machine learning - How to reduce a fully-connected (`"InnerProduct"`) layer using truncated SVD

Question

In the paper Girshick, R Fast-RCNN (ICCV 2015), section "3.1 Truncated SVD for faster detection", the author proposes to use SVD trick to reduce the size and computation time of a fully connected layer.

Given a trained model (deploy.prototxt and weights.caffemodel), how can I use this trick to replace a fully connected layer with a truncated one?

Answer 1

Some linear-algebra background
Singular Value Decomposition (SVD) is a decomposition of any matrix W into three matrices:

W = U S V*

Where U and V are ortho-normal matrices, and S is diagonal with elements in decreasing magnitude on the diagonal. One of the interesting properties of SVD is that it allows to easily approximate W with a lower rank matrix: Suppose you truncate S to have only its k leading elements (instead of all elements on the diagonal) then

W_app = U S_trunc V*

is a rank k approximation of W.

Using SVD to approximate a fully connected layer
Suppose we have a model deploy_full.prototxt with a fully connected layer

# ... 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...

Furthermore, we have trained_weights_full.caffemodel - trained parameters for deploy_full.prototxt model.

1. Copy deploy_full.protoxt to deploy_svd.protoxt and open it in editor of your choice. Replace the fully connected layer with these two layers:

   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. In python, a little net surgery:

   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')
   

Now we have deploy_svd.prototxt with trained_weights_svd.caffemodel that approximate the original net with far less multiplications, and weights.