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If I train a simple CNN with an MNIST dataset for digit classification. Is it possible to get a similar performance if I replace the max-pooling layers with the min-pooling layers?

This problem may be trivial in MNIST due to the small image size of 28x28. What about applications with FG-BG segmentation? Essentially I can say that if max-pooling layers are supposed to look at the foreground class by focusing on the "important features" then min pooling is supposed to look at the background class by focusing on the "least important" features.

Is it possible that subsequent layers will learn to ignore "least important" features and to focus on the "most important" features by simply learning the inverse?

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Min pooling most likely result in zero activations (layer's activations will be equal to 0), with ReLU activations this is almost guaranteed to be the case, thus, there will be no learning as all gradients will be equal to 0.

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