# Signal recovery from Pooling Representations

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Joan Bruna Estrach,
Arthur Szlam,
Yann LeCun
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Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):307-315, 2014.

#### Abstract

Pooling operators construct non-linear representations by cascading a redundant linear transform, followed by a point-wise nonlinearity and a local aggregation, typically implemented with a \ell_p norm. Their efficiency in recognition architectures is based on their ability to locally contract the input space, but also on their capacity to retain as much stable information as possible. We address this latter question by computing the upper and lower Lipschitz bounds of \ell_p pooling operators for p=1, 2, ∞as well as their half-rectified equivalents, which give sufficient conditions for the design of invertible pooling layers. Numerical experiments on MNIST and image patches confirm that pooling layers can be inverted with phase recovery algorithms. Moreover, the regularity of the inverse pooling, controlled by the lower Lipschitz constant, is empirically verified with a nearest neighbor regression.

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