Notes on Fractional Max-Pooling

FMP.md

Fractional Max-Pooling (FMP)

Introduction

• Link to Paper
• Spatial pooling layers are building blocks for Convolutional Neural Networks (CNNs).
• Input to pooling operation is a N_in x N_in matrix and output is a smaller matrix N_out x N_out.
• Pooling operation divides N_in x N_in square into N²_out pooling regions P_{i, j}.
• P_{i, j} ⊂ {1, 2, . . . , N_in} ∀ (i, j) ∈ {1, . . . , N_out}²

MP2

• 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.

FMP

• 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 a_i and b_i be 2 increasing sequences of integers, starting at 1 and ending at N_in.
• Increments are either 1 or 2.
• For disjoint regions, P = [a_i−1, a_i − 1] × [b_j−1, b_j − 1]
• For overlapping regions, P = [a_i−1, a_i] × [b_j−1, b_j1]
• Pooling regions can be generated randomly by choosing the increment randomly at each step.
• To generate pooling regions in a peusdorandom manner, choose a_i = 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.

Observations

• 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.

Weakness

• 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.