Video Tutorial
Introduction to Computer Programming for beginners || Coding for beginners
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as analysis, generating algorithms, profiling algorit
From playlist Programming
Abstract Factory Design Pattern
Get the Code Here: http://goo.gl/AZU3C Best Design Patterns Book : http://goo.gl/W0wyie Here is my Abstract Factory design pattern tutorial. I also take a second look at the Factory design pattern here as well. This is considered a hard pattern to understand, but I consider it to be a c
From playlist Java Video Tutorial
Learn how to factor a perfect square trinomial to a binomial squared
👉Learn how to factor quadratics. A quadratic is an algebraic expression having two as the highest power of its variable(s). To factor an algebraic expression means to break it up into expressions that can be multiplied together to get the original expression. To factor a quadratic with th
From playlist Factor Quadratic Expressions