Linear Algebra and Probability for Computer Science Applications

Linear Algebra and Probability for Computer Science Applications

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Features

  • Focuses on mathematical techniques that are most relevant to computer scientists
  • Assumes as little mathematical background as possible
  • Covers applications from computer graphics, web search, machine learning, cryptography, and a host of other computer science areas
  • Includes MATLAB functions, MATLAB programming assignments, and problems in each chapter
  • Offers MATLAB code at www.cs.nyu.edu/faculty/davise/MathTechniques/index.html

Summary

Based on the author’s course at NYU, Linear Algebra and Probability for Computer Science Applications gives an introduction to two mathematical fields that are fundamental in many areas of computer science. The course and the text are addressed to students with a very weak mathematical background. Most of the chapters discuss relevant MATLAB® functions and features and give sample assignments in MATLAB; the author’s website provides the MATLAB code from the book.

After an introductory chapter on MATLAB, the text is divided into two sections. The section on linear algebra gives an introduction to the theory of vectors, matrices, and linear transformations over the reals. It includes an extensive discussion on Gaussian elimination, geometric applications, and change of basis. It also introduces the issues of numerical stability and round-off error, the discrete Fourier transform, and singular value decomposition. The section on probability presents an introduction to the basic theory of probability and numerical random variables; later chapters discuss Markov models, Monte Carlo methods, information theory, and basic statistical techniques. The focus throughout is on topics and examples that are particularly relevant to computer science applications; for example, there is an extensive discussion on the use of hidden Markov models for tagging text and a discussion of the Zipf (inverse power law) distribution.

Examples and Programming Assignments
The examples and programming assignments focus on computer science applications. The applications covered are drawn from a range of computer science areas, including computer graphics, computer vision, robotics, natural language processing, web search, machine learning, statistical analysis, game playing, graph theory, scientific computing, decision theory, coding, cryptography, network analysis, data compression, and signal processing.

Homework Problems
Comprehensive problem sections include traditional calculation exercises, thought problems such as proofs, and programming assignments that involve creating MATLAB functions.

Table of Contents

MATLAB
Desk calculator operations
Booleans
Nonstandard numbers
Loops and conditionals
Script file
Functions
Variable scope and parameter passing

I: Linear Algebra
Vectors

Definition of vectors
Applications of vectors
Basic operations on vectors
Dot product
Vectors in MATLAB: Basic operations
Plotting vectors in MATLAB
Vectors in other programming languages

Matrices
Definition of matrices
Applications of matrices
Simple operations on matrices
Multiplying a matrix times a vector
Linear transformation
Systems of linear equations
Matrix multiplication
Vectors as matrices
Algebraic properties of matrix multiplication
Matrices in MATLAB

Vector Spaces
Subspaces
Coordinates, bases, linear independence
Orthogonal and orthonormal basis
Operations on vector spaces
Null space, image space, and rank
Systems of linear equations
Inverses
Null space and Rank in MATLAB
Vector spaces
Linear independence and bases
Sum of vector spaces
Orthogonality
Functions
Linear transformations
Inverses
Systems of linear equations
The general definition of vector spaces

Algorithms
Gaussian elimination: Examples
Gaussian elimination: Discussion
Computing a matrix inverse
Inverse and systems of equations in MATLAB
Ill-conditioned matrices
Computational complexity

Geometry
Arrows
Coordinate systems
Simple geometric calculations
Geometric transformations

Change of Basis, DFT, and SVD
Change of coordinate system
The formula for basis change
Confusion and how to avoid it
Nongeometric change of basis
Color graphics
Discrete Fourier transform (Optional)
Singular value decomposition
Further properties of the SVD
Applications of the SVD
MATLAB

II: Probability
Probability
The interpretations of probability theory
Finite sample spaces
Basic combinatorial formulas
The axioms of probability theory
Conditional probability
The likelihood interpretation
Relation between likelihood and sample space probability
Bayes’ law
Independence
Random variables
Application: Naive Bayes’ classification

Numerical Random Variables
Marginal distribution
Expected value
Decision theory
Variance and standard deviation
Random variables over infinite sets of integers
Three important discrete distributions
Continuous random variables
Two important continuous distributions
MATLAB

Markov Models
Stationary probability distribution
PageRank and link analysis
Hidden Markov models and the k-gram model

Confidence Intervals
The basic formula for confidence intervals
Application: Evaluating a classifier
Bayesian statistical inference (Optional)
Confidence intervals in the frequentist viewpoint: (Optional)
Hypothesis testing and statistical significance
Statistical inference and ESP

Monte Carlo Methods
Finding area
Generating distributions
Counting
Counting solutions to DNF (Optional)
Sums, expected values, integrals
Probabilistic problems
Resampling
Pseudo-random numbers
Other probabilistic algorithms
MATLAB

Information and Entropy
Information
Entropy
Conditional entropy and mutual information
Coding
Entropy of numeric and continuous random variables
The principle of maximum entropy
Statistical inference

Maximum Likelihood Estimation
Sampling
Uniform distribution
Gaussian distribution: Known variance
Gaussian distribution: Unknown variance
Least squares estimates
Principal component analysis
Applications of PCA

References

Notation

Index

Author Bio(s)

Ernest Davis is a computer science professor in the Courant Institute of Mathematical Sciences at New York University. He earned a Ph.D. in computer science from Yale University. Dr. Davis is a member of the American Association of Artificial Intelligence and is a reviewer for many journals. His research primarily focuses on spatial and physical reasoning.

Downloads / Updates

Resource OS Platform Updated Description Instructions
Platform type September 19, 2012 Additional Exercises Link click on http://www.cs.nyu.edu/faculty/davise/MathTechniques/MoreAssigs/MoreAssigs.html

 
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