The Gram Matrix arises from a function in a finite-dimensional space; the Gram matrix entries are then the inner products of the essential services of the finite-dimensional subspace. We have to compute the style loss. But we haven’t been shown “why the style loss is computed using the Gram matrix.” The Gram matrix captures the “distribution of features” of a set of feature maps in a given layer.

Note: We don’t think the above question has been answered satisfactorily. For example, let us take a shot explaining it more intuitively. Suppose we have the following feature map. For simplicity, we consider only three feature maps, and two of them are entirely passive. We have a feature map set where the first feature map looks like a nature picture, and in the second feature map, the 1st feature map looks like a dark cloud. Then if we try to calculate the content and style loss manually, we will get these values.

This means that we have not lost style information between the two feature map sets. However, the content is different.

Understanding the style loss

Final loss

It is defined as,

Where α and β are user-defined hyperparameters. Here β has absorbed the M^l normalization factor defined earlier. By controlling α and β, we can control the amount of content and style injected into the generated image. We can also see a beautiful visualization of the different effects of different α and β values in the paper.

Defining the optimizer

Next, we use Adam optimizer to optimize the loss of the network.

Defining the input pipeline

Here we describe the full input pipeline. tf.data provided a very easy to use and intuitive interface to implement the input pipelines. For most of the image manipulation tasks, we use the tf. Image api, still, the ability of tf.image to dynamically sized images is minimal.

For example, if we want to crop and resize images dynamically, it is better to do in the form of the generator, as implemented below.

We have defined two input pipelines; one for style and one for content. The content input pipeline looks for only jpg images that start with a word content_, where the style pipeline looks for models beginning with style_.

def image_gen_function(data_dir, file_match_str, do_shuffle=True):
"""
"   The function returns a produced image, and the color channel is like values.
    This is a generator function that is used by the service of tf.data api.
""""# Load the filenames
    files = [f for f in os.listdir(data_dir) if f.startswith(file_match_str)]
if do_shuffle:
    shuffle(files)
        mean = np.array([[vgg_mean]])
     # For each file preprocess the image
for f in file:
    img = Image.open(os.path.join(data_dir, f))
    width, height = img.size
    #Here, we crop the image to a square by cropping on the longer axis
if width < height:
    left,right = 0, width
top, bottom = (height-width)/2, ((height-width)/2) + width
    elif width > height:
top, bottom = 0, height
    left, right = (width - height)/2, ((width-height)/2) + height
else:
    arr = np.array(img.resize((image_size,image_size))).astype(np.float32)
yield (arr, mean)
arr = np.array(img.crop((left, top, right, bottom)).resize((image_size,image_size))).astype(np.float32)
    yield (arr, mean)
def load_images_iterator(gen_func, zero_mean=False):
"""This function returns a dataset iterator of tf.data API.
        """
image_dataset = tf.data.Dataset.from_generator(
            gen_func,
output_types=(tf.float32, tf.float32),
            output_shapes=(tf.TensorShape(input_shape[1:]), tf.TensorShape([1, 1, 3]))
)
# If true, the mean will be subtracted

Defining the computational graph

We will be representing the full computational graph.

Define iterators which provide inputs
Define input and CNN variable
Define the content, style, and total loss
Define the optimization operation

config = tf.ConfigProto(allow_soft_placement=True)
# 1. Define the input pipeline in thisstep
part_style_gen_func = partial(image_gen_func, 'data', "style_")
part_content_gen_func = partial(image_gen_func, 'data', "content_")
style_iter = load_images_iterator(part_style_gen_func, zero_mean=True)
content_iter = load_images_iterator(part_content_gen_func, zero_mean=True)
# 2. Defining the inputs and weights
inputs = define_inputs(input_shape)
define_tf_weights()
layer_ids = list(vgg_layers.keys())
## gen_ph is used forinitializing the generated image with the pixel value
##trying initializing with white noise
.
## The init_generate gives the initialization operation.
gen_ph = tf.placeholder(shape=input_shape, dtype=tf.float32)
init_generated = tf.assign(inputs["generated"], gen_ph)
# 3. Loss
# 3.1 Content loss in tf
c_loss = define_content_loss(
inputs=inputs,
layer_ids=layer_ids, pool_inds=pool_inds, c_weight=alpha
)
# 3.2 Style loss
layer_weights_ph = tf.placeholder(shape=[len(layer_ids)], dtype=tf.float32, name='layer_weights')
s_loss = define_style_loss(
inputs=inputs,
layer_ids=layer_ids, pool_inds=pool_inds, s_weight=beta, layer_weights=None
)

So, this brings us to the end of blog. This Tecklearn ‘Concept of Gram Matrix’ blog helps you with commonly asked questions if you are looking out for a job in Artificial Intelligence. If you wish to learn Artificial Intelligence and build a career in AI or Machine Learning domain, then check out our interactive, Artificial Intelligence and Deep Learning with TensorFlow Training, that comes with 24*7 support to guide you throughout your learning period. Please find the link for course details:

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Introduction to Deep Learning and AI

What is Deep Learning?
Advantage of Deep Learning over Machine learning
Real-Life use cases of Deep Learning
Review of Machine Learning: Regression, Classification, Clustering, Reinforcement Learning, Underfitting and Overfitting, Optimization
Pre-requisites for AI & DL
Python Programming Language
Installation & IDE

Environment Set Up and Essentials

Installation
Python – NumPy
Python for Data Science and AI
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Python Libraries – Numpy and Pandas
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More Prerequisites for Deep Learning and AI

Pandas for Data Analysis
Machine Learning Basic Concepts
Normalization
Data Set
Machine Learning Concepts
Regression
Logistic Regression
SVM – Support Vector Machines
Decision Trees
Python Libraries for Data Science and AI

Introduction to Neural Networks

Creating Module
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Sigmoid Function
Multi-layered perception
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Activation Functions
Gradient Decent or Error function
Epoch, Forward & backword propagation
What is TensorFlow?
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Graph Visualization
Constants, Placeholders, Variables

Multi-layered Neural Networks

Error Back propagation issues
Drop outs

Regularization techniques in Deep Learning

Deep Learning Libraries

Tensorflow
Keras
OpenCV
SkImage
PIL

Building of Simple Neural Network from Scratch from Simple Equation

Training the model

Dual Equation Neural Network

TensorFlow
Predicting Algorithm

Introduction to Keras API

Define Keras
How to compose Models in Keras
Sequential Composition
Functional Composition
Predefined Neural Network Layers
What is Batch Normalization
Saving and Loading a model with Keras
Customizing the Training Process
Using TensorBoard with Keras
Use-Case Implementation with Keras

GPU in Deep Learning

Introduction to GPUs and how they differ from CPUs
Importance of GPUs in training Deep Learning Networks
The GPU constituent with simpler core and concurrent hardware
Keras Model Saving and Reusing
Deploying Keras with TensorBoard

Keras Cat Vs Dog Modelling

Activation Functions in Neural Network

Optimization Techniques

Some Examples for Neural Network

Convolutional Neural Networks (CNN)

Introduction to CNNs
CNNs Application
Architecture of a CNN
Convolution and Pooling layers in a CNN
Understanding and Visualizing a CNN

RNN: Recurrent Neural Networks

Introduction to RNN Model
Application use cases of RNN
Modelling sequences
Training RNNs with Backpropagation
Long Short-Term memory (LSTM)
Recursive Neural Tensor Network Theory
Recurrent Neural Network Model

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