TY - CONF
T1 - From hard to soft
T2 - 7th International Conference on Learning Representations, ICLR 2019
AU - Balestriero, Randall
AU - Baraniuk, Richard G.
N1 - Funding Information:
Our development of the SMASO model opens the door to several new research questions. First, we have merely scratched the surface in the exploration of new nonlinear activation functions and pooling operators based on the SVQ and β-VQ. For example, the soft-or β-VQ versions of leaky-ReLU, absolute value, and other piecewise affine and convex nonlinearities could outperform the new swish nonlinearity. Second, replacing the entropy penalty in the (7) and (8) with a different penalty will create entirely new classes of nonlinearities that inherit the rich analytical properties of MASO DNs. Third, orthogonal DN filters will enable new analysis techniques and DN probing methods, since from a signal processing point of view problems such as denoising, reconstruction, compression have been extensively studied in terms of orthogonal filters. This work was partially supported by NSF grants IIS-17-30574 and IIS-18-38177, AFOSR grant FA9550-18-1-0478, ARO grant W911NF-15-1-0316, ONR grants N00014-17-1-2551 and N00014-18-12571, DARPA grant G001534-7500, and a DOD Vannevar Bush Faculty Fellowship (NSSEFF) grant N00014-18-1-2047.
Publisher Copyright:
© 7th International Conference on Learning Representations, ICLR 2019. All Rights Reserved.
PY - 2019
Y1 - 2019
N2 - Nonlinearity is crucial to the performance of a deep (neural) network (DN). To date there has been little progress understanding the menagerie of available nonlinearities, but recently progress has been made on understanding the rôle played by piecewise affine and convex nonlinearities like the ReLU and absolute value activation functions and max-pooling. In particular, DN layers constructed from these operations can be interpreted as max-affine spline operators (MASOs) that have an elegant link to vector quantization (VQ) and K-means. While this is good theoretical progress, the entire MASO approach is predicated on the requirement that the nonlinearities be piecewise affine and convex, which precludes important activation functions like the sigmoid, hyperbolic tangent, and softmax. This paper extends the MASO framework to these and an infinitely large class of new nonlinearities by linking deterministic MASOs with probabilistic Gaussian Mixture Models (GMMs). We show that, under a GMM, piecewise affine, convex nonlinearities like ReLU, absolute value, and max-pooling can be interpreted as solutions to certain natural “hard” VQ inference problems, while sigmoid, hyperbolic tangent, and softmax can be interpreted as solutions to corresponding “soft” VQ inference problems. We further extend the framework by hybridizing the hard and soft VQ optimizations to create a β-VQ inference that interpolates between hard, soft, and linear VQ inference. A prime example of a β-VQ DN nonlinearity is the swish nonlinearity, which offers state-of-the-art performance in a range of computer vision tasks but was developed ad hoc by experimentation. Finally, we validate with experiments an important assertion of our theory, namely that DN performance can be significantly improved by enforcing orthogonality in its linear filters.
AB - Nonlinearity is crucial to the performance of a deep (neural) network (DN). To date there has been little progress understanding the menagerie of available nonlinearities, but recently progress has been made on understanding the rôle played by piecewise affine and convex nonlinearities like the ReLU and absolute value activation functions and max-pooling. In particular, DN layers constructed from these operations can be interpreted as max-affine spline operators (MASOs) that have an elegant link to vector quantization (VQ) and K-means. While this is good theoretical progress, the entire MASO approach is predicated on the requirement that the nonlinearities be piecewise affine and convex, which precludes important activation functions like the sigmoid, hyperbolic tangent, and softmax. This paper extends the MASO framework to these and an infinitely large class of new nonlinearities by linking deterministic MASOs with probabilistic Gaussian Mixture Models (GMMs). We show that, under a GMM, piecewise affine, convex nonlinearities like ReLU, absolute value, and max-pooling can be interpreted as solutions to certain natural “hard” VQ inference problems, while sigmoid, hyperbolic tangent, and softmax can be interpreted as solutions to corresponding “soft” VQ inference problems. We further extend the framework by hybridizing the hard and soft VQ optimizations to create a β-VQ inference that interpolates between hard, soft, and linear VQ inference. A prime example of a β-VQ DN nonlinearity is the swish nonlinearity, which offers state-of-the-art performance in a range of computer vision tasks but was developed ad hoc by experimentation. Finally, we validate with experiments an important assertion of our theory, namely that DN performance can be significantly improved by enforcing orthogonality in its linear filters.
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M3 - Paper
AN - SCOPUS:85083950229
Y2 - 6 May 2019 through 9 May 2019
ER -