# Deep Polynomial Neural Networks

Deep Convolutional Neural Networks (DCNNs) are currently the method of choice both for generative, as well as for discriminative learning in computer vision and machine learning. The success of DCNNs can be attributed to the careful selection of their building blocks (e.g., residual blocks, rectifiers, sophisticated normalization schemes, to mention but a few). In this paper, we propose $\Pi$-Nets, a new class of function approximators based on polynomial expansions. $\Pi$-Nets are polynomial neural networks, i.e., the output is a high-order polynomial of the input. The unknown parameters, which are naturally represented by high-order tensors, are estimated through a collective tensor factorization with factors sharing. We introduce three tensor decompositions that significantly reduce the number of parameters and show how they can be efficiently implemented by hierarchical neural networks. We empirically demonstrate that $\Pi$-Nets are very expressive and they even produce good results without the use of non-linear activation functions in a large battery of tasks and signals, i.e., images, graphs, and audio. When used in conjunction with activation functions, $\Pi$-Nets produce state-of-the-art results in three challenging tasks, i.e. image generation, face verification and 3D mesh representation learning. The source code is available at \url{https://github.com/grigorisg9gr/polynomial_nets}.

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## Results from the Paper Edit

Task Dataset Model Metric Name Metric Value Global Rank Result Benchmark
Face Recognition CFP-FF Prodpoly Accuracy 99.886 # 1
Face Recognition CFP-FP Prodpoly Accuracy 0.98986 # 1
Image Generation CIFAR-10 ProdPoly no activation functions Inception score 6.95 # 58
FID 40.45 # 89
Image Classification CIFAR-10 Prodpoly Percentage correct 94.9 # 119
Image Generation CIFAR-10 ProdPoly Inception score 8.49 # 37
FID 16.79 # 69
Conditional Image Generation CIFAR-10 ProdPoly no activation functions Inception score 7.5 # 15
FID 36.77 # 14
Image Classification ImageNet Prodpoly Top 1 Accuracy 77.17% # 548
Top 5 Accuracy 93.56% # 180
Hardware Burden None # 1
Operations per network pass None # 1
Face Recognition LFW Prodpoly Accuracy 0.99833 # 1
Face Verification MegaFace Prodpoly Accuracy 98.95% # 1
Face Identification MegaFace Prodpoly Accuracy 98.78% # 4