Reconstructing Numbers

I’m working on finishing up the code for the final 30 Days of Python project, saving the whales, but I took a detour to work with the MNIST handwritten digits again. I did this because they are great dataset for testing because it’s obvious what you are training on and its also well known how well an algorithm should do.

I’ve been implementing python code that trains restricted boltzmann machines (RBMs), deep belief nets (DBNs), and deep neural networks (DNNs). I originally implemented this in Matlab (as part of a coursera course) and then ported it to R (because it’s free, open source, and good for stats) and am now bringing it into Python. I chose to bring this code into python for a couple of reasons: I’m using python as my go to language, I want learn how to use numpy effectively, and I’ve become a fan of how python manages code. There are plenty of other reasons but I will caveat all of this with the fact that I know there are available RBM/DBN/DNN implementations available but I want to make my own as a learning exercise and because I like having control of the finer details of the model.

A Brief Intro to RBMs, DBNs, and DNNs

A Restricted Boltzmann Machine (RBM) is a model that has two layers: an input layer (called the visible units) and a hidden layer. Each layer is made up of “units” which can take on either a 1 or a 0. For a given input, the hidden layer can be computed using a weight matrix, a bias term, and the logistic function: h = logistic(W*v+b_fwd). Because the inputs are often real valued and the computation ends up with real values on the interval from 0.0 to 1.0 the values are treated as probabilities that a given unit will be ‘on’ or ‘off.’

The interesting thing about RBMs is they are trained to reconstruct their inputs. So instead of trying to tie the output layer to a digit label, we try to reconstruct the digit image. The theory being that if the model can reconstruct the input then the features learned will be useful in determining the label later. To calculate a reconstruction of the visible layer use the transpose of the weight matrix, a reverse bias term, and the logistic function: v = logistic(t(W)*h + b_rev). Calculating the hidden units from the visible is a forward pass and calculating the visible from the hidden is a reverse pass. The optimal way to train these models is quite slow but Geoffery Hinton found a shortcut called contrastive divergence:

  1. Compute a forward pass (v0 -> h0)
  2. Compute a reverse pass (h0 -> v1)
  3. Compute another forward pass (v1 -> h1)
  4. Calculate a weight update rule: delta_W = h0*t(v0) - h1*t(v1)

The full version of contrastive divergence requires the back and forth to go onto infinity but it turns out even one pass gives a good weight update rule that can be used in gradient descent.

RBMs can be stacked by using the output of one layer as the input to the next. Stacked RBMs are mathematically equivalent to Deep Belief Nets (DBNs) and it’s much faster to train them by doing “greedy” training one layer at a time. There are ways to train the DBN to improve the generative ability of the model and tie in the labels but I’m not interested in them for that.

The reason I use RBMs is you can initialize a deep neural network (DNN) with the same coefficients as the DBN. The two changes to make are discarding the reverse biases since the DNN is one way and adding an extra layer to connect to the labels. I follow what I learned in Hinton’s coursera class and use a softmax layer. A softmax layer is like a logistic layer except that it is normalized so that you can interpret it as a probability distribution. For the 10 MNIST digits there will always be 10 units in the softmax layer. For a digit of 2, unit 2 will have a high value (near 1.0) and the others will be near 0.0, which can be interpreted as high probability of it being a digit 2. For any input the sum of all the output units will be 1.0. The computation of a forward pass in the DNN is just a succession of forward passes of each layer. The model can be updated through gradient descent using the backprop method.

Crucially because the DNN is initialized with pre-trained RBMs only a few passes of backprop are needed to fine tune the model for discrimination. Virtually all of the training time is spent in the pre-training without the labels minimizing the amount of time the model has to overfit the training dataset.

Visualizing the Learning: Reconstructed Digits

One way to see how well the model learned from the data is to reconstruct the input data through the model. What I mean is set the input data to be an image of a digit, compute the hidden layer through a forward pass of the model, and then compute a reverse pass to get a reconstructed image. For a random model this results in white noise (think snow on the TV). Below is the result for a forward pass to the first layer and into the second layer followed by reverse pass by both. The reconstructed image is quite good meaning the model has the ability to represent the data well. Having worked with RBMs on multiple projects, I highly recommend this technique to verify that the model has learned something. It very easily points bugs in the code (my first reconstructions looked identical) or points to issues with trying to reconstruct the data this way.

Reconstruction of a digit through the RBMs

Reconstruction of digits through the RBMs

To explore what the model actually learned we can reconstruct an image from just the bias unit (i.e. turn off all of the hidden units). The reconstructed image is similar to the average of all of the images the model sees:

Digit Reconstruction from just the Bias compared to the Average Digit

Digit reconstruction from just the bias compared to the mean digit

Another way to see what’s going on is to turn on a single hidden unit in a layer and see what it constructs. The resulting image is essentially what would drive this hidden unit as close to 1 as possible. It shows what the model is paying attention too:

Reconstruction from a single hidden unit in the first layer

Reconstruction from a single hidden unit in the first layer

A dark area means it’s a higher value so you can see that these units tend have a very active curve somewhere in the image and very inactive spot next to it. What that means is each unit is triggering off of part of a digit. Notice that there aren’t really dark areas around the edges because the digits are centered in the image. Here’s a reconstruction from a single unit in the second layer:

Reconstruction from a single hidden unit in the seond layer

Reconstruction from a single hidden unit in the second layer

It gets harder to see what’s going on at this point but the main thing that is visible is that some of these have more than one area active and others have very tight localized spots that are active. Getting these plots for the first time wasn’t exactly straight forward so I thought I’d share my code:

# %% Probe the bias units
probe1 = dbn.rbm_rev(model_rbm1, np.matrix(np.zeros((N_hid1, 1))))
probe2 = dbn.rbm_rev(model_rbm1, dbn.rbm_rev(model_rbm2, np.matrix(np.zeros((N_hid2, 1)))))

plt.subplot(1, 3, 1)
plt.axis('off')
plt.imshow(np.reshape(np.mean(train['data'],axis=1),image_shape), cmap=plt.cm.gray_r, interpolation='nearest')
plt.title('Mean Image')
plt.subplot(1, 3, 2)
plt.axis('off')

plt.imshow(np.reshape(probe1,image_shape), cmap=plt.cm.gray_r, interpolation='nearest')
plt.title('Rev Bias 1')
plt.subplot(1, 3, 3)
plt.axis('off')
plt.imshow(np.reshape(probe2,image_shape), cmap=plt.cm.gray_r, interpolation='nearest')
plt.title('Rev Bias 2')

# %% Probe the first RBM
bias1_probe = dbn.rbm_rev(model_rbm1, np.matrix(np.zeros((N_hid1, N_hid1))))
W1_probe = dbn.rbm_rev(model_rbm1, np.matrix(np.eye(N_hid1)))
rbm1_probe = W1_probe - bias1_probe

plt_shape1 = (6,12)
plt.figure(figsize=(15.,9.))
for i in range(np.prod(plt_shape1)):
    plt.subplot(plt_shape1[0], plt_shape1[1], i+1)
    plt.axis('off')
    plt.imshow(np.reshape(rbm1_probe[:,i],image_shape), cmap=plt.cm.gray_r, interpolation='nearest')

# %% Probe the second RBM
bias2_probe = dbn.rbm_rev(model_rbm1, dbn.rbm_rev(model_rbm2, np.matrix(np.zeros((N_hid2, N_hid2)))))
W2_probe = dbn.rbm_rev(model_rbm1, dbn.rbm_rev(model_rbm2, np.matrix(np.eye(N_hid2))))
rbm2_probe = W2_probe - bias2_probe

plt_shape2 = (6,12)
plt.figure(figsize=(15.,9.))
for i in range(np.prod(plt_shape2)):
    plt.subplot(plt_shape2[0], plt_shape2[1], i+1)
    plt.axis('off')
    plt.imshow(np.reshape(rbm2_probe[:,i],image_shape), cmap=plt.cm.gray_r, interpolation='nearest')

Results

I trained each RBM layer with 30 passes through the data, 10 passes for the softmax layer, and then finished up with 10 passes of backpropagation for the whole DNN. The resulting model got me my highest score on the MNIST digit challenge on Kaggle, 97% accurate, which I believe is on par with humans. At the very least it beats my 96% that I got with K Nearest Neighbors.

Stay tuned for a comparison of the Python implementation to the R implementation with a bit about optimization.

p.s. Once my code structure settles down, I plan to update this with a link to github repo for the full thing.

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30 Days of Python: Day 30 Saving the Whales

I’ve made a small project every day in python for the past 30 days (minus some vacation days). I’ve learned many new packages and  made a wide variety of projects, including games, computer tools, machine learning, and some science. It was a lot of fun.

Day 30: Saving the Whales

For my last project, I thought I would revisit the Whale Detection Competition on Kaggle that I competed in last year. The goal was to train a model that could detect the presence of a whale call in an audio file. The data was provided by Cornell and consisted of 2 second sound recordings  from buoys out in Massachusetts Bay of either background ocean noise or a Right whale’s “up call” which starts low and rises up. The Right whale is endangered (only 400 left) and doesn’t call out very often so it can be harder to detect than, say, a Humpback whale, so better detection algorithms will help save the whales from being hit by shipping traffic.

I did pretty well last year in the competition, scoring 0.95 Area Under the Curve score (AUC) where perfect would be 1.0. I utilized the deep learning models that I learned in Geoffery Hinton’s Coursera course on Neural Networks to build the models but I did all of the work in R, which brings me to the present project.

My main tool over the years for data analysis has been Matlab at school and then at work, but last year, I learned R as an open source alternative. I took the deep belief net code that I learned from the Hinton course and retooled it from Matlab to R. I added evaluation features and hyper-parameters  for controlling various learning rates and just in general kept developing that code to work on further projects.

But the R code was clunky and messy. It got harder and harder to add new features each building on previous functions. Additionally a lot of the algorithms I wanted to try out and learn were in Matlab or Python (for instance the winning solution to the whale detection challenge). This was one of the big motivations behind learning Python. So in order to fully transition from R to Python I thought I would take the time to rework the Whale detection code into Python and learn about the various data tools in the process.

Fair warning: I didn’t finish the deep learning portion of the project, but I walk through what I did complete and show how a simple model fairs with the feature set that I constructed.

Pre-processing and Feature Extraction:

The data is a zipped up folder of .aiff files, so the first thing that’s necessary to build a program to read in the files and extract whatever features are needed for the model. In R there was no direct way to read in a .aiff file so I had to run the sox tool in a .bat file to convert the files into .wav files. To my delight not only is there an easy way to read .aiff files in Python, but it is part of the standard modules – batteries included so to speak.

With the file ingested, I converted it to a numpy array and then used matplotlib to plot a spectrogram of the audio file. A spectrogram is way of examining the spectrum of a time signal as it evolves over time. Specifically it takes short chunks of time, computes the FFT, and then plots these snapshots of the spectrum on the y axis versus time on the x axis (amplitude of the spectrum is intensity in the image).

Here’s how to do that in code:

plt.figure(figsize=(18.,12.))
for i, file_name in enumerate(file_names[j*N_plot:(j+1)*N_plot]):
    f = aifc.open(os.path.join(data_loc,train_folder,file_name), 'r')
    str_frames = f.readframes(f.getnframes())
    Fs = f.getframerate()
    time_data = np.fromstring(str_frames, np.short).byteswap()
    f.close()

    # spectrogram of file
    plt.subplot(N_plot/4, 4, i+1)
    Pxx, freqs, bins, im = plt.specgram(time_data,Fs=Fs,noverlap=90,cmap=plt.cm.gist_heat)
    plt.title(file_name+' '+file_name_to_labels[file_name])

Instead of just looking at one file, which might not be a great example and would only show either a whale call or not a whale call, I used matplotlib to tile multiple images to get a better sense of the data. This was way easier to do then my experiences with R and being able to easily control its size was easier than Matlab.

Here’s what some of those look like:

Whale Call Spectrograms

Right Whale Call Spectrograms (calls are labelled 1)

To make this useable as inputs to the data model, I needed the raw data that went into the image that matplotlib created, which was readily available in the data returned by the specgram function. As is this would yield almost 3000 features per audio file which is too much for my computer to handle (there are 30,000 audio files). So I used the frequency vector and bin vector (time) to eliminate the lowest and highest frequencies as well as the beginning and ending of each clip. The result was reduced to 600 features per clip, which is more manageable.

I turned the plotting routine into a function, wrapped that in a list comprehension which looped over each file in the the list of files and finally constructed a numpy array out of the resulting list. I used cPickle to save this to disk so I wouldn’t need to repeat it. This portion of the project took me a while to do since I had never done any of these operations before and my original whales project was quite a while ago.

Building Restricted Boltzmann Machines and Deep Belief Nets

Unfortunately, I ran out of time and was unable to complete the conversion of the stacked RBM code. I did however complete the optimize function that could perform the model updates and I was able to verify that the RBM executed properly (although I couldn’t test its efficacy).

My deep learning model is a Neural Network constructed from a Deep Belief Net, which in turn is made of stacked Restricted Boltzmann machines. Restricted Boltzmann machines (RBM) are like one layer of a neural network but they are trained in a special way, Contrastive Divergence, that doesn’t require the data labels. This unsupervised learning algorithm seeks to improve the ability of the RBM to represent the data by training it to reconstruct the the input data from the hidden layer of the network. For a better explanation of why this works, I recommend Hinton’s homepage which is full of his papers and lectures.

Here’s the Python version of the Contrastive Divergence algorithm:


def logistic(x):
    '''Computes the logistic'''
    return 1./(1 + np.exp(-x))

def sample_bernoulli(probabilities):
    '''Samples from a bernoulli distribution for each element of the matrix'''
    return np.greater(probabilities, np.random.rand(*np.shape(probabilities))).astype(np.float)

def cd1(model, visible_data):
    '''Computes one iteration of contrastive divergence on the rbm model'''
    N_cases = np.shape(visible_data)[1]

    #forward propagation of the inputs
    vis_prob_0 = visible_data
    vis_states_0 = sample_bernoulli(vis_prob_0)

    hid_prob_0 = logistic(model['W']*vis_states_0 + model['fwd_bias'])
    hid_states_0 = sample_bernoulli(hid_prob_0)

    #reverse propagation to reconstruct the inputs
    vis_prob_n = logistic(model['W'].T*hid_states_0 + model['rev_bias'])
    vis_states_n = sample_bernoulli(vis_prob_n)

    hid_prob_n = logistic(model['W']*vis_states_n + model['fwd_bias'])

    #compute how good the reconstruction was
    vh0 = hid_states_0 * vis_states_0.T / N_cases
    vh1 = hid_prob_n * vis_states_n.T / N_cases

    cd1_value = vh0 - vh1

    model_delta = dict([('W', cd1_value),
                        ('fwd_bias',np.mean(hid_states_0 - hid_prob_n, axis=1)),
                        ('rev_bias',np.mean(visible_data - vis_prob_n, axis=1))])
    return model_delta

A Deep Belief Network (DBN) is a stacked up version of pre-trained RBM models which can then be treated as a Neural Network and fine tuned by the standard back propagation algorithm using the data labels. Because the model is pre-trained on the data, the back propagation step doesn’t have to change as much to get a good model and because the pre-training didn’t use the labels it is less likely to overfit.

K Nearest Neighbors

To achieve some closure in this project, I ran the feature set that I built through the K nearest neighbors algorithm in scikit-learn. The results weren’t great but they were about the same as the Conrell Benchmark for the competition. I really like how easy it is to get all of the reporting tools so easily in python:

Classification report for classifier KNeighborsClassifier(algorithm=auto, leaf_size=30,
metric=minkowski, n_neighbors=5, p=2, weights=uniform):
            precision recall f1-score support
          0      0.84   0.90     0.87   11286
          1      0.61   0.48     0.54    3714

avg / total      0.78   0.79     0.79   15000

Confusion matrix:
    [[10119 1167]
     [ 1923 1791]]
AUC:
0.689413478377

Conclusions

Overall I am very pleased with writing these algorithms in Python. I had to jump through many hoops to get the matrices to work right when I wrote it in R. For Python, the ability to use numpy algorithms over and over again in clear and simple ways was quite nice. I think I was more slowed down by reading my old code in R then writing the Python version, although testing each bit of code to make sure it was right did also take some time.  I decided to end this project early because I knew I wouldn’t be able to write good Python code if I rushed it any more than I already had, and I plan on using Python for a while so it was better to get it right. Rest assured I will follow up with the completion of the conversion to Python; after all, the Whale Detection Challenge inspired half of this blog’s name.

Final thoughts for 30 Days of Python

These 30 days have been a great experience. I did find the process quite exhausting at times and I wasn’t always sure I would get through it. I came out the other side though with a lot more knowledge of how to do useful things in Python. I’ve already started applying this knowledge at work. I hope to continue this learning process at a slower pace and also take the time to dive into some deeper projects that I thought of while doing my 30 days. I want to thank everyone who left comments, liked a post (here or on google+), or even just read what I wrote. Knowing that people were paying attention really kept me to my schedule and I learned a lot of useful information from people’s feedback.

Thanks,
Robb

30 Days of Python: Day 20 MNIST Digit Recognition

I’m making a small project every day in python for the next 30 days (minus some vacation days). I’m hoping to learn many new packages and  make a wide variety of projects, including games, computer tools, machine learning, and maybe some science. It should be a good variety and I think it will be a lot of fun.

Day 20: MNIST Digit Recognition

I took a crack at the digit recognition task on Kaggle today. I’m using Python’s scikit-learn package to learn a model for recognizing which hand written digit is present. I included in my code a preview of what the images look like:

Sample MNIST Digits

Sample MNIST Digits

 

The data features for each image are the 28×28 pixels unwrapped into 784 element array. I tried three different models: Logistic Regression, SVM, and KNN. Something went wrong with the SVM model (I’m guessing it couldn’t handle the integer values I originally gave it), so I didn’t end up using it. Because the other two were working and taking a while to train I didn’t spend much time on debugging it. I got fairly good results initially: LR had 88% precision and KNN had 96% precision.

A feature I used the metrics module’s classification_report to estimate how well my model did before I submitted it. I split the training data into training and validation data, predicted on the validation data without any extra training on it, and the used the classification report to tell me how the model did. For the classification report, the LR came out with 89% and the KNN with 96% which is very close to the actual scores I achieved. Another feature of the metrics module is the confusion_matrix which can show you where the classifications go awry. Here’s the output from the KNN classification report and confusion matrix:

Classification report for classifier KNeighborsClassifier(algorithm=auto, leaf_size=30,
 metric=minkowski, n_neighbors=5, p=2, weights=uniform):

  precision recall f1-score support

0      0.97   0.99     0.98    2013
1      0.94   1.00     0.97    2349
2      0.98   0.95     0.96    2042
3      0.95   0.96     0.96    2199
4      0.97   0.96     0.96    1999
5      0.96   0.95     0.96    1877
6      0.97   0.98     0.98    2122
7      0.95   0.97     0.96    2261
8      0.98   0.91     0.95    2019
9      0.94   0.94     0.94    2119

avg / total 0.96 0.96 0.96 21000

Confusion matrix:
[[2000    2    1    0    0    3    5    1    0    1]
 [   0 2338    4    0    1    1    0    4    0    1]
 [  17   29 1936    5    2    3    4   38    5    3]
 [   3    8   13 2121    0   22    1   10   11   10]
 [   2   23    0    0 1918    0    8    4    0   44]
 [  12    7    0   30    3 1778   27    2    6   12]
 [  16    8    1    0    3    7 2086    0    1    0]
 [   0   30    7    1    4    0    0 2189    0   30]
 [   7   23    6   53   16   29   12    8 1846   19]
 [   7   10    3   22   32    3    0   38    7 1997]]

Another feature of scikit-learn that I decided to check out was the preprocessing module, namely the StandardScaler which can learn the mean and variance of the training data and then can be used to center and scale the data to have a mean of 0 and variance of 1. This is useful to avoid fitting to spurious effects in the training data (say all of the ones just happened to have a particular lighting effect). The StandardScaler is as easy to use as the classifiers:

...
scaler = preprocessing.StandardScaler()
scaler.fit(processed_data)
processed_data = scaler.transform(processed_data)
...
test_data = scaler.transform(np.array(line).astype(np.int))

The results for this were mixed. It improved the logistic regression fit to 89% but the KNN scored only a 93%. Again the classification report predicted the results correctly (LR 90%, KNN 93%). Even though the StandardScaler didn’t help much in this case, I’m glad I took the time to learn how to use it. I’m also really glad to find out about the metrics module. I’ve had to make that sort of function before and it’s tricky to get it robust and useful. I’m really glad to see that it’s just included (a great example of the batteries included philosophy of Python). The classification report is a must for Kaggle competitions given the limited number of submissions you get per day. Knowing your performance before you submit is definitely an edge. I’d like to try out either Neural Nets, Restricted Boltzmann Machines, or something else from that family of algorithms on the dataset but that will have to wait for another project.

This will be my last post for a few days as I will be on vacation. But I will be back with the rest of my 30 days!