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Notes on Fractional Max-Pooling

Fractional Max-Pooling (FMP)


  • Link to Paper
  • Spatial pooling layers are building blocks for Convolutional Neural Networks (CNNs).
  • Input to pooling operation is a Nin x Nin matrix and output is a smaller matrix Nout x Nout.
  • Pooling operation divides Nin x Nin square into N2out pooling regions Pi, j.
  • Pi, j ⊂ {1, 2, . . . , Nin} ∀ (i, j) ∈ {1, . . . , Nout}2


  • Refers to 2x2 max-pooling layer.
  • Popular choice for max-pooling operation.

Advantages of MP2

  • Fast.
  • Quickly reduces the size of the hidden layer.
  • Encodes a degree of invariance with respect to translations and elastic distortions.

Issues with MP2

  • Disjoint nature of pooling regions.
  • Since size decreases rapidly, stacks of back-to-back CNNs are needed to build deep networks.


  • Reduces the spatial size of the image by a factor of α, where α ∈ (1, 2).
  • Introduces randomness in terms of choice of pooling region.
  • Pooling regions can be chosen in a random or pseudorandom manner.
  • Pooling regions can be disjoint or overlapping.

Generating Pooling Regions

  • Let ai and bi be 2 increasing sequences of integers, starting at 1 and ending at Nin.
  • Increments are either 1 or 2.
  • For disjoint regions, P = [ai−1, ai − 1] × [bj−1, bj − 1]
  • For overlapping regions, P = [ai−1, ai] × [bj−1, bj1]
  • Pooling regions can be generated randomly by choosing the increment randomly at each step.
  • To generate pooling regions in a peusdorandom manner, choose ai = ceil(α*(i+u)), where α ∈ (1, 2) with some u ∈ (0, 1).
  • Each FMP layer uses a different pair of sequence.
  • An FMP network can be thought of as an ensemble of similar networks, with each different pooling-region configuration defining a different member of the ensemble.


  • Random FMP is good on its own but may underfit when combined with dropout or training data augmentation.
  • Pseudorandom approach generates more stable pooling regions.
  • Overlapping FMP performs better than disjoint FMP.


  • No justification is provided for the observations mentioned above.
  • It needs to be seen how performance is affected if the pooling layer in architectures like GoogLeNet.
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