An Introduction Linear Algebra for Neural Networks and Deep learning

Talk: Hack Linear Algebra with Python for Deep Neural Network
Speaker: Chetan Khatri, Department of Computer Science, University of
Kachchh alumnus.
Lead - Technology, Accionlabs Inc.
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from IPython.display import Image
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Image(filename="/home/chetan/Pictures/Screenshot from 2018-01-12 14-24-40.png")
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Image(filename="/home/chetan/Downloads/screenshot-towardsdatascience.com-2018-01
-13-07-49-05-835.png")
common linear algebra operations used in Neural Network
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Say What ? Linear Algebra for Neural Network
In the context of deep learning / Neural network, linear algebra is a mathematical toolbox that offers helpful
techniques for manipulating groups of numbers simultaneously. It provides structures like vectors and
matrices (spreadsheets) to hold these numbers and new rules for how to add, subtract, multiply, and divide
them.
Why is it useful?
It turns complicated problems into simple, intuitive, efficiently calculated problems. Here is an example of
how linear algebra can achieve greater speed and simplicity.
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import numpy as np
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# Multiply two arrays
x = [1,2,3]
y = [2,3,4]
product = []
for i in range(len(x)):
product.append(x[i]*y[i])
print product
In [6]:
# Linear algebra version
x = np.array([1,2,3])
y = np.array([2,3,4])
x * y
After initializing the arrays, the linear algebra approach was 3x faster.
How is it used in Artificial Neural Network / deep learning ?
Neural networks store weights in matrices. Linear algebra makes matrix operations fast and easy, especially
when training on GPUs. In fact, GPUs were created with vector and matrix operations in mind. Similar to how
images can be represented as arrays of pixels, video games generate compelling gaming experiences using
enormous, constantly evolving matrices. Instead of processing pixels one-by-one, GPUs manipulate entire
matrices of pixels in parallel.
Vectors
[2, 6, 12]
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array([ 2, 6, 12])

Vectors are 1-dimensional arrays of numbers or terms. In geometry, vectors store the magnitude and
direction of a potential change to a point. The vector [3, -2] says go right 3 and down 2. A vector with more
than one dimension is called a matrix.
Vector -> is arrow pointing in a space. What defines a vector: length and direction it's pointing, as long as
length and direction are same you can move all around.
Vectors in geometry
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Vector lives in flat plane is 2 dimensional.
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If vector sits in broader space where you and I leave they are 3 Dimensional.
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vector addition & vector multiplication plays important role in linear algebra
positive numbers indicating rightward and upward motion negative numbers indicating leftward and
downward motion
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Every pair of number gives 1 and only 1 vector. And every vector is associated with 1 and only 1 pair of
number.
3 Dimensions - Z
Z - Parpendicular to both X & Y axis, which orders triplet numbers.
[2 1 3] Move along to how far to X axis. How far to move parellel to Y axis. How far to move parellel to Z axis
In Brief: Vectors typically represent movement from a point. They store both the magnitude and direction of
potential changes to a point. The vector [-2,5] says move left 2 units and up 5 units.
A vector can be applied to any point in space. The vector’s direction equals the slope of the hypotenuse
created moving up 5 and left 2. Its magnitude equals the length of the hypotenuse.
What's going on to space & Ask computer to figure it out numerically.
Scalar operations
Scalar operations involve a vector and a number. You modify the vector in-place by adding, subtracting, or
multiplying the number from all the values in the vector.
Examples: Streaching vector by scalar, Scaling Vector by scalar, Squash in vector by Scalar, Shear vector in
plane geometry
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Span: What does the span of two 3d vectors look like ?
All possible linear combination of two vectors. If one vector is already span of other then it's called "Linearly
dependent" for some values of a and b. Each vector adds another dimension to span that's "Linearly
independent" w /= av for all values of a.
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Basis The basis of a vector space is a set of linearly independent vectors that span the full space. The word
"transformation" suggests that you think using "Movement"
Transformation is linear: If all lines must remain lines, origin remains fixed. Not linear: Some lines get
curved, grid lines remain parellel and evenly spaced.
Elementwise operations
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Image(filename="/home/chetan/Music/images/1_g7cc0GkA7xrkhoh-__Ok3g.png")
In elementwise operations like addition, subtraction, and division, values that correspond positionally are
combined to produce a new vector. The 1st value in vector A is paired with the 1st value in vector B. The 2nd
value is paired with the 2nd, and so on. This means the vectors must have equal dimensions to complete the
operation.*
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Image(filename="/home/chetan/Music/images/2.png")
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y = np.array([1,2,3])
x = np.array([2,3,4])
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y + x
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array([3, 5, 7])

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y - x
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y / x
Vector multiplication
There are two types of vector multiplication: Dot product and Hadamard product.
Dot product
The dot product of two vectors is a scalar. Dot product of vectors and matrices (matrix multiplication) is one of
the most important operations in deep learning.
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y = np.array([1,2,3])
x = np.array([2,3,4])
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np.dot(y,x)
Hadamard product
Hadamard Product is elementwise multiplication and it outputs a vector.
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array([-1, -1, -1])
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array([0, 0, 0])
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20

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y = np.array([1,2,3])
x = np.array([2,3,4])
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y * x
Vector fields
A vector field shows how far the point (x,y) would hypothetically move if we applied a vector function to it like
addition or multiplication. Given a point in space, a vector field shows the power and direction of our
proposed change at a variety of points in a graph.
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Matrices
A matrix is a rectangular grid of numbers or terms (like an Excel spreadsheet) with special rules for addition,
subtraction, and multiplication.
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array([ 2, 6, 12])
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Matrix dimensions
We describe the dimensions of a matrix in terms of rows by columns.
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a = np.array([
[1,2,3],
[4,5,6]
])
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a.shape
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b = np.array([
[1,2,3]
])
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b.shape
Matrix scalar operations
Scalar operations with matrices work the same way as they do for vectors. Simply apply the scalar to every
element in the matrix—add, subtract, divide, multiply, etc.
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(2, 3)
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(1, 3)

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a = np.array(
[[1,2],
[3,4]])
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a + 1
Matrix elementwise operations
In order to add, subtract, or divide two matrices they must have equal dimensions.* We combine
corresponding values in an elementwise fashion to produce a new matrix.
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a = np.array([
[1,2],
[3,4]
])
b = np.array([
[1,2],
[3,4]
])
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array([[2, 3],
[4, 5]])
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a + b
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a - b
Numpy broadcasting*
In numpy the dimension requirements for elementwise operations are relaxed via a mechanism called
broadcasting. Two matrices are compatible if the corresponding dimensions in each matrix (rows vs rows,
columns vs columns) meet the following requirements:
1. The dimensions are equal, or
2. One dimension is of size 1
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a = np.array([
[1],
[2]
])
b = np.array([
[3,4],
[5,6]
])
c = np.array([
[1,2]
])
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# Same no. of rows
# Different no. of columns
# but a has one column so this works
a * b
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array([[2, 4],
[6, 8]])
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array([[0, 0],
[0, 0]])
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array([[ 3, 4],
[10, 12]])

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# Same no. of columns
# Different no. of rows
# but c has one row so this works
b * c
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# Different no. of columns
# Different no. of rows
# but both a and c meet the
# size 1 requirement rule
a + c
Things get weirder in higher dimensions—3D, 4D, but for now we won’t worry about that. Understanding 2D
operations is a good start.
Matrix Hadamard product
Hadamard product of matrices is an elementwise operation. Values that correspond positionally are
multiplied to produce a new matrix.
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array([[ 3, 8],
[ 5, 12]])
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array([[2, 3],
[3, 4]])
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a = np.array(
[[2,3],
[2,3]])
b = np.array(
[[3,4],
[5,6]])
# Uses python's multiply operator
a * b
In numpy you can take the Hadamard product of a matrix and vector as long as their dimensions meet the
requirements of broadcasting.
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Matrix transpose
Neural networks frequently process weights and inputs of different sizes where the dimensions do not meet
the requirements of matrix multiplication. Matrix transpose provides a way to “rotate” one of the matrices so
that the operation complies with multiplication requirements and can continue. There are two steps to
transpose a matrix: Rotate the matrix right 90° Reverse the order of elements in each row (e.g. [a b c]
becomes [c b a]) As an example, transpose matrix M into T:
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array([[ 6, 12],
[10, 18]])
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a = np.array([
[1, 2],
[3, 4]])
a.T
Matrix multiplication
Matrix multiplication specifies a set of rules for multiplying matrices together to produce a new matrix.
Rules
Not all matrices are eligible for multiplication. In addition, there is a requirement on the dimensions of the
resulting matrix output.
1. The number of columns of the 1st matrix must equal the number of rows of the 2nd
2. The product of an M x N matrix and an N x K matrix is an M x K matrix. The new matrix takes the
rows of the 1st and columns of the 2nd
Steps
Matrix multiplication relies on dot product to multiply various combinations of rows and columns.
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array([[1, 3],
[2, 4]])

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The operation a1 · b1 means we take the dot product of the 1st row in matrix A (1, 7) and the 1st column in
matrix B (3, 5).
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Here’s another way to look at it:
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Test yourself with these examples
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An Introduction Linear Algebra for Neural Networks and Deep learning

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An Introduction Linear Algebra for Neural Networks and Deep learning