Word embeddings and high-dimensional data are ubiquitous in many facets of deep learning research such as natural language processing, building recommendation systems. Google in the recent times has open sourced the embedding project that works as an interactive collaborative visualization tool to visualize the high-dimensional data. The embedding works as a navigational map from the points to the data inputs in Euclidean space. Many types of big data require visualization of graphs, neural nets, writing summaries on the parameters of weights, sigmoid activation functions, and the accuracy of the machine learning models.

In this tutorial, we use the MNIST handwritten digits’ dataset. More explanation on the mechanics of the MNIST handwritten digits dataset can be found on Yann Lecun’s website. As described by Yann Lecun.

**“THE IDX FILE FORMAT**

the IDX file format is a simple format for vectors and multidimensional matrices of various numerical types.

The basic format is

magic number

size in dimension 0

size in dimension 1

size in dimension 2

…..

size in dimension N

data

The magic number is an integer (MSB first). The first 2 bytes are always 0.

The third byte codes the type of the data:

0x08: unsigned byte

0x09: signed byte

0x0B: short (2 bytes)

0x0C: int (4 bytes)

0x0D: float (4 bytes)

0x0E: double (8 bytes)

The 4-th byte codes the number of dimensions of the vector/matrix: 1 for vectors, 2 for matrices….

The sizes in each dimension are 4-byte integers (MSB first, high endian, like in most non-Intel processors).

The data is stored like in a C array, i.e. the index in the last dimension changes the fastest.”

Several deep learning frameworks have measured the benchmarks leveraging MNIST handwritten digits. MNIST dataset is easily accessible to TensorFlow framework. TensorFlow represents the data as tensors in the graph. Tensors most of the time are representatives of high-dimensional data. MNIST dataset has around Z^{784 }dimensions. The world we live only has Z^{3 }space. Hence, visualizing the data in 784 dimensions for human eye is a daunting task. Therefore, various techniques such as Principle Component Analysis or t-distributed Stochastic Neighbor Embedding (t-SN3) can be adapted to reduce the curse of the dimensionality to reduce the dimensions and visualize the data in lower dimensional space interpretable to the human eye. ^{ }

Building a Neural Net to visualize the tensors of the hidden layer activation is possible through TensorBoard’s embedded projector. For this, numpy, matplotlib libraries will be imported.

Figure 1. Importing libraries.

The input data for this example will be leveraged from MNIST dataset of handwritten digits.

Figure 2. Importing Input data from MNIST dataset.

Extracting MNIST_data/train-images-idx3-ubyte.gz

Extracting MNIST_data/train-labels-idx1-ubyte.gz

Extracting MNIST_data/t10k-images-idx3-ubyte.gz

Extracting MNIST_data/t10k-labels-idx1-ubyte.gz

Size of:

– Training-set: 55000

– Test-set: 10000

– Validation-set: 5000

The hyper parameters will be specified externally as constant parameters as these are not learned by the neural networks.

Figure 3. Hyper-parameters.

The graph needs to be constructed here, but a fully convolutional neural network or FC Layer (LeNet) needs to be defined along with weights, and bias variables.

Figure 4. Definition of weights, convolutional neural network, and bias variables.

Since the helper functions have been activated, the graph can now be created:

Figure 5. Creation of graph.

An interactive graph session will be launched in the Jupyter notebook after executing the following code:

Figure 6. Launching the graph session.

After executing the code, it displays the training accuracy for the neural network.

Training epoch: 1

iter 0: Loss=2.28, Training Accuracy=37.0%

iter 100: Loss=0.45, Training Accuracy=85.0%

iter 200: Loss=0.21, Training Accuracy=95.0%

iter 300: Loss=0.34, Training Accuracy=89.0%

iter 400: Loss=0.29, Training Accuracy=90.0%

iter 500: Loss=0.18, Training Accuracy=97.0%

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Epoch: 1, validation loss: 0.21, validation accuracy: 94.1%

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Training epoch: 2

iter 0: Loss=0.14, Training Accuracy=97.0%

iter 100: Loss=0.19, Training Accuracy=94.0%

iter 200: Loss=0.11, Training Accuracy=95.0%

iter 300: Loss=0.06, Training Accuracy=99.0%

iter 400: Loss=0.08, Training Accuracy=96.0%

iter 500: Loss=0.12, Training Accuracy=97.0%

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Epoch: 2, validation loss: 0.14, validation accuracy: 96.0%

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Training epoch: 3

iter 0: Loss=0.16, Training Accuracy=96.0%

iter 100: Loss=0.15, Training Accuracy=96.0%

iter 200: Loss=0.08, Training Accuracy=97.0%

iter 300: Loss=0.11, Training Accuracy=94.0%

iter 400: Loss=0.10, Training Accuracy=96.0%

iter 500: Loss=0.09, Training Accuracy=97.0%

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Epoch: 3, validation loss: 0.11, validation accuracy: 96.7%

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Training epoch: 4

iter 0: Loss=0.03, Training Accuracy=100.0%

iter 100: Loss=0.04, Training Accuracy=99.0%

iter 200: Loss=0.11, Training Accuracy=94.0%

iter 300: Loss=0.05, Training Accuracy=98.0%

iter 400: Loss=0.11, Training Accuracy=97.0%

iter 500: Loss=0.11, Training Accuracy=95.0%

---------------------------------------------------------

Epoch: 4, validation loss: 0.09, validation accuracy: 97.2%

---------------------------------------------------------

Training epoch: 5

iter 0: Loss=0.11, Training Accuracy=97.0%

iter 100: Loss=0.04, Training Accuracy=98.0%

iter 200: Loss=0.04, Training Accuracy=100.0%

iter 300: Loss=0.02, Training Accuracy=100.0%

iter 400: Loss=0.11, Training Accuracy=97.0%

iter 500: Loss=0.04, Training Accuracy=99.0%

---------------------------------------------------------

Epoch: 5, validation loss: 0.08, validation accuracy: 97.7%

---------------------------------------------------------

Training epoch: 6

iter 0: Loss=0.04, Training Accuracy=100.0%

iter 100: Loss=0.07, Training Accuracy=98.0%

iter 200: Loss=0.08, Training Accuracy=98.0%

iter 300: Loss=0.04, Training Accuracy=99.0%

iter 400: Loss=0.04, Training Accuracy=99.0%

iter 500: Loss=0.05, Training Accuracy=97.0%

---------------------------------------------------------

Epoch: 6, validation loss: 0.07, validation accuracy: 97.9%

---------------------------------------------------------

Training epoch: 7

iter 0: Loss=0.04, Training Accuracy=99.0%

iter 100: Loss=0.03, Training Accuracy=99.0%

iter 200: Loss=0.07, Training Accuracy=99.0%

iter 300: Loss=0.06, Training Accuracy=98.0%

iter 400: Loss=0.13, Training Accuracy=96.0%

iter 500: Loss=0.07, Training Accuracy=97.0%

---------------------------------------------------------

Epoch: 7, validation loss: 0.07, validation accuracy: 97.8%

---------------------------------------------------------

Training epoch: 8

iter 0: Loss=0.03, Training Accuracy=99.0%

iter 100: Loss=0.01, Training Accuracy=100.0%

iter 200: Loss=0.03, Training Accuracy=99.0%

iter 300: Loss=0.02, Training Accuracy=100.0%

iter 400: Loss=0.02, Training Accuracy=100.0%

iter 500: Loss=0.05, Training Accuracy=98.0%

---------------------------------------------------------

Epoch: 8, validation loss: 0.07, validation accuracy: 97.8%

---------------------------------------------------------

Training epoch: 9

iter 0: Loss=0.04, Training Accuracy=99.0%

iter 100: Loss=0.01, Training Accuracy=100.0%

iter 200: Loss=0.07, Training Accuracy=98.0%

iter 300: Loss=0.03, Training Accuracy=99.0%

iter 400: Loss=0.03, Training Accuracy=99.0%

iter 500: Loss=0.09, Training Accuracy=97.0%

---------------------------------------------------------

Epoch: 9, validation loss: 0.07, validation accuracy: 97.9%

---------------------------------------------------------

Training epoch: 10

iter 0: Loss=0.02, Training Accuracy=100.0%

iter 100: Loss=0.03, Training Accuracy=100.0%

iter 200: Loss=0.02, Training Accuracy=100.0%

iter 300: Loss=0.02, Training Accuracy=100.0%

iter 400: Loss=0.01, Training Accuracy=100.0%

iter 500: Loss=0.01, Training Accuracy=100.0%

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Epoch: 10, validation loss: 0.07, validation accuracy: 98.1%

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Considering the dataset has around 10,000 samples, the hidden layer in the neural network has around 200 nodes. The tensor output data has around 10,000 x 200 layers.

Figure 7. Assigning of hidden layer of the convolutional neural network to the variables.

The objective of this code is to visualize the high-dimensional data in a low dimensional way to make it interpretable for the human eye. The metadata can be downloaded into metadata.tsv from the TensorBoard that can store the index and label for individual samples. This file has to be added to the points embedding config. File.

Figure 8. Creating config. object.

Running the session in Jupyter Notebook will evaluate the tensor for the embeddings and save the data.

Figure 9. It creates the tensor in the deep learning model cockpit configuration log file.

I have shared the code on GitHub at GPSingularity:

Once the session is executed, it can now be visualized in TensorBoard with the following command:

tensorboard –logdir=logs/embedding/ –host localhost The data will be displayed as follows in TensorBoard. **Results from the program**

Figure 10. TensorBoard Embedded Projector to visualize High-dimensional data in low dimension.

This program I’ve written is inspired by Easy TensorFlow http://www.easy-tensorflow.com/’sembedded visualization.

I’ve shared my program in GPSingularity Github.

References

Easy TensorFlow (2018). Easy TensorFlow. Retrieved from http://www.easy-tensorflow.com/

Smilkov, D., Thorat, N., Nicholson, C., Reif, E., Viegas, F. B., & Watternberg, M. (2016). Embedding Projector: Interactive Visualization and Interpretation of Embeddings. Retrieved from https://arxiv.org/pdf/1611.05469v1.pdf

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