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Linear Classifier

Introduction

A linear classifier is a classifier that uses a linear function of its inputs to base its decision on. That is, if the input feature vector to the classifier is a real vector

\vec x

, then the estimated output score (or probability) is :

y = f(\vec{w}\cdot\vec{x}) = f\left(\sum_j w_j x_j\right),

where

\vec w

is a real vector of weights and ''f'' is a function that converts the dot product of the two vectors into the desired output. Often ''f'' is a simple function that maps all values above a certain threshold to "yes" and all other values to "no". For a two-class classification problem, one can visualize the operation of a linear classifier as splitting a high-dimensional input space with a hyperplane: all points on one side of the hyperplane are classified as "yes", while the others are classified as "no". A linear classifier is often used in situations where the speed of classification is an issue, since it is often the fastest classifier, especially when

\vec x

is sparse. However, decision trees can be faster. Also, linear classifiers often work very well when the number of dimensions in

\vec x

is large, as in document classification, where each element in

\vec x

is typically the number of counts of a word in a document (see document-term matrix). In such cases, the classifier should be well-regularized.

Generative models vs. discriminative models

There are two main approaches for determining the parameters of a linear classifier

\vec w

2,3. The first is by modeling conditional density functions

P(\vec x|{\rm class})

. Examples of such algorithms include:

Linear Discriminant Analysis (or Fisher's linear discriminator) (LDA) --- assumes Gaussian conditional density models

Naive Bayes classifier --- assumes independent binomial conditional density models. The second approach is called discriminative training, which

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Classified Information

Classified information is secret information to which access is restricted by law or corporate rules to a particular hierarchical class of people. A security clearance is permission to handle classified documents or class of information, often requiring a satisfactory background check. This sort of hierarchical system of secrecy is used by virtually every national government, and by many corporations as well.

Purpose

The purpose of classification and secrecy is to protect information from being used to damage (or endanger) national security objectives. In the United States, information cannot be classified merely because it would be embarrassing; information can only be classified in relationship to protecting national security objectives of the state.

U.S. Government's Classification system

Most of the information about clearances and classification relates to the United States Government classification systems established by the Director of Central Intelligence — utilized under Executive Order 13292 (amending EO 12958) issued by President George W. Bush in 2003; this executive order lays out the system of classification for information handled by the United States Government and its employees, contractors, and industrial firms, handling classified information. The desired degree of secrecy about such information is known as its sensitivity. Sensitivity is based upon a calculation as to the damage to "national security". The United States has three levels of classification — confidential, secret, and top secret. Each level of classification indicates an increasing degree of sensitivity — top-secret being the highest, and confidential being the lowest. If one holds a "top-secret" clearance, one is allowed to handle information up to the level of "top-secret" (thus, secret, and confidential information). If one holds a "secret" clearance, one may not then handle "top-secret" information, but may handle confidential

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Subobject Classifier

In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object ''X'' correspond to the morphisms from ''X'' to Ω.

Introductory example

As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions: to every subset ''U'' of ''X'' we can assign the function from ''X'' to Ω that maps precisely the elements of ''U'' to 1 (see characteristic function). Every function from ''X'' to Ω arises in this fashion from precisely one subset ''U''.

Definition

For the general definition, we start with a category C that has a terminal object, which we denote by 1. The object Ω of C is a subobject classifier for C if there exists a morphism :1 → Ω with the following property: :for each monomorphism ''j'': ''U'' → ''X'' there is a unique morphism ''g'': ''X'' -> Ω such that the following commutative diagram ''U'' -> 1 | | v v ''X'' -> Ω :is a pullback diagram - that is, ''U'' is the limit of the diagram: 1 | v ''g'': ''X'' -> Ω The morphism ''g'' is then called the classifying morphism for the subobject ''j''.

Further examples

Every topos has a subobject classifier. For the topos of sheaves of sets on a topological space ''X'', it can be described in these terms: take the disjoint union Ω of all the open sets ''U'' of ''X'', and its natural mapping π to ''X'' coming from all the inclusion maps. Then π is a local homeomorphism, and the sheaf corresponding is the required subobject classifier (in other words the construction of Ω is by means of its espace étalé). One can also consider Ω to be, in a (tautological) sense, the graph of the membership relation obtaining between points ''x'' of ''X'' and open sets ''U'' category:category

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