Variational Auto-Encoder (VAE)

Zhenwen Dai (2018-8-21)

import mxfusion as mf
import mxnet as mx
import numpy as np
import mxnet.gluon.nn as nn
import mxfusion.components
import mxfusion.inference
%matplotlib inline
from pylab import *

/Users/zhenwend/anaconda3/lib/python3.6/site-packages/h5py/__init__.py:36: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  from ._conv import register_converters as _register_converters

Load a toy dataset

import GPy
data = GPy.util.datasets.oil_100()
Y = data['X']
label = data['Y'].argmax(1)

N, D = Y.shape

Model Defintion

Q = 2

H = 50
encoder = nn.HybridSequential(prefix='encoder_')
with encoder.name_scope():
    encoder.add(nn.Dense(H, activation="tanh"))
    encoder.add(nn.Dense(H, activation="tanh"))
    encoder.add(nn.Dense(Q, flatten=True))
encoder.initialize(mx.init.Xavier(magnitude=3))
_=encoder(mx.nd.array(np.random.rand(5,D)))

H = 50
decoder = nn.HybridSequential(prefix='decoder_')
with decoder.name_scope():
    decoder.add(nn.Dense(H, activation="tanh"))
    decoder.add(nn.Dense(H, activation="tanh"))
    decoder.add(nn.Dense(D, flatten=True))
decoder.initialize(mx.init.Xavier(magnitude=3))
_=decoder(mx.nd.array(np.random.rand(5,Q)))

from mxfusion.components.var_trans import PositiveTransformation

m = mf.models.Model()
m.N = mf.components.Variable()
m.decoder = mf.components.functions.MXFusionGluonFunction(decoder, nOutputs=1,broadcastable=False)
m.x = mf.components.distributions.Normal.define_variable(mean=mx.nd.array([0]), variance=mx.nd.array([1]), shape=(m.N, Q))
m.f = m.decoder(m.x)
m.noise_var = mf.components.Variable(shape=(1,), transformation=PositiveTransformation(), initial_value=mx.nd.array([0.01]))
m.y = mf.components.distributions.Normal.define_variable(mean=m.f, variance=m.noise_var, shape=(m.N, D))
m.show()

x ~ Normal(mean=Variable(61c66), variance=Variable(3581c))
f = GluonFunctionEvaluation(decoder_dense0_weight=Variable(1e112), decoder_dense0_bias=Variable(12aa9), decoder_dense1_weight=Variable(d74ee), decoder_dense1_bias=Variable(e16ac), decoder_dense2_weight=Variable(9a998), decoder_dense2_bias=Variable(f53b6), decoder_input_0=x)
y ~ Normal(mean=f, variance=noise_var)

q = mf.models.Posterior(m)
q.x_var = mf.components.Variable(shape=(1,), transformation=PositiveTransformation(), initial_value=mx.nd.array([1e-6]))
q.encoder = mf.components.functions.MXFusionGluonFunction(encoder, nOutputs=1, broadcastable=False)
q.x_mean = q.encoder(q.y)
q.x.set_prior(mf.components.distributions.Normal(mean=q.x_mean, variance=q.x_var))
q.show()

x_mean = GluonFunctionEvaluation(encoder_dense0_weight=Variable(2b695), encoder_dense0_bias=Variable(dc41f), encoder_dense1_weight=Variable(a3272), encoder_dense1_bias=Variable(5387d), encoder_dense2_weight=Variable(75bd2), encoder_dense2_bias=Variable(b470d), encoder_input_0=y)
x ~ Normal(mean=x_mean, variance=x_var)

Variational Inference

from mxfusion.inference import BatchInferenceLoop, StochasticVariationalInference, GradBasedInference

observed = [m.y]
alg = StochasticVariationalInference(num_samples=3, model=m, posterior=q, observed=observed)
infr = GradBasedInference(inference_algorithm=alg, grad_loop=BatchInferenceLoop())

infr.run(max_iter=2000, learning_rate=1e-2, y=mx.nd.array(Y), verbose=True)

 /Users/zhenwend/mxfusion/src/MXFusion/mxfusion/inference/inference_parameters.py:52: UserWarning:InferenceParameters has already been initialized.  The existing one will be overwritten.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'decoder_dense0_weight' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'decoder_dense0_bias' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'decoder_dense1_weight' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'decoder_dense1_bias' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'decoder_dense2_weight' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'decoder_dense2_bias' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'encoder_dense0_weight' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'encoder_dense0_bias' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'encoder_dense1_weight' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'encoder_dense1_bias' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'encoder_dense2_weight' is already initialized, ignoring. Set force_reinit=True to re-initialize.
 /Users/zhenwend/anaconda3/lib/python3.6/site-packages/mxnet/gluon/parameter.py:689: UserWarning:Parameter 'encoder_dense2_bias' is already initialized, ignoring. Set force_reinit=True to re-initialize.

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Iteration 1823 logL: 1184.36865234375
Iteration 1824 logL: 1179.5299072265625
Iteration 1825 logL: 1195.7923583984375
Iteration 1826 logL: 1207.087890625
Iteration 1827 logL: 1137.3955078125
Iteration 1828 logL: 1137.027587890625
Iteration 1829 logL: 1150.9666748046875
Iteration 1830 logL: 1153.437255859375
Iteration 1831 logL: 1195.476318359375
Iteration 1832 logL: 1201.533935546875
Iteration 1833 logL: 1209.5994873046875
Iteration 1834 logL: 1133.618896484375
Iteration 1835 logL: 1119.32373046875
Iteration 1836 logL: 1163.4991455078125
Iteration 1837 logL: 1176.7540283203125
Iteration 1838 logL: 1182.48046875
Iteration 1839 logL: 1180.406982421875
Iteration 1840 logL: 1139.6094970703125
Iteration 1841 logL: 1161.464599609375
Iteration 1842 logL: 1150.23486328125
Iteration 1843 logL: 1201.280517578125
Iteration 1844 logL: 1171.5853271484375
Iteration 1845 logL: 1145.40234375
Iteration 1846 logL: 1153.2955322265625
Iteration 1847 logL: 1182.4085693359375
Iteration 1848 logL: 1182.6571044921875
Iteration 1849 logL: 1121.10498046875
Iteration 1850 logL: 1168.7552490234375
Iteration 1851 logL: 1215.7532958984375
Iteration 1852 logL: 1168.606201171875
Iteration 1853 logL: 1156.6822509765625
Iteration 1854 logL: 1185.0814208984375
Iteration 1855 logL: 1208.6463623046875
Iteration 1856 logL: 1177.5478515625
Iteration 1857 logL: 1141.1156005859375
Iteration 1858 logL: 1191.260498046875
Iteration 1859 logL: 1197.61767578125
Iteration 1860 logL: 1227.127197265625
Iteration 1861 logL: 1163.735107421875
Iteration 1862 logL: 1154.60986328125
Iteration 1863 logL: 1187.0550537109375
Iteration 1864 logL: 1213.536865234375
Iteration 1865 logL: 1185.064453125
Iteration 1866 logL: 1157.4537353515625
Iteration 1867 logL: 1200.761962890625
Iteration 1868 logL: 1222.02783203125
Iteration 1869 logL: 1147.2158203125
Iteration 1870 logL: 1144.282958984375
Iteration 1871 logL: 1199.695068359375
Iteration 1872 logL: 1214.810791015625
Iteration 1873 logL: 1194.4130859375
Iteration 1874 logL: 1198.478515625
Iteration 1875 logL: 1210.9346923828125
Iteration 1876 logL: 1211.574462890625
Iteration 1877 logL: 1205.450439453125
Iteration 1878 logL: 1172.6024169921875
Iteration 1879 logL: 1174.087646484375
Iteration 1880 logL: 1203.639892578125
Iteration 1881 logL: 1190.721435546875
Iteration 1882 logL: 1192.437744140625
Iteration 1883 logL: 1220.7801513671875
Iteration 1884 logL: 1226.697509765625
Iteration 1885 logL: 1211.2330322265625
Iteration 1886 logL: 1160.687255859375
Iteration 1887 logL: 1158.32666015625
Iteration 1888 logL: 1117.36279296875
Iteration 1889 logL: 1082.3958740234375
Iteration 1890 logL: 1176.63623046875
Iteration 1891 logL: 1232.69580078125
Iteration 1892 logL: 1152.8900146484375
Iteration 1893 logL: 1107.21142578125
Iteration 1894 logL: 1147.4429931640625
Iteration 1895 logL: 1213.348388671875
Iteration 1896 logL: 1174.9425048828125
Iteration 1897 logL: 1107.7596435546875
Iteration 1898 logL: 1107.3375244140625
Iteration 1899 logL: 1192.389404296875
Iteration 1900 logL: 1214.0885009765625
Iteration 1901 logL: 1117.0269775390625
Iteration 1902 logL: 1143.033447265625
Iteration 1903 logL: 1161.20361328125
Iteration 1904 logL: 1222.047119140625
Iteration 1905 logL: 1183.93896484375
Iteration 1906 logL: 1175.2034912109375
Iteration 1907 logL: 1191.1968994140625
Iteration 1908 logL: 1223.955810546875
Iteration 1909 logL: 1210.2122802734375
Iteration 1910 logL: 1181.78759765625
Iteration 1911 logL: 1210.9541015625
Iteration 1912 logL: 1204.458984375
Iteration 1913 logL: 1196.3955078125
Iteration 1914 logL: 1206.798095703125
Iteration 1915 logL: 1198.698974609375
Iteration 1916 logL: 1188.166259765625
Iteration 1917 logL: 1203.7275390625
Iteration 1918 logL: 1186.60546875
Iteration 1919 logL: 1183.64013671875
Iteration 1920 logL: 1230.890380859375
Iteration 1921 logL: 1194.9090576171875
Iteration 1922 logL: 1123.248779296875
Iteration 1923 logL: 1112.940673828125
Iteration 1924 logL: 1139.887939453125
Iteration 1925 logL: 1239.1982421875
Iteration 1926 logL: 1157.0389404296875
Iteration 1927 logL: 1124.8778076171875
Iteration 1928 logL: 1168.7947998046875
Iteration 1929 logL: 1134.8958740234375
Iteration 1930 logL: 1171.053955078125
Iteration 1931 logL: 1164.15185546875
Iteration 1932 logL: 1136.9044189453125
Iteration 1933 logL: 1129.55078125
Iteration 1934 logL: 1136.8323974609375
Iteration 1935 logL: 1210.345703125
Iteration 1936 logL: 1214.7027587890625
Iteration 1937 logL: 1189.9776611328125
Iteration 1938 logL: 1247.0433349609375
Iteration 1939 logL: 1231.4537353515625
Iteration 1940 logL: 1184.2755126953125
Iteration 1941 logL: 1211.376708984375
Iteration 1942 logL: 1223.527587890625
Iteration 1943 logL: 1248.0943603515625
Iteration 1944 logL: 1256.9500732421875
Iteration 1945 logL: 1241.0760498046875
Iteration 1946 logL: 1246.47509765625
Iteration 1947 logL: 1227.07080078125
Iteration 1948 logL: 1244.85693359375
Iteration 1949 logL: 1223.26611328125
Iteration 1950 logL: 1218.07666015625
Iteration 1951 logL: 1259.81201171875
Iteration 1952 logL: 1214.4500732421875
Iteration 1953 logL: 1218.712646484375
Iteration 1954 logL: 1271.29638671875
Iteration 1955 logL: 1250.2908935546875
Iteration 1956 logL: 1225.498291015625
Iteration 1957 logL: 1218.0775146484375
Iteration 1958 logL: 1243.34521484375
Iteration 1959 logL: 1228.5477294921875
Iteration 1960 logL: 1213.529052734375
Iteration 1961 logL: 1214.8626708984375
Iteration 1962 logL: 1231.824462890625
Iteration 1963 logL: 1230.359619140625
Iteration 1964 logL: 1250.611572265625
Iteration 1965 logL: 1241.3177490234375
Iteration 1966 logL: 1239.142822265625
Iteration 1967 logL: 1231.14208984375
Iteration 1968 logL: 1259.7193603515625
Iteration 1969 logL: 1221.607421875
Iteration 1970 logL: 1226.014404296875
Iteration 1971 logL: 1216.6728515625
Iteration 1972 logL: 1205.4998779296875
Iteration 1973 logL: 1214.89208984375
Iteration 1974 logL: 1222.856689453125
Iteration 1975 logL: 1237.81689453125
Iteration 1976 logL: 1208.667236328125
Iteration 1977 logL: 1146.57763671875
Iteration 1978 logL: 1131.6876220703125
Iteration 1979 logL: 1097.2244873046875
Iteration 1980 logL: 1074.828369140625
Iteration 1981 logL: 1098.2362060546875
Iteration 1982 logL: 1106.648681640625
Iteration 1983 logL: 1064.2252197265625
Iteration 1984 logL: 1102.831787109375
Iteration 1985 logL: 1179.8662109375
Iteration 1986 logL: 1138.1878662109375
Iteration 1987 logL: 1111.7362060546875
Iteration 1988 logL: 1141.1280517578125
Iteration 1989 logL: 1190.904296875
Iteration 1990 logL: 1222.82470703125
Iteration 1991 logL: 1200.0384521484375
Iteration 1992 logL: 1195.4183349609375
Iteration 1993 logL: 1223.42919921875
Iteration 1994 logL: 1202.7012939453125
Iteration 1995 logL: 1181.566650390625
Iteration 1996 logL: 1211.32568359375
Iteration 1997 logL: 1247.471435546875
Iteration 1998 logL: 1243.31005859375
Iteration 1999 logL: 1203.407470703125
Iteration 2000 logL: 1221.434326171875

Plot the training data in the latent space

q_x_mean = q.encoder.block(mx.nd.array(Y)).asnumpy()

for i in range(3):
    plot(q_x_mean[label==i,0], q_x_mean[label==i,1], '.')