Introduction

Transcription

Introduction

Practical Deep Neural Networks
GPU computing perspective
Introduction
Yuhuang Hu
Chu Kiong Loo
Advanced Robotic Lab
Department of Artificial Intelligence
Faculty of Computer Science & IT
University of Malaya
Yuhuang Hu, Chu Kiong Loo (UM)
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Outline
1
Introduction
2
Linear Algebra
3
Dataset Preparation
Data Standardization
Principle Components Analysis
ZCA Whitening
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Introduction
Outline
1
Introduction
2
Linear Algebra
3
Dataset Preparation
ZCA Whitening
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Introduction
Objectives
â Light introduction of numerical computation.
â Fundamentals of Machine Learning.
â Support Vector Machine, Softmax Regression.
â Feed-forward Neural Network.
â Convolutional Networks.
â Recurrent Neural Networks.
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Introduction
Prerequisites
? Basic training in Calculus
? Basic training in Linear Algebra
Matrix operations
Matrix properties: transform, rank, norm, determinant, etc
Eigendecomposition, Singular Value Decomposition.
? Basic programming skills
If-else conditioning
Loops
Function, class, library
Source code control: Git (optional)
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Introduction
References
© Deep Learning: An MIT Press book in preparation
Main reference in this workshop, still in development, awesome structure,
awesome contents.
© Machine Learning: A probabilistic perspective
One of the best Machine Learning books on the market.
© CS231n Convolutional Neural Networks for Visual Recognition Notes
Nice structured, well written, loads of pictures.
© CS229 Machine Learning Course Materials
For basic knowledge, well written, easy-to-read.
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Introduction
Software and Tools
? Ubuntu 14.04
? CUDA Toolkit 7
? Python 2.7.9 (Why not 3.*?)
? Theano
? numpy, scipy, etc
? Eclipse+PyDev
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Introduction
Reading List
A reading list is prepared for this workshop, all reading materials can be
found at:
http://rt.dgyblog.com/ref/ref-learning-deep-learning.html
The list keeps expanding!!
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Linear Algebra
Outline
1
Introduction
2
Linear Algebra
3
Dataset Preparation
ZCA Whitening
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Linear Algebra
Scalar, Vector, Matrix and Tensor
Scalars A scalar is a single number.
Vectors A vector is an array of numbers.
Matrices A matrix is a 2-D array of numbers.
Tensors A tensor is an array of numbers arranged on a regular grid
with a variable number of axes.
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Linear Algebra
Matrix Operations and Properties
Identity In ∈ Rn×n , IA = AI = A.
Transpose (A> )ij = Aji .
Addition C = A + B where Cij = Aij + Bij .
Scalar D = a · B + c where Dij = a · Bij + c.
P
Multiply C = AB where Cij = k Aik Bkj .
Element-wise Product C = A B where Cij = Aij Bij .
Distributive A(B + C) = AB + AC.
Associative A(BC) = (AB)C.
Transpose of Matrix Product (AB)> = B > A> .
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Linear Algebra
Inverse Matrices
The matrix inverse of A is denoted as A−1 , and it is defined as the matrix
such that
A−1 A = AA−1 = In
if A is not singular matrix.
If A is not square or is square but singular, it’s still possible to find it’s
generalized inverse or pseudo-inverse.
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Linear Algebra
Norms
We usually measure the size of vectors using an Lp norm:
!1
p
X
p
kxkp =
|xi |
i
There are few commonly used norms
L1 norm
X
kxk1 =
|xi |.
i
l∞ (Max norm)
kxk∞ = max |xi |.
i
Frobenius norm: Measure the size of matrix, analogy to the L2 norm.
sX
A2ij
kAkF =
i,j
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Linear Algebra
Unit vector, Orthogonal, Orthonormal, Orthogonal Matrix
A unit vector is a vector with unit norm:
kxk2 = 1
A vector x and y are orthogonal to each other if x> y = 0. In Rn , at most n
vectors may be mutually orthogonal with nonzero norm. If vectors are not only
orthogonal but also have unit norm, we call them orthonormal.
An orthogonal matrix is a square matrix whose rows are mutually orthonormal
and whose columns are mutually orthonormal:
A> A = AA> = I
This implies that
A−1 = A>
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Linear Algebra
Eigendecomposition
An eigenvector of a square matrix A is a non-zero vector v such that
Av = λv
where scalar λ is known as the eigenvalue corresponding to the eigenvector. If v
is an eigenvector of A, so is any rescaled vector sv for s ∈ R, s 6= 0. Therefore,
we usually only look for unit eigenvectors.
We can represent the matrix A using an eigendecomposition.
A = V diag(λ)V −1
where V = [v (1) , v (2) , . . . , v (n) ] (one column per eigenvector) and
λ = [λ1 , . . . , λn ].
Every real symmetric matrix can be decomposed into
A = QΛQ>
where Q is an orthogonal matrix composed of eigenvectors of A, and Λ is a
diagonal matrix, with λii being the eigenvalues.
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Linear Algebra
Singular Value Decomposition (SVD)
SVD is a general purpose matrix factorization method that decompose an n × m
matrix X into:
X = U ΣW >
where U is an n × n matrix whose columns are mutually orthonormal, Σ is a
retangular diagonal matrix and W is an m × m matrix whose columns are
mutually orthonormal. Elements alone the main diagonal of Σ are referred to as
the singular values. Columns of U and W are referred as the left-singular vectors
and right-singular vectors of X respectively.
Left-singular vectors of X are the eigenvectors of XX > .
Right-singular vectors of X are the eigenvectors of X > X.
Non-zero singular values of X are the square roots of the non-zero
eigenvalues for both XX > and X > X.
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Linear Algebra
Trace, Determinant
The trace operator is defined as
Tr(A) =
X
Aii
i
Frobenius norm in trace operator:
kAkF =
q
Tr(A> A)
The determinant of a square matrix, denoted det(A) is a function
mapping matrices to real scalars. The determinant is equal to the product
of all matrix’s eigenvalues.
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Dataset Preparation
Outline
1
Introduction
2
Linear Algebra
3
Dataset Preparation
ZCA Whitening
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Dataset Preparation
MNIST
MNIST dataset contains 60,000 training samples and 10,000 testing
samples (Lecun, Bottou, Bengio, & Haffner, 1998). Each sample is a
hand-written digit from 0-9.
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Dataset Preparation
CIFAR-10
CIFAR-10 dataset consists 60,000 images in 10 classes (Krizhevsky, 2009).
There are 50,000 training images and 10,000 testing images.
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Dataset Preparation
Dataset: How to split the data?
, If dataset has been splitted initially, please keep that way.
, If dataset comes as a whole, usually use 60% as trianing data, 20% as
cross-validation and 20% as testing data.
, Another alternative is using 70% as training data, 30% as testing
data.
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Dataset Preparation
Dataset: Cross validation
v Cross validation is used to keep track the performance of learning
algorithm (overfitting? underfitting?).
v Used to be a small portion of training dataset, randomly selected.
v Serve as testing data when testing data is not visible.
v Help to choose and adjust parameters of the learning model.
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Dataset Preparation
Mean subtraction, Unit Variance
Let µ as mean image
m
1 X (i)
x
µ=
m
i=1
And then subtract mean image on every image:
x(i) = x(i) − µ
Let
σj2 =
1 X (i) 2
xj
m
i
and then
(i)
(i)
xj
=
xj
σj
This is also called PCA whitening if it’s applied to rotated data, the
variance here can also be viewed as eigenvalues.
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Dataset Preparation
PCA
Principle Components Analysis (PCA) is a dimension reduction algorithm
that can be used to significantly speed up unsupervised feature learning
algorithm. PCA will find a lower-dimensional subspace onto which to
project our data.
Given a set of data X = {x(1) , x(2) , . . . , x(m) }, where x(i) ∈ Rn ,
covariance is computed by
XX >
Σ=
m
Use eigendecomposition, we can obtain the eigenvectors U :


|
| ··· |
U = u1 u2 · · · un 
|
| ··· |
and corresponding eigenvalues Λ = {λ1 , . . . , λn }. Note that usually Λ is a
sorted in descending order.
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Dataset Preparation
PCA
Columns in U are principle bases, so that we can rotate our data to the
new bases:
Xrot = U T X;
Noted that X = U Xrot .
To one of the data x, we can reduce the dimension by

 

xrot,1
xrot,1
 ..   .. 
 .   . 
 


xrot,k   xrot,k 




x
˜=
 ≈ xrot,k+1  = xrot
0

 

 ..   .. 
 .   . 
0
xrot,n
˜ can approximate the distribution of Xrot since it’s
The resulted X
dropping only small components. And we reduced n − k dimensions of the
original data
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Dataset Preparation
PCA
ˆ from X:
˜
We can recover the approximation of original signal X
 
x
˜1
 .. 
.
 
x
˜k 

x
ˆ=U
0
 
 .. 
.
0
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Dataset Preparation
PCA: Number of components to retain
To determine k, we usually look at the percentage of variance retained,
Pk
λj
p = Pni=1
j=1 λj
(1)
One common heuristic is to choose k so as to retain 99% of the variance
(p ≥ 0.99).
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Dataset Preparation
ZCA Whitening
ZCA Whitening
ZCA whitening is just a recover version of PCA whitening:
xZCA = U xPCA = U
xrot
Λ+
where is a regularization term. When x takes values around [−1, 1], a
value of ≈ 10−5 might be typical.
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Q&A
Q&A
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Q&A
References
Krizhevsky, A. (2009). Learning multiple layers of features from tiny
images (Master Thesis).
Lecun, Y., Bottou, L., Bengio, Y., & Haffner, P. (1998, Nov).
Gradient-based learning applied to document recognition.
Proceedings of the IEEE, 86(11), 2278-2324.
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Introduction

Transcription

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