如何使用截断的SVD减少完全连接的("InnerProduct"`)层 [英] How to reduce a fully-connected (`"InnerProduct"`) layer using truncated SVD

问题描述

在论文中 Girshick，R 快速RCNN (ICCV 2015)，"3.1截断的SVD以加快检测速度"部分，作者建议使用 SVD 技巧可减少全连接层的大小和计算时间.

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.

给定受过训练的模型(deploy.prototxt和weights.caffemodel)，如何使用此技巧将完整连接的层替换为截断的层?

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?

推荐答案

一些线性代数背景
奇异值分解( SVD )是将任何矩阵W分解为三个矩阵:

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

W = U S V*

其中U和V是正交法线矩阵，而S是对角线，其对角线上的元素的数量递减. SVD有趣的特性之一是它允许使用较低的秩矩阵轻松地近似W:假设您将S截断为仅具有k个前导元素(而不是对角线上的所有元素)

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*

是W的等级k近似值.

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

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...

此外，我们还有trained_weights_full.caffemodel-为deploy_full.prototxt模型训练的参数.

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

1. 将deploy_full.protoxt复制到deploy_svd.protoxt并在您选择的编辑器中将其打开.用以下两层替换完全连接的层:

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...
}

在python中，进行了净手术:

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，我们有了deploy_svd.prototxt和trained_weights_svd.caffemodel，它们的乘积和权重都远小于原始网络.

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

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