concat 层后 googlenet 中的输出维度是多少？

Question

查看 googlenet 的 prototxt 时，发现初始层的末尾有一个 concat 层，它需要几个底部输入。

例如：

layer {
  name: "inception_3a/output"
  type: "Concat"
  bottom: "inception_3a/1x1"
  bottom: "inception_3a/3x3"
  bottom: "inception_3a/5x5"
  bottom: "inception_3a/pool_proj"
  top: "inception_3a/output"
}

可以看出，有一个 1x1 的卷积层，一个 3x3 的卷积层，一个 5x5 的卷积层，最后是一个池化层。这些层描述如下：

layer {
  name: "inception_3a/1x1"
  type: "Convolution"
  bottom: "pool2/3x3_s2"
  top: "inception_3a/1x1"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 2
    decay_mult: 0
  }
  convolution_param {
    num_output: 64
    kernel_size: 1
    weight_filler {
      type: "xavier"
      std: 0.03
    }
    bias_filler {
      type: "constant"
      value: 0.2
    }
  }
}
layer {
  name: "inception_3a/relu_1x1"
  type: "ReLU"
  bottom: "inception_3a/1x1"
  top: "inception_3a/1x1"
}
layer {
  name: "inception_3a/3x3_reduce"
  type: "Convolution"
  bottom: "pool2/3x3_s2"
  top: "inception_3a/3x3_reduce"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 2
    decay_mult: 0
  }
  convolution_param {
    num_output: 96
    kernel_size: 1
    weight_filler {
      type: "xavier"
      std: 0.09
    }
    bias_filler {
      type: "constant"
      value: 0.2
    }
  }
}
layer {
  name: "inception_3a/relu_3x3_reduce"
  type: "ReLU"
  bottom: "inception_3a/3x3_reduce"
  top: "inception_3a/3x3_reduce"
}
layer {
  name: "inception_3a/3x3"
  type: "Convolution"
  bottom: "inception_3a/3x3_reduce"
  top: "inception_3a/3x3"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 2
    decay_mult: 0
  }
  convolution_param {
    num_output: 128
    pad: 1
    kernel_size: 3
    weight_filler {
      type: "xavier"
      std: 0.03
    }
    bias_filler {
      type: "constant"
      value: 0.2
    }
  }
}
layer {
  name: "inception_3a/relu_3x3"
  type: "ReLU"
  bottom: "inception_3a/3x3"
  top: "inception_3a/3x3"
}
layer {
  name: "inception_3a/5x5_reduce"
  type: "Convolution"
  bottom: "pool2/3x3_s2"
  top: "inception_3a/5x5_reduce"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 2
    decay_mult: 0
  }
  convolution_param {
    num_output: 16
    kernel_size: 1
    weight_filler {
      type: "xavier"
      std: 0.2
    }
    bias_filler {
      type: "constant"
      value: 0.2
    }
  }
}
layer {
  name: "inception_3a/relu_5x5_reduce"
  type: "ReLU"
  bottom: "inception_3a/5x5_reduce"
  top: "inception_3a/5x5_reduce"
}
layer {
  name: "inception_3a/5x5"
  type: "Convolution"
  bottom: "inception_3a/5x5_reduce"
  top: "inception_3a/5x5"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 2
    decay_mult: 0
  }
  convolution_param {
    num_output: 32
    pad: 2
    kernel_size: 5
    weight_filler {
      type: "xavier"
      std: 0.03
    }
    bias_filler {
      type: "constant"
      value: 0.2
    }
  }
}
layer {
  name: "inception_3a/relu_5x5"
  type: "ReLU"
  bottom: "inception_3a/5x5"
  top: "inception_3a/5x5"
}
layer {
  name: "inception_3a/pool"
  type: "Pooling"
  bottom: "pool2/3x3_s2"
  top: "inception_3a/pool"
  pooling_param {
    pool: MAX
    kernel_size: 3
    stride: 1
    pad: 1
  }
}
layer {
  name: "inception_3a/pool_proj"
  type: "Convolution"
  bottom: "inception_3a/pool"
  top: "inception_3a/pool_proj"
  param {
    lr_mult: 1
    decay_mult: 1
  }
  param {
    lr_mult: 2
    decay_mult: 0
  }
  convolution_param {
    num_output: 32
    kernel_size: 1
    weight_filler {
      type: "xavier"
      std: 0.1
    }
    bias_filler {
      type: "constant"
      value: 0.2
    }
  }
}

可以看出，这些有不同的输出数量和不同的过滤器大小，无论如何concat层的文档如下：

输入：

n_i * c_i * h * w for each input blob i from 1 to K.

输出：

如果axis = 0：(n_1 + n_2 + ... + n_K) * c_1 * h * w，并且所有输入c_i 应该是一样的。

如果axis = 1：n_1 * (c_1 + c_2 + ... + c_K) * h * w，并且所有输入n_i都应该相同。

首先，我不确定默认值是什么，其次我不确定哪个维度将具有输出 Volume，因为宽度和高度应该保持不变，但所有 thre 卷积层都会产生不同的输出。任何指针将不胜感激

Answer 1

'Concat' 轴的默认值为 1，因此通过通道维度连接。为了做到这一点，所有连接的层都应该具有相同的高度和宽度。查看日志，尺寸为（假设批量大小为 32）：

inception_3a/1x1 -> [32, 64, 28, 28]
inception_3a/3x3 -> [32, 128, 28, 28]
inception_3a/5x5 -> [32, 32, 28, 28]
inception_3a/pool_proj -> [32, 32, 28, 28]

因此最终输出将具有以下维度：
inception_3a/输出 -> [32 (64+128+32+32) 28, 28] -> [32, 256, 28, 28]

正如 Caffe 日志中所预期的那样：

Creating Layer inception_3a/output
inception_3a/output <- inception_3a/1x1
inception_3a/output <- inception_3a/3x3
inception_3a/output <- inception_3a/5x5
inception_3a/output <- inception_3a/pool_proj
inception_3a/output -> inception_3a/output
Setting up inception_3a/output
Top shape: 32 256 28 28 (6422528)