谷歌&MIT:计算机科学的数学

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Mathematics for Computer Science.pdf

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2017年3月,谷歌与MIT联袂巨著《计算机科学的数学》(Mathematics for Computer Science)发布。这是谷歌工程师EricLehman,与MIT两位教授ThomsonLeighton和AlbertMeyer合著的教科书。如同书名,为计算机专业的学生提供数学基础知识。“计算机科学的数学”(MathematicsforComputerScience)本是MIT计算机专业的本科课程。其实,对于机器学习入门来说,不管是自学数学、统计学基础还是其它领域,都有很多优秀又不贵的资源。

这本书涵盖了国外计算机科学专业涉及的基础数学知识。它特别强调数学定义、概念、证明以及应用方法。其涉及的话题包含:证明方法、归纳、图论(graph theory)、formal logic notation、良序 (well-ordering)、集合与关系、整数同余(integer congruences) 、计数原理、函数增长率、离散概率等等。另外,这本书还介绍了一些延伸话题,比如递归、结构归纳、状态机与不变量、还有生成函数。对于机器学习开发者最有价值的是其中涉及的统计学知识点,可作为深入学习统计学之前的甜点。

《计算机科学的数学》Mathematics for Computer Science 目录
I Proofs
Introduction
0.1 References
1 What is a Proof?
1.1 Propositions
1.2 Predicates
1.3 The Axiomatic Method
1.4 Our Axioms
1.5 Proving an Implication
1.6 Proving an “If and Only If”
1.7 Proof by Cases
1.8 Proof by Contradiction
1.9 Good Proofs in Practice
1.10 References
2 The Well Ordering Principle
2.1 Well Ordering Proofs
2.2 Template for Well Ordering Proofs
2.3 Factoring into Primes
2.4 Well Ordered Sets
3 Logical Formulas
3.1 Propositions from Propositions
3.2 Propositional Logic in Computer Programs
3.3 Equivalence and Validity
3.4 The Algebra of Propositions
3.5 The SAT Problem
3.6 Predicate Formulas
3.7 References
4 Mathematical Data Types
4.1 Sets
4.2 Sequences
4.3 Functions
4.4 Binary Relations
4.5 Finite Cardinality
5 Induction
5.1 Ordinary Induction
5.2 Strong Induction
5.3 Strong Induction vs. Induction vs. Well Ordering
6 State Machines
6.1 States and Transitions
6.2 The Invariant Principle
6.3 Partial Correctness & Termination
6.4 The Stable Marriage Problem
7 Recursive Data Types
7.1 Recursive Definitions and Structural Induction
7.2 Strings of Matched Brackets
7.3 Recursive Functions on Nonnegative Integers
7.4 Arithmetic Expressions
7.5 Induction in Computer Science
8 Infinite Sets
8.1 Infinite Cardinality
8.2 The Halting Problem
8.3 The Logic of Sets
8.4 Does All This Really Work?

II Structures
Introduction
9 Number Theory
9.1 Divisibility
9.2 The Greatest Common Divisor
9.3 Prime Mysteries
9.4 The Fundamental Theorem of Arithmetic
9.5 Alan Turing
9.6 Modular Arithmetic
9.7 Remainder Arithmetic
9.8 Turing’s Code (Version 2.0)
9.9 Multiplicative Inverses and Cancelling
9.10 Euler’s Theorem
9.11 RSA Public Key Encryption
9.12 What has SAT got to do with it?
9.13 References
10 Directed graphs & Partial Orders
10.1 Vertex Degrees
10.2 Walks and Paths
10.3 Adjacency Matrices
10.4 Walk Relations
10.5 Directed Acyclic Graphs & Scheduling
10.6 Partial Orders
10.7 Representing Partial Orders by Set Containment
10.8 Linear Orders
10.9 Product Orders
10.10 Equivalence Relations
10.11 Summary of Relational Properties
11 Communication Networks
11.1 Routing
11.2 Routing Measures
11.3 Network Designs
12 Simple Graphs
12.1 Vertex Adjacency and Degrees
12.2 Sexual Demographics in America
12.3 Some Common Graphs
12.4 Isomorphism
12.5 Bipartite Graphs & Matchings
12.6 Coloring
12.7 Simple Walks
12.8 Connectivity
12.9 Forests & Trees
12.10 References
13 Planar Graphs
13.1 Drawing Graphs in the Plane
13.2 Definitions of Planar Graphs
13.3 Euler’s Formula
13.4 Bounding the Number of Edges in a Planar Graph
13.5 Returning to K5 and K3;3
13.6 Coloring Planar Graphs
13.7 Classifying Polyhedra
13.8 Another Characterization for Planar Graphs

III Counting
Introduction
14 Sums and Asymptotics
14.1 The Value of an Annuity
14.2 Sums of Powers
14.3 Approximating Sums
14.4 Hanging Out Over the Edge
14.5 Products
14.6 Double Trouble
14.7 Asymptotic Notation
15 Cardinality Rules
15.1 Counting One Thing by Counting Another
15.2 Counting Sequences
15.3 The Generalized Product Rule
15.4 The Division Rule
15.5 Counting Subsets
15.6 Sequences with Repetitions
15.7 Counting Practice: Poker Hands
15.8 The Pigeonhole Principle
15.9 Inclusion-Exclusion
15.10 Combinatorial Proofs
15.11 References
16 Generating Functions
16.1 Infinite Series
16.2 Counting with Generating Functions
16.3 Partial Fractions
16.4 Solving Linear Recurrences
16.5 Formal Power Series
16.6 References

IV Probability
Introduction
17 Events and Probability Spaces
17.1 Let’s Make a Deal
17.2 The Four Step Method
17.3 Strange Dice
17.4 The Birthday Principle
17.5 Set Theory and Probability
17.6 References
18 Conditional Probability
18.1 Monty Hall Confusion
18.2 Definition and Notation
18.3 The Four-Step Method for Conditional Probability
18.4 Why Tree Diagrams Work
18.5 The Law of Total Probability
18.6 Simpson’s Paradox
18.7 Independence
18.8 Mutual Independence
18.9 Probability versus Confidence
19 Random Variables
19.1 Random Variable Examples
19.2 Independence
19.3 Distribution Functions
19.4 Great Expectations
19.5 Linearity of Expectation
20 Deviation from the Mean
20.1 Markov’s Theorem
20.2 Chebyshev’s Theorem
20.3 Properties of Variance
20.4 Estimation by Random Sampling
20.5 Confidence in an Estimation
20.6 Sums of Random Variables
20.7 Really Great Expectations
21 Random Walks
21.1 Gambler’s Ruin
21.2 Random Walks on Graphs

V Recurrences
Introduction
22 Recurrences
22.1 The Towers of Hanoi
22.2 Merge Sort
22.3 Linear Recurrences
22.4 Divide-and-Conquer Recurrences
22.5 A Feel for Recurrences


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