SEBASTIAN RASCHKA. Introduction to Artificial Neural Networks and Deep Learning. with Applications in Python

Transcription

1 SEBASTIAN RASCHKA Introduction to Artificial Neural Networks and Deep Learning with Applications in Python

2 Introduction to Artificial Neural Networks with Applications in Python Sebastian Raschka Last updated: May 25, 2018 This book will be available at Please visit for more information, supporting material, and code examples. c Sebastian Raschka

3 Contents A Mathematical Notation Reference 4 A.1 Sets and Intervals A.2 Sequences A.3 Functions A.4 Linear Algebra A.5 Calculus A.6 Probability and Statistics A.7 Numbers A.8 Approximation A.9 Logic i

4 Website Please visit the GitHub repository to download the code examples accompanying this book and other supplementary material. If you like the content, please consider supporting the work by buying a copy of the book on Leanpub. Also, I would appreciate hearing your opinion and feedback about the book, and if you have any questions about the contents, please don t hesitate to get in touch with me via mail@sebastianraschka.com. Happy learning! Sebastian Raschka 1

5 About the Author Sebastian Raschka received his doctorate from Michigan State University developing novel computational methods in the field of computational biology. In summer 2018, he joined the University of Wisconsin Madison as Assistant Professor of Statistics. Among others, his research activities include the development of new deep learning architectures to solve problems in the field of biometrics. Among his other works is his book "Python Machine Learning," a bestselling title at Packt and on Amazon.com, which received the ACM Best of Computing award in 2016 and was translated into many different languages, including German, Korean, Italian, traditional Chinese, simplified Chinese, Russian, Polish, and Japanese. Sebastian is also an avid open-source contributor and likes to contribute to the scientific Python ecosystem in his free-time. If you like to find more about what Sebastian is currently up to or like to get in touch, you can find his personal website at 2

6 Acknowledgements I would like to give my special thanks to the readers, who provided feedback, caught various typos and errors, and offered suggestions for clarifying my writing. Appendix A: Artem Sobolev, Ryan Sun Appendix B: Brett Miller, Ryan Sun Appendix D: Marcel Blattner, Ignacio Campabadal, Ryan Sun, Denis Parra Santander Appendix F: Guillermo Monecchi, Ged Ridgway, Ryan Sun, Patric Hindenberger Appendix H: Brett Miller, Ryan Sun, Nicolas Palopoli, Kevin Zakka 3

7 Appendix A Mathematical Notation Reference This appendix provides a brief overview of the mathematical notation used throughout this book. The following appendices describe most of the corresponding concepts in more detail, and additional information is provided in the context of the applications in the main chapters. 4

8 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 5 A.1 Sets and Intervals Z set of integers, {..., 2, 1, 0, 1, 2,...} N set of natural numbers, {0, 1, 2, 3,...} N + set of natural numbers excluding zero, {1, 2, 3,...} R set of real numbers element of symbol; for example, x A translates to "x is an element of set A" / not an element of symbol A B A B A B A B null set, empty set union of two sets, A and B intersection of two sets, A and B A is a subset of B or included in B symmetric difference between two sets A and B A cardinality of a set A (number of elements in a set A) (a, b) [a, b] [a, b) (a, b] open interval from a to b, excluding a and b closed interval from a to b, including a and b half-open interval from a to b, including a but not b half-open interval from a to b, including b but not a

9 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 6 A.2 Sequences n x i i=1 n x i i=1 summation of an indexed variable x i, defined as n x i = x 1 + x x n product over an indexed variable x i, defined as n x i = x 1 x 2... x n i=1 i=1 A.3 Functions f : A B function f with domain A and codomain B (g f)(x) composition of two functions g and f alternative form: g[f(x)] f 1 (x) inverse of a function f, such that f(y) = x if f 1 stands for y x absolute value of x; for example, 2 = 2 log b log base-b logarithm natural logarithm (base-e logarithm) n! n-factorial, where 0! = 1 and n! = n(n 1)(n 2) 2 1 for n > 0 ( n k) binomial coefficient ("n choose k"); ( n) k = n! k!(n k)! for 0 k n arg max f(x) arg min f(x) the x value that makes f(x) as large as possible the x value that makes f(x) as small as possible

10 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 7 A.4 Linear Algebra x x scalar (lower-case italics notation) column vector (lower-case bold notation) or n 1-matrix a b dot product of two vectors, a and b; if a and b are n 1-matrices, also written as a T b; a b = a T b = i a ib i = a 1 b 1 + a 2 b a n b n X X R n m n-matrix (upper-case bold notation) 3D-tensor (upper-case italics notation) real coordinate space, written as a column vector with length n x = x 1 x 2. x n x T transpose of a n 1-matrix ] x T = [x 1 x 2... x n = T x 1 x 2. x n x p x L p norm, vector p-norm, x p = ( x p 1 + xp xp n ) 1/p L norm, max norm; largest absolute value of a vector x = max i x i

11 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 8 x vector norm, L 2 -norm, x = x 2 x = x x x2 n A i,: A :,j A T ith row of matrix A jth column of matrix A transpose of a matrix, matrix element A i,j becomes A T j,i 1 2 T [ ] for example, 3 4 = I n n n identity matrix I 3 = A 1 tr A det A diag(a 1, a 2,..., a n ) A B inverse of a matrix A, such that AA 1 = A 1 A = I trace of a matrix A (sum of the diagonal elements) tr A = n A i,i i=1 determinant of a matrix A diagonal matrix, matrix whose diagonal have the values a 1, a 2,..., a n and all other elements are zero Hadamard product, element-wise matrix multiplication

12 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 9 A.5 Calculus lim f(x) x a lim f(x) x a lim f(x) x a+ df dx d n f dx n limit of f(x) as x approaches a limit of f(x) as x approaches a from the left limit of f(x) as x approaches a from the right derivative of f n-th derivative of f f x partial derivative of f(x, y,...) with respect to variable x, where x is a scalar f gradient of a function f : R n R f x 1 f x f(x 1, x 2,..., x n ) = 2. f x n f Laplacian of a function f : R n R f = n i=1 2 f x 2 i

13 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 10 Hf Hessian of a function f : R n R Hf = 2 f x 1 x 1 2 f x 2 x 1. 2 f x n x 1 2 f x 1 x f 2 f x 1 x n 2 f x 2 x n 2 f x n x n x 2 x f x n x 2... f j x i partial derivative of component function f j and the variable x j, where f : R n R m, such that f 1 (x) f 2 (x) f(x) =. f m (x) f x i = f 1 x i f 2 x i. f m x i Df Jacobian matrix of f. f 1 f 1 f x 1 x x n f 2 f 2 f x Df = 1 x x n f m f m f x 1 x 2... m x n f(x)dx b a f(x)dx indefinite integral of f (derivative of F ) with f : R R definite integral of f (derivative of F ) with f : R R

14 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 11 A.6 Probability and Statistics P (A B) P (A B) P (A B) E(X), µ X X var(x), σx 2 probability that event A and B occur probability that event A or B occurs conditional probability of A given B expected value (mean) of a random variable X E(X) = p i x i for a discrete random variable X i=1 with values x 1, x 2,... and probabilities p 1, p 2,.... E(X) = xf(x)dx for a continuos random variable and probability density function f(x). sample average of numerical data X 1,..., X n X = 1 n n X i i=1 variance of a random variable X var(x) = E [ (X µ X ) 2] = E(X 2 ) E(X) 2 s 2 X sample variance of numerical data X 1,..., X n s 2 X = 1 n n (X i X) 2 i=1

15 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 12 std(x), σ x standard deviation of a random variable, square root of the variance s X cov(x, Y ) sample standard deviation, the square root of the sample variance s 2 X covariance of two random variables X and Y cov(xy ) = E[(X E(X))(Y E(Y ))] = E(XY ) E(X)E(Y ) s XY sample covariance of numerical data X 1,..., X n, and Y 1,..., Y n s XY = 1 n n (X i X)(Y i Ȳ ) i=1 corr(x, Y ) correlation coefficient of two random variables X and Y, corr(x, Y ) = cov(x,y ) σ X σ Y H(X) entropy of a random variable X discrete: H(X) = P (X = x) log b P (X = x) x continuous: H(X) = f(x) log b f(x)dx PMF probability mass function of a discrete random variable, f(x) = P (X = x) CDF PDF X D ˆθ cumulative distribution function of a continuous random variable, F (x) = P (X x) probability density function of a continuous random variable, P (X [a, b]) = b f(x)dx a random variable X has a distribution D estimator of a parameter θ N(x, µ, σ 2 ) normal (Gaussian) distribution over x with mean µ and variance σ 2

16 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 13 A.7 Numbers e Euler s number, mathematical constant approximated by π "pi", mathematical constant approximated by infinity symbol scientific notation for 123, 400 or 1.234E05 < less than sign, for example, x < 10 means that x is smaller than 10 much less than sign > greater than sign, for example, x > 10 means that x is larger than 10 much greater than sign much less than sign A.8 Approximation approximate equality, for instance, e is the approximation of Euler s number f(x) g(x) symbol to assert that the ratio of two functions approaches 1 f(x) = 1, if x is small lim x 0 lim x g(x) f(x) g(x) = 1, if x is large f(x) g(x) the two functions f(x) and g(x) are proportional to each other T (n) O(n 2 ) big-o notation, an algorithm is asymptotically bounded by n 2 ; an algorithm has an order of n 2 time complexity

17 APPENDIX A. MATHEMATICAL NOTATION REFERENCE 14 A.9 Logic implication operator for example, A B translates to "if A implies B" or "if A then B" (or "B only if A") equality operator (if and only if (iff)) for example, A B translates to "A if and only if B" or "if A then B and if B then A" logical conjunction, and for example, A B means "A and B" logical (inclusive) disjunction, or for example, A B means "A or B" negation, not for example, A means "not A" or "if A is true then A is false" and vice versa universal quantifier, means for all for example, " x R, x > 1" translates to "for all real numbers x, x is greater than one" existential quantifier, means there exists for example, " x A, f(x)" translates to "there is an element in set A for which the predicate f(x) holds true"

18 Bibliography 15

19 Abbreviations and Terms CNN [Convolutional Neural Network] 16

20 Index 17