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Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great successes. Unfortunately, the understanding on how it works remains unclear. It has the central importance to lay down the theoretic foundation for deep learning.In this work, we give a geometric view to understand deep learning: we show that the fundamental principle attributing to the success is the manifold structure in data, namely natural high dimensional data concentrates close to a low-dimensional manifold, deep learning learns the manifold and the probability distribution on it.We further introduce the concepts of rectified linear complexity for deep neural network measuring its learning capability, rectified linear complexity of an embedding manifold describing the difficulty to be learned. Then we show for any deep neural network with fixed architecture, there exists a manifold that cannot be learned by the network. Finally, we propose to apply optimal mass transportation theory to control the probability distribution in the latent space.

In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This leads to a geometric interpretation to generative models, and leads to a novel framework for generative models.By using the optimal transportation view of GAN model, we show that the discriminator computes the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. Therefore, it is sufficient to solely optimize the discriminator. This shows the adversarial competition can be avoided, and the computational architecture can be simplified.Preliminary experimental results show the geometric method outperforms WGAN for approximating probability measures with multiple clusters in low dimensional space.

Background Burns are life-threatening with high morbidity and mortality. Reliable diagnosis supported by accurate burn area and depth assessment is critical to the success of the treatment decision and, in some cases, can save the patient’s life. Current techniques such as straight-ruler method, aseptic film trimming method, and digital camera photography method are not repeatable and comparable, which lead to a great difference in the judgment of burn wounds and impede the establishment of the same evaluation criteria. Hence, in order to semi-automate the burn diagnosis process, reduce the impact of human error, and improve the accuracy of burn diagnosis, we include the deep learning technology into the diagnosis of burns. Method This article proposes a novel method employing a state-of-the-art deep learning technique to segment the burn wounds in the images. We designed this deep learning segmentation framework based on the Mask Regions with Convolutional Neural Network (Mask R-CNN). For training our framework, we labeled 1150 pictures with the format of the Common Objects in Context (COCO) data set and trained our model on 1000 pictures. In the evaluation, we compared the different backbone networks in our framework. These backbone networks are Residual Network-101 with Atrous Convolution in Feature Pyramid Network (R101FA), Residual Network-101 with Atrous Convolution (R101A), and InceptionV2-Residual Network with Atrous Convolution (IV2RA). Finally, we used the Dice coefficient (DC) value to assess the model accuracy. Result The R101FA backbone network gains the highest accuracy 84.51% in 150 pictures. Moreover, we chose different burn depth pictures to evaluate these three backbone networks. The R101FA backbone network gains the best segmentation effect in superficial, superficial thickness, and deep partial thickness. The R101A backbone network gains the best segmentation effect in full-thickness burn. Conclusion This deep learning framework shows excellent segmentation in burn wound and extremely robust in different burn wound depths. Moreover, this framework just needs a suitable burn wound image when analyzing the burn wound. It is more convenient and more suitable when using in clinics compared with the traditional methods. And it also contributes more to the calculation of total body surface area (TBSA) burned.