| # Method |
|
|
| All the architectures were built using Keras. The loss is categorical cross entropy, adam optimizer and learning rate of 0.001 was used |
|
|
| ``` {.python language="Python" style="mystyle"} |
| model.add(Dense(128, activation='relu', input_dim=input_shape)) |
| model.add(Dropout(0.5)) |
| model.add(Dense(256, activation='relu')) |
| model.add(Dense(256, activation='relu')) |
| model.add(Dropout(0.5)) |
| model.add(Dense(128, activation='relu')) |
| model.add(Dropout(0.5)) |
| model.add(Dense(64, activation='relu')) |
| model.add(Dropout(0.5)) |
| model.add(Dense(output_shape, activation='softmax')) |
| ``` |
|
|
| ``` {.python language="Python" style="mystyle"} |
| model = Sequential() |
| model.add(Dense(25, |
| kernel_initializer=keras.initializers.glorot_normal(seed=0), kernel_regularizer=keras.regularizers.l2(1e-4))) |
| model.add(Activation('relu')) |
| model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=0))), kernel_regularizer=keras.regularizers.l2(1e-4))) |
| model.add(BatchNormalization()) |
| model.add(Activation('relu')) |
| model.add(Dropout(0.3)) |
| model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='softmax')) |
| ``` |
|
|
| ``` {.python language="Python" style="mystyle"} |
| model = Sequential() |
| model.add(Dense(20, kernel_initializer=keras.initializers.glorot_normal(seed=5), activation='relu')) |
| model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=5), activation='relu')) |
| model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=5)))#, activation='softmax')) |
| model.add(Activation('softmax')) |
| ``` |
|
|
| ``` {.python language="Python" style="mystyle"} |
| model = Sequential() |
| model.add(Dense(35, kernel_initializer=keras.initializers.glorot_normal(seed=5))) |
| model.add(Activation('relu')) |
| model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=5))) |
| model.add(BatchNormalization()) |
| model.add(Activation('relu')) |
| model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=5)))#, activation='softmax')) |
| model.add(Dropout(0.5)) |
| model.add(Activation('softmax')) |
| ``` |
|
|
| ``` {.python language="Python" style="mystyle"} |
| model = Sequential() |
| model.add(Dense(15, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) |
| model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) |
| model.add(Dropout(0.2)) |
| model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='softmax')) |
| ``` |
|
|
| ``` {.python language="Python" style="mystyle"} |
| model = Sequential() |
| model.add(Dense(40, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) |
| model.add(Dense(25, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) |
| model.add(Dense(10, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='relu')) |
| model.add(Dropout(0.2)) |
| model.add(Dense(num_classes, kernel_initializer=keras.initializers.glorot_normal(seed=0), activation='softmax')) |
| ``` |
|
|
| ::: small |
| ``` {caption="We present below the top 5 boolean rule based features used by Logistic regression (with L1 penalty) on the Sky Survey Dataset along with their feature importances."} |
| |
| GBFL rank 1 feature |
| |
| 1.51 >= dirf1 >= 0.0 & 0.66 >= dirf2 >= 0.0 & |
| 0.26 >= dirf3 >= 0.0 & 629.05 >= fiberid >= 419.36 & |
| 0.80 >= redshift >= 0.0 & 233.35 >= ra >= 137.26 & |
| 57481 >= mjd >= 42354.42 & 14.42 >= dec >= 0.0 & |
| 1770.52 >= plate >= 0.0 |
| |
| |
| GBFL rank 2 feature |
| |
| 2.53 >= dirf1 >= 0.50 & |
| 0.49 >= dirf2 >= 0.0 & |
| 0.35 >= dirf3 >= 0.0 & |
| 2213.15 >= plate >= 442.63 & |
| 14.42 >= dec >= 0.0 & |
| 0.80 redshift >= 0.0 & |
| 262.10 >= fiberid >= 52.42 & |
| 164.72 >= ra >= 109.81 & |
| 57481 >= mjd >= 42354.42 |
| |
| |
| GBFL rank 3 feature |
| |
| 1.51 >= dirf1 >= 0.0 & |
| 0.49 >= dirf2 >= 0.0 & |
| 0.35 >= dirf3 >= 0.0 & |
| 260.81 >= ra >= 233.35 & |
| 0.80 >= redshift >= 0.0 & |
| 524.21 >= fiberid >= 157.26 & |
| 14.42 >= dec >= 0.0 & |
| 1770.52 >= plate >= 0.0 & |
| 57481 >= mjd >= 42354.42 |
| |
| |
| GBFL rank 4 feature |
| |
| 2.53 >= dirf1 >= 0.50 & |
| 0.49 >= dirf2 >= 0.0 & |
| 0.44 >= dirf3 >= 0.089 & |
| 576.63 >= fiberid >= 366.94 & |
| 260.81 >= ra >= 233.35 & |
| 14.42 >= dec >= 0.0 & |
| 0.80 >= redshift >= 0.0 & |
| 1770.52 >= plate >= 0.0 & |
| 57481.0 >= mjd >= 42354.42 |
| |
| GBFL rank 5 feature |
| |
| 2.53 >= dirf1 >= 0.50 & |
| 0.49 >= dirf2 >= 0.0 & |
| 0.35 >= dirf3 >= 0.0 & |
| 10.82 >= dec >= 0.0 & |
| 52.42 >= fiberid >= 0.0 & |
| 178.44 >= ra >= 123.54 & |
| 0.80 >= redshift >= 0.0 & |
| 1770.52 >= plate >= 0.0 & |
| 57481.0 >= mjd >= 42354.42 |
| ``` |
| ::: |
|
|
| ::: small |
| ``` {caption="We present below the top 5 boolean rule based features used by the decision tree on the WDBC dataset ranked by their importance."} |
| |
| |
| GBFL rank 1 feature |
| |
| 3575.86>=n2_area>=863.33 & |
| 36.04>=n2_radius>=17.29 & |
| n1_fractald<=0.01 & n0_concavity<=0.28 & |
| n1_area<=363.87 & n1_compactness<=0.09 & |
| n1_concavepts<=0.03 & n0_symmetry>=0.13 & |
| n2_concavity<=0.83 & n2_fractald<=0.15 & |
| n0_area<=2108.08 & n0_smoothness<=0.14 & |
| n0_fractald>=0.04 & n0_concavepts<=0.16 & |
| n2_texture<=43.28 & n2_smoothness<=0.19 & |
| n0_perimeter<=188.5 & n2_symmetry>=0.15 & |
| n2_concavepts<=0.27 & n0_texture>=9.71 & |
| n0_compactness<=0.29 & n1_radius>=0.11 & |
| n1_texture>=0.36 & n1_perimeter>=0.75 & |
| n1_smoothness>=0 & n1_concavity>=0 & |
| n1_symmetry>=0 & n2_perimeter<=251.2 & |
| n0_radius>=10.5 & n2_compactness<=0.71 |
| |
| |
| GBFL rank 2 feature |
| |
| n0_concavity>=0.07 & n2_concavepts>=0.13 & |
| 7.22<=n1_area<=274.71 & |
| 0<=n1_fractald<=0.01 & |
| 185.2<=n2_area<=2219.6 & |
| n0_perimeter<=140.26 & |
| 0<=n1_compactness<=0.09 & |
| 0<=n1_concavepts<=0.03 & |
| 7.93<=n2_radius<=26.66 & |
| 0.01<=n0_compactness<=0.29 & |
| 0<=n0_concavity<=0.35 & |
| 12.02<=n2_texture<=43.28 & |
| 0.02<=n2_compactness<=0.88 & |
| 0.05<=n2_fractald<=0.18 & |
| 0.07<=n0_smoothness<=0.16 & |
| 0.12<=n2_smoothness<=0.22 & |
| 0.2<=n2_concavity<=1.25 & |
| 0.09<=n2_concavepts<=0.27 & |
| 21.06>=n0_radius>=6.98 & |
| n0_texture>=9.71 & n0_concavepts>=0.0 & |
| 1322.25>=n0_area>=143.5 & |
| n0_fractald>=0.04 & 1.49>=n1_radius>=0.11 & |
| 2.62>=n1_texture>=0.36 & |
| 11.36>=n1_perimeter>=0.75 & |
| n1_smoothness>=0 & n1_concavity>=0 & |
| 0.04>=n1_symmetry>=0 & |
| n2_perimeter>=50.41 & |
| n2_symmetry>=0.15 & n0_symmetry>=0.13 |
| |
| |
| GBFL rank 3 feature |
| |
| n0_concavity>=0.07 & n0_texture>=19.56 & |
| n1_area<=274.71 & n1_compactness<=0.06 & |
| n1_fractald<=0.01 & n2_compactness<=0.54 & |
| n2_fractald<=0.13 & n0_compactness<=0.23 & |
| n1_concavepts<=0.03 & n2_area<=2897.73 & |
| n2_concavity<=0.83 & n0_area<=2108.08 & |
| n0_smoothness<=0.14 & n0_concavity<=0.35 & |
| n0_concavepts<=0.16 & n1_smoothness<=0.01 & |
| n2_radius<=31.35 & n0_texture<=39.28 & |
| n0_perimeter<=188.5 & n2_texture<=49.54 & |
| n2_smoothness<=0.2226 & n0_symmetry>=0.106 & |
| n0_fractald>=0.04 & n1_radius>=0.11 & |
| n1_texture>=0.36 & n1_perimeter>=0.75 & |
| n1_concavity>=0 & n1_symmetry>=0 & |
| n2_perimeter>=50.41 & n2_symmetry>=0.15 & |
| n0_radius>=10.50 & n2_concavepts>=0.04 |
| |
| GBFL rank 4 feature |
| |
| nucleus2_area >= 863.33 & nucleus2_radius >= 17.29 & |
| nucleus1_compactness <= 0.06 & nucleus1_fractal_dim <= 0.01 & |
| nucleus2_fractal_dim <= 0.13 & nucleus1_area <= 363.87 & |
| nucleus1_concave_pts <= 0.03 & nucleus2_compactness <= 0.71 & |
| nucleus0_smoothness <= 0.14 & nucleus0_compactness <= 0.29 & |
| nucleus2_texture <= 43.28 & nucleus2_concavity <= 1.04 & |
| nucleus0_perimeter <= 188.5 & nucleus0_area <= 2501.0 & |
| nucleus0_concavity <= 0.42 & nucleus0_concave_pts <= 0.20 & |
| nucleus2_radius <= 36.04 & nucleus2_area <= 4254.0 & |
| nucleus2_smoothness <= 0.22 & nucleus0_texture >= 9.71 & |
| nucleus0_symmetry >= 0.10 & nucleus0_fractal_dim >= 0.04 & |
| nucleus1_radius >= 0.11 & nucleus1_texture >= 0.36 & |
| nucleus1_perimeter >= 0.75 & nucleus1_smoothness >= 0 & |
| nucleus1_concavity >= 0 & nucleus1_symmetry >= 0 & |
| nucleus2_symmetry >= 0.15 & nucleus0_radius >= 14.02 & |
| nucleus2_perimeter >= 117.33 & nucleus2_concave_pts >= 0.09 |
| |
| GBFL rank 5 feature |
| |
| nucleus2_concave_pts >= 0.13 & 7.22 <= nucleus1_area <= 274.71 & |
| 0 <= nucleus1_fractal_dim <= 0.01 & 185.2 <= nucleus2_area <= 2219.6 & |
| 0.01 <= nucleus0_compactness <= 0.23 & 0 <= nucleus0_concavity <= 0.28 & |
| 0 <= nucleus0_concave_pts <= 0.13 & 0 <= nucleus1_smoothness <= 0.01 & |
| 0 <= nucleus1_compactness <= 0.09 & 0 <= nucleus1_concave_pts <= 0.03 & |
| 7.93 <= nucleus2_radius <= 26.66 & 0.02 <= nucleus2_compactness <= 0.71 & |
| 0 <= nucleus2_concavity <= 0.83 & 0.05 <= nucleus2_fractal_dim <= 0.15 & |
| 43.79 <= nucleus0_perimeter <= 164.38 & 0.05 <= nucleus0_smoothness <= 0.14 & |
| 0.07 <= nucleus2_smoothness <= 0.19 & 18.27 <= nucleus2_texture <= 49.54 & |
| 0.04 <= nucleus2_concave_pts <= 0.27 & nucleus0_radius >= 6.98 & |
| nucleus0_texture >= 9.71 & nucleus0_area >= 143.5 & |
| nucleus0_symmetry >= 0.10 & nucleus0_fractal_dim >= 0.04 & |
| 1.49 >= nucleus1_radius >= 0.11 & nucleus1_texture >= 0.36 & |
| nucleus1_perimeter >= 0.75 & nucleus1_concavity >= 0.0 & |
| 0.04 >= nucleus1_symmetry >= 0 & nucleus2_perimeter >= 50.41 & |
| 0.42 >= nucleus2_symmetry >= 0.15 |
| ``` |
| ::: |
|
|
| ::: small |
| ``` {caption="Top 5 boolean rule based features used by the decision tree on the Magic dataset ranked by their importance."} |
| |
| GBFL rank 1 feature |
| |
| fSize >= 2.937951724137931 & |
| fSize <= 3.629234482758621 & |
| fAlpha <= 6.206896551724138 & |
| fLength >= 0.0 & |
| fWidth >= 0.0 & |
| fM3Long >= 0.0 & |
| fAlpha >= 0.0 |
| |
| GBFL rank 2 feature |
| |
| fM3Long <= 0.0 & |
| fAlpha <= 9.310344827586206 & |
| fLength >= 0.0 & |
| fWidth >= 0.0 & |
| fAlpha >= 0.0 & |
| fSize >= 2.073848275862069 |
| |
| GBFL rank 3 feature |
| |
| fSize >= 2.7651310344827587 & |
| fAlpha <= 34.13793103448276 & |
| fSize <= 3.456413793103448 & |
| fWidth <= 15.207889655172414 & |
| fLength >= 0.0 & |
| fWidth >= 0.0 & |
| fM3Long >= 0.0 |
| |
| GBFL rank 4 feature |
| |
| fWidth >= 30.415779310344828 & |
| fSize >= 2.937951724137931 & |
| fWidth <= 60.831558620689655 & |
| fSize <= 3.629234482758621 & |
| fM3Long <= 0.0 & |
| fLength >= 0.0 & |
| fAlpha >= 12.413793103448276 |
| |
| GBFL rank 5 features |
| |
| fM3Long >= 15.961793103448276 & |
| fM3Long <= 47.88537931034483 & |
| fWidth <= 7.603944827586207 & |
| fLength <= 34.57003448275862 & |
| fAlpha <= 21.724137931034484 & |
| fLength >= 0.0 & |
| fWidth >= 0.0 & |
| fAlpha >= 9.310344827586206 & |
| fSize >= 2.073848275862069 |
| ``` |
| ::: |
|
|
| **Remark:** Although, the method in [@macem] does not really perform the constrained optimization but uses regularization like 'Elasticnet' penalty to impose sparsity, we will assume that our PPs and PNs are the result of these optimizations just for simplicity of exposition. The only difference is that the sparsity $k$ cannot be pre-determined but is typically a constant for many training samples in practice. |
|
|
