如何使用截断的SVD减少完全连接("InnerProduct")层

Question

在Girshick,R Fast-RCNN(ICCV 2015),"3.1截断SVD以便更快检测"一节中,作者提出使用SVD技巧来减小完全连接层的大小和计算时间.

给定一个训练有素的模型(deploy.prototxt和weights.caffemodel),我如何使用这个技巧用截断的模型替换完全连接的层？

Answer 1

一些线性代数背景
奇异值分解(SVD)是将任何矩阵分解W为三个矩阵:

W = U S V*

其中U和V是正交正规矩阵,并且S对角线上的元素的幅度逐渐减小.SVD的一个有趣特性是它允许W使用较低秩矩阵轻松逼近:假设您截断S只有其k前导元素(而不是对角线上的所有元素)然后

W_app = U S_trunc V*

是一个等级k近似值W.

使用SVD逼近完全连接的层
假设我们有一个deploy_full.prototxt具有完全连接层的模型

# ... some layers here
layer {
  name: "fc_orig"
  type: "InnerProduct"
  bottom: "in"
  top: "out"
  inner_product_param {
    num_output: 1000
    # more params...
  }
  # some more...
}
# more layers...

此外,我们已经trained_weights_full.caffemodel- 训练过的deploy_full.prototxt模型参数.

1. 复制deploy_full.protoxt到deploy_svd.protoxt和您选择的编辑器打开它.用这两层替换完全连接的层:

   layer {
     name: "fc_svd_U"
     type: "InnerProduct"
     bottom: "in" # same input
     top: "svd_interim"
     inner_product_param {
       num_output: 20  # approximate with k = 20 rank matrix
       bias_term: false
       # more params...
     }
     # some more...
   }
   # NO activation layer here!
   layer {
     name: "fc_svd_V"
     type: "InnerProduct"
     bottom: "svd_interim"
     top: "out"   # same output
     inner_product_param {
       num_output: 1000  # original number of outputs
       # more params...
     }
     # some more...
   }
   

2. 在python中,有点网手术:

   import caffe
   import numpy as np
   
   orig_net = caffe.Net('deploy_full.prototxt', 'trained_weights_full.caffemodel', caffe.TEST)
   svd_net = caffe.Net('deploy_svd.prototxt', 'trained_weights_full.caffemodel', caffe.TEST)
   # get the original weight matrix
   W = np.array( orig_net.params['fc_orig'][0].data )
   # SVD decomposition
   k = 20 # same as num_ouput of fc_svd_U
   U, s, V = np.linalg.svd(W)
   S = np.zeros((U.shape[0], k), dtype='f4')
   S[:k,:k] = s[:k]  # taking only leading k singular values
   # assign weight to svd net
   svd_net.params['fc_svd_U'][0].data[...] = np.dot(U,S)
   svd_net.params['fc_svd_V'][0].data[...] = V[:k,:]
   svd_net.params['fc_svd_V'][1].data[...] = orig_net.params['fc_orig'][1].data # same bias
   # save the new weights
   svd_net.save('trained_weights_svd.caffemodel')
   

现在我们已经deploy_svd.prototxt得到trained_weights_svd.caffemodel了近似于原始网络的乘法和权重.