# global dimensionality-reduction (GDR) algorithm 示例



In [1]:

import numpy as np
import pandas as pd
from scipy import linalg
from scipy import optimize
import functools

import tensorly
from tensorly.decomposition import partial_tucker
from tensorly.decomposition import tucker
tensorly.set_backend('numpy')




Using mxnet backend.
Using numpy backend.



### 生成数据集



In [2]:

tensor_steam_length = 300

factors_tensor_list = []
for i in np.arange(tensor_steam_length):
a = np.random.normal(size=[69], scale=0.5)
b = np.random.normal(size=[16], scale=0.5)
c = np.random.normal(size=[32], scale=0.5)
x = np.zeros([1, 69, 16, 32])
x[0,:,0,0] = a
x[0,1,:,1] = b
x[0,2,2,:] = c
factors_tensor_list.append(x)
factors_tensor = np.concatenate(factors_tensor_list)

targets = np.random.normal(scale=0.01, size=[300,1])



### 3.3 $\sum$ 循环向量化

$U^i_k\in\mathbb{R}^{batch \times I_k \times J_k}$

$U^{iT}_k\in\mathbb{R}^{batch \times J_k \times I_k}$

$M^i_k = U^i_k U^{iT}_k\in \mathbb{R}^{batch \times I_k \times I_k}$

$M^i_{k(0)} \in \mathbb{R}^{batch \times I_kI_k}$

• $D_{U_k} = \sum\limits ^N_{i=1}d_{i,i}U^{i}_kU^{iT}_k \\=\sum\limits ^N_{i=1}d_{i,i}M^i_k\\= mat \{(vec(diag(D)))^T M_{k(0)}\}$
• $W_{U_k} = \sum\limits^N_{i=1}\sum\limits ^N_{j=1}w_{i,j}U^i_kU^{jT}_k\\= \sum\limits^N_{i=1}(\sum\limits ^N_{j=i}w_{i,j})U^i_kU^{jT}_k\\= \sum\limits^N_{i=1}(\sum\limits ^N_{j=i}w_{i,j})U^i_kU^{jT}_k\\=\sum\limits^N_{i=1}w_iU^i_kU^{jT}_k\\=\sum\limits^N_{i=1}w_i M^i_{k}\\=mat\{w M_{k(0)} \}$

• $w_{i,j} = \begin{cases}1,\ if\ \ i \le j \ and \ \|y_i -y_j \| \le 5\% \\0, \ otherwise \end{cases}$
• $\sum\limits ^N_{j=1}w_{i,j} = w_i= sum(w_{i,:}, axis=1)$


In [3]:

def get_weighting_of_geometric_structure (targets):
W = (targets - targets.T) / targets.T # 广播
W = np.abs(W) - 0.05 # 转换绝对值并判断相似度
W[W>0.0]=0
W[W<0.0]=1
upper_traingular_matrix = np.eye(W.shape[0]).cumsum(1) # 上三角矩阵掩码
return W * upper_traingular_matrix

W = get_weighting_of_geometric_structure(targets)
return np.expand_dims(W.sum(0),axis=0)

W = get_weighting_of_geometric_structure(targets)



### 循环tucker分解



In [4]:

factors_tensor = tensorly.tensor(factors_tensor)

core_list = []
mode_factors_list = []
for i in range(factors_tensor.shape[0]):
print (i)
core, mode_factors= tucker(factors_tensor[i])
core = np.expand_dims(core, axis=0)
core_list.append(core)
mode_factors_list.append(mode_factors)




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##### 连接张量流


In [5]:

batch_length = tensor_steam_length
a_list = []
b_list = []
c_list = []
for i in range(batch_length):
a = np.expand_dims(mode_factors_list[i][0], axis=0)
b = np.expand_dims(mode_factors_list[i][1], axis=0)
c = np.expand_dims(mode_factors_list[i][2], axis=0)
a_list.append(a)
b_list.append(b)
c_list.append(c)

U1 = np.concatenate(a_list)
U2 = np.concatenate(b_list)
U3 = np.concatenate(c_list)

M1 = U1 * np.transpose(U1, axes=[0,2,1])
M2 = U2 * np.transpose(U2, axes=[0,2,1])
M3 = U3 * np.transpose(U3, axes=[0,2,1])



#### 暂时没有找到较好的批次处理tucker分解的方法，这里特例处理



In [6]:

D_U1_core = np.matmul(D, tensorly.base.unfold(M1, mode=0))
I_k = np.int(np.sqrt(D_U1_core.shape[1]))
D_U1 = tensorly.base.fold(D_U1_core, mode=0, shape=[I_k, I_k])

D_U2_core = np.matmul(D, tensorly.base.unfold(M2, mode=0))
I_k = np.int(np.sqrt(D_U2_core.shape[1]))
D_U2 = tensorly.base.fold(D_U2_core, mode=0, shape=[I_k, I_k])

D_U3_core = np.matmul(D, tensorly.base.unfold(M3, mode=0))
I_k = np.int(np.sqrt(D_U3_core.shape[1]))
D_U3 = tensorly.base.fold(D_U3_core, mode=0, shape=[I_k, I_k])




In [7]:

vec_W = np.expand_dims(W.sum(axis=0), axis=0)

W_U1_core = np.matmul(vec_W, tensorly.base.unfold(M1, mode=0))
I_k = np.int(np.sqrt(W_U1_core.shape[1]))
W_U1 = tensorly.base.fold(W_U1_core, mode=0, shape=[I_k, I_k])

W_U2_core = np.matmul(vec_W, tensorly.base.unfold(M2, mode=0))
I_k = np.int(np.sqrt(W_U2_core.shape[1]))
W_U2 = tensorly.base.fold(W_U2_core, mode=0, shape=[I_k, I_k])

W_U3_core = np.matmul(vec_W, tensorly.base.unfold(M3, mode=0))
I_k = np.int(np.sqrt(W_U3_core.shape[1]))
W_U3 = tensorly.base.fold(W_U3_core, mode=0, shape=[I_k, I_k])



### 二次规划优化估计修正矩阵$V_k$

• $min\ J(V) = trace(V^TD_UV-V^TW_UV)$
• $s.t. \ trace(V^TD_UV)=1$


In [8]:

def objective_function(V, D_U, W_U, J_K):
newshape = [D_U.shape[0], J_K]
V = np.reshape(V,newshape=newshape)
left = np.matmul(np.matmul(V.T, D_U),V)
right = np.matmul(np.matmul(V.T, W_U),V)
return np.trace(left + right)

def constraints(V, D_U):
newshape = [D_U.shape[0], J_K]
V = np.reshape(V,newshape=newshape)
left = np.matmul(np.matmul(V.T, D_U),V)
return np.trace(left)- 1.0



#### $V_1$



In [9]:

objective_function_1 = functools.partial(
objective_function, D_U = D_U1, W_U = W_U1, J_K = 10)

constraints_1 = functools.partial(constraints, D_U = D_U1)

cons_1 = ({'type':'ineq', 'fun':constraints_1})
initial_1 = np.random.normal(scale=0.1, size=D_U1.shape[0]*10).reshape(D_U1.shape[0],10)

V1 = optimize.minimize(objective_function_1, initial_1).x.reshape(D_U1.shape[0],10)



#### $V_2$



In [10]:

objective_function_2 = functools.partial(
objective_function, D_U = D_U2, W_U = W_U2, J_K = 10)

constraints_2 = functools.partial(constraints, D_U = D_U2)

cons_2 = ({'type':'ineq', 'fun':constraints_2})
initial_2 = np.random.normal(scale=0.1, size=D_U2.shape[0]*10).reshape(D_U2.shape[0],10)

V2 = optimize.minimize(objective_function_2, initial_2).x.reshape(D_U2.shape[0],10)



#### $V_3$



In [11]:

objective_function_3 = functools.partial(
objective_function, D_U = D_U3, W_U = W_U3, J_K = 10)

constraints_3 = functools.partial(constraints, D_U = D_U3)

cons_3 = ({'type':'ineq', 'fun':constraints_3})
initial_3 = np.random.normal(scale=0.1, size=D_U3.shape[0]*10).reshape(D_U3.shape[0],10)

V3 = optimize.minimize(objective_function_3, initial_3).x.reshape(D_U3.shape[0],10)



#### $\bar{\mathcal{X}_i} = \mathcal{C}_i \times_1 (V^T_1U^i_1) \times_2 (V^T_2U^i_2) \times_3 (V^T_3U^i_3)$



In [12]:

new_U1 =  np.matmul(V1.T, U1)
new_U2 =  np.matmul(V2.T, U2)
new_U3 =  np.matmul(V3.T, U3)




In [13]:

unfold_mode1 = tensorly.base.partial_unfold(factors_tensor, mode=0, skip_begin=1)
times_mode1 = np.matmul(new_U1, unfold_mode1)
times1_shape = (factors_tensor.shape[0] ,new_U1.shape[1], factors_tensor.shape[2], factors_tensor.shape[3])
times1 = tensorly.base.partial_fold(times_mode1, 0, times1_shape, skip_begin=1, skip_end=0)

unfold_mode2 = tensorly.base.partial_unfold(times1, mode=1, skip_begin=1)
times_mode2 = np.matmul(new_U2, unfold_mode2)
times2_shape = (factors_tensor.shape[0] ,new_U1.shape[1] ,new_U2.shape[1], factors_tensor.shape[3])
times2 = tensorly.base.partial_fold(times_mode2, 1, times2_shape, skip_begin=1, skip_end=0)




In [14]:

unfold_mode3 = tensorly.base.partial_unfold(times2, mode=2, skip_begin=1)
times_mode3 = np.matmul(new_U3, unfold_mode3)
times3_shape = (factors_tensor.shape[0] ,new_U1.shape[1], new_U2.shape[1], new_U3.shape[1])
times3 = tensorly.base.partial_fold(times_mode3, 2, times3_shape, skip_begin=1, skip_end=0)




In [15]:

new_factors_tensor = times3



#### 动态关系捕获和降维 $\mathcal{X}\in\mathbb{R}^{I_1 \times I_2 \times I_3} \to \bar{\mathcal{X}} \in \mathbb{R}^{J_1 \times J_2 \times J_3}$



In [16]:

factors_tensor.shape




Out[16]:

(300, 69, 16, 32)




In [17]:

new_factors_tensor.shape




Out[17]:

(300, 10, 10, 10)




In [18]:

new_factors_tensor




Out[18]:

array([[[[  1.20088711e+18,  -9.39257236e+18,   4.17323386e+18, ...,
1.22671463e+19,   2.83522318e+18,  -8.96128282e+18],
[ -1.85584727e+18,   1.48577156e+19,  -6.74303253e+18, ...,
-1.81974692e+19,  -4.55194623e+18,   1.27374179e+19],
[ -3.03252593e+18,   2.51420172e+19,  -1.17587833e+19, ...,
-2.78207473e+19,  -7.86770073e+18,   1.80138729e+19],
...,
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3.30711160e+19,   7.40156045e+18,  -2.45479422e+19],
[  1.61892482e+18,  -1.39250973e+19,   6.70826945e+18, ...,
1.37387725e+19,   4.45022985e+18,  -7.98855173e+18],
[ -2.59008049e+18,   1.89475885e+19,  -7.87689904e+18, ...,
-2.93666789e+19,  -5.46299794e+18,   2.35803904e+19]],

[[ -4.72346653e+17,   2.48191702e+18,  -5.90022662e+17, ...,
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-3.22334445e+19,  -3.42590301e+19,  -1.95282395e+19],
...,
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[[  1.18964404e+18,  -9.59180496e+18,   4.38543203e+18, ...,
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...,
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[  3.75626767e+18,  -3.61469614e+19,   1.88458727e+19, ...,
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[ -3.30910349e+17,  -3.40409802e+18,   3.96944550e+18, ...,
-1.65781076e+19,   2.19252494e+18,   2.17781476e+19]],

...,
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-3.36030164e+19,  -9.53628125e+18,   2.16939340e+19],
[  9.14226186e+18,  -8.34420107e+19,   4.20318595e+19, ...,
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[  2.36749612e+19,  -2.28084972e+20,   1.19081775e+20, ...,
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...,
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-7.88293326e+19,  -1.63679534e+19,   6.05472872e+19],
[ -1.77219466e+19,   1.75151194e+20,  -9.28977985e+19, ...,
-1.00073942e+20,  -6.00110462e+19,   1.38663302e+19],
[ -5.33980040e+18,   7.35387112e+19,  -4.57486148e+19, ...,
1.56718755e+19,  -2.83935213e+19,  -6.28530030e+19]],

[[ -1.08810315e+18,   8.69004209e+18,  -3.93808307e+18, ...,
-1.07023909e+19,  -2.65937524e+18,   7.51822581e+18],
[  2.04563573e+18,  -1.74528068e+19,   8.35812749e+18, ...,
1.76559523e+19,   5.55380918e+18,  -1.05295809e+19],
[  4.26050395e+18,  -3.86658143e+19,   1.93861425e+19, ...,
3.16734382e+19,   1.27164174e+19,  -1.44697500e+19],
...,
[ -2.63211172e+18,   2.02489745e+19,  -8.86059078e+18, ...,
-2.76198151e+19,  -6.04751085e+18,   2.07147273e+19],
[ -2.80857329e+18,   2.65482404e+19,  -1.36829188e+19, ...,
-1.85547853e+19,  -8.90786783e+18,   6.13638629e+18],
[  9.53668459e+17,  -3.34836631e+18,  -2.00892497e+17, ...,
1.88114874e+19,   2.10347771e+17,  -2.03835513e+19]],

[[  1.11274186e+18,  -8.18365535e+18,   3.41600126e+18, ...,
1.25470807e+19,   2.36655464e+18,  -1.00255531e+19],
[ -6.75230113e+17,   2.31667523e+18,   2.02908351e+17, ...,
-1.35120670e+19,  -1.16482796e+17,   1.47045607e+19],
[  1.52916856e+18,  -2.22601845e+19,   1.41502911e+19, ...,
-7.25157280e+18,   8.73605580e+18,   2.20265930e+19],
...,
[  3.68086921e+18,  -2.89116445e+19,   1.28881109e+19, ...,
3.73709118e+19,   8.74794881e+18,  -2.71264583e+19],
[ -2.34802290e+18,   2.70404882e+19,  -1.55945908e+19, ...,
-4.76016603e+18,  -9.85795247e+18,  -1.05896794e+19],
[ -6.39636645e+18,   5.71789459e+19,  -2.83444499e+19, ...,
-4.95533416e+19,  -1.86528996e+19,   2.46406581e+19]]],

[[[  9.99455160e+17,   8.65125923e+17,  -2.68625403e+18, ...,
-2.18778907e+18,   1.42901845e+18,  -1.27919095e+18],
[ -1.10065411e+18,  -9.60352860e+17,   2.98362937e+18, ...,
2.48296215e+18,  -1.61102757e+18,   1.41055774e+18],
[ -1.79957527e+18,  -1.45551541e+18,   4.35469593e+18, ...,
2.40609091e+18,  -1.72565919e+18,   2.21648939e+18],
...,
[  3.70591350e+18,   2.92578134e+18,  -8.79586073e+18, ...,
-4.51885785e+18,   3.36571094e+18,  -4.57614727e+18],
[ -1.87938900e+18,  -1.54546924e+18,   4.76568989e+18, ...,
3.27105173e+18,  -2.25262312e+18,   2.37919187e+18],
[ -2.94408845e+18,  -2.40830791e+18,   7.34259970e+18, ...,
4.69117343e+18,  -3.26930117e+18,   3.68871457e+18]],

[[ -3.92668097e+18,  -2.35039366e+18,   6.08316818e+18, ...,
-5.30618967e+18,   1.91706083e+18,   4.34316165e+18],
[  5.24423028e+18,   3.08491411e+18,  -7.88785164e+18, ...,
7.82604621e+18,  -2.96295745e+18,  -5.76274375e+18],
[ -1.57015409e+19,  -8.26216952e+18,   2.01070955e+19, ...,
-3.38696359e+19,   1.42931128e+19,   1.69012901e+19],
...,
[  3.60912404e+19,   1.89471365e+19,  -4.54980455e+19, ...,
8.04821468e+19,  -3.44710752e+19,  -3.85879948e+19],
[ -3.52777455e+18,  -1.49116633e+18,   2.83365743e+18, ...,
-1.29397110e+19,   6.14489281e+18,   3.50413288e+18],
[ -1.29913589e+19,  -6.48064447e+18,   1.50138320e+19, ...,
-3.31483266e+19,   1.46412647e+19,   1.37058126e+19]],

[[  2.53868674e+16,   2.65924445e+17,  -1.09195748e+18, ...,
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[  1.92714816e+17,  -1.78937774e+17,   9.27773088e+17, ...,
4.05273207e+18,  -2.13833805e+18,  -3.40124983e+16],
[ -5.12621357e+18,  -3.15606660e+18,   8.35340787e+18, ...,
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...,
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...,
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...,
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...,
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...,
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...,
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...,
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[[[ -2.34662545e+18,   3.78842621e+18,  -2.92935785e+18, ...,
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[[  1.77077763e+17,  -1.14389851e+18,  -6.51758634e+17, ...,
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...,
[[ -7.43874516e+18,   1.70861834e+19,  -4.11195992e+18, ...,
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...,
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