MANIFOLDNET: A DEEP NEURAL NETWORK FOR MANIFOLD-VALUED DATA
Developing deep neural networks (DNNs) for manifold-valued data sets has gained much interest of late in the deep learning research community. Examples of manifold-valued data include data from omnidirectional cameras on automobiles, drones etc., diffusion magnetic resonance imaging, elastography and others. In this paper, we present a novel theoretical framework for DNNs to cope with manifold-valued data inputs. In doing this generalization, we draw parallels to the widely popular convolutional neural networks (CNNs). We call our network the ManifoldNet. As in vector spaces where convolutions are equivalent to computing the weighted mean of functions, an analogous definition for manifold-valued data can be constructed involving the computation of the weighted Fr*{e}chet Mean (wFM). To this end, we present a provably convergent recursive computation of the wFM of the given data, where the weights makeup the convolution mask, to be learned. Further, we prove that the proposed wFM layer achieves a contraction mapping and hence the ManifoldNet does not need the additional non-linear ReLU unit used in standard CNNs. Operations such as pooling in traditional CNN are no longer necessary in this setting since wFM is already a pooling type operation. Analogous to the equivariance of convolution in Euclidean space to translations, we prove that the wFM is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside. This equivariance property facilitates weight sharing within the network. We present experiments, using the ManifoldNet framework, to achieve video classification and image reconstruction using an auto-encoder+decoder setting. Experimental results demonstrate the efficacy of ManifoldNet in the context of classification and reconstruction accuracy.
 This paper proposes a method to use weighted Frechet Mean (wFM) for the operation on Manifold valued data for CNN. The novel point is to view wFM as a convolutional layer. Overall, this paper is mathematically well written, however, how each theory improves CNN and the model used in experiments are not clear enough. Pros + The use of wFM instead of a convolutional layer is an interesting idea. + This paper is mathematically well written. Cos - It is hard to understand how each theory presented in this paper helps to improve CNN. For example, the invariance to group operation. Some experimental results would help to understand the advantage of the group invariance. - It is also unclear why the authors constructed the invariant last layer although the inputs of the last layer are invariant under group operations. - In the introduction section, the authors raised the omnidirectional camera, diffusion magnetic resonance imaging, elastography as examples of manifold-valued data. However, experiments are limited to standard video sequences. - It is unclear how to obtain the weights {w_i} of wFM by backpropagation. - Since the contribution of this paper is to to use wFM instead of a convolutional layer, it is more interesting to visualize the weights {w_i}. - More explanation needs for the model used for experiments. Especially in dimensional reduction experiments, I could not understand how each subspace is obtained and averaged. If each frame is a subspace, by averaging frames, the reconstruction would be blurred.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
MANIFOLDNET: A DEEP NEURAL NETWORK FOR MANIFOLD-VALUED DATA
Developing deep neural networks (DNNs) for manifold-valued data sets has gained much interest of late in the deep learning research community. Examples of manifold-valued data include data from omnidirectional cameras on automobiles, drones etc., diffusion magnetic resonance imaging, elastography and others. In this paper, we present a novel theoretical framework for DNNs to cope with manifold-valued data inputs. In doing this generalization, we draw parallels to the widely popular convolutional neural networks (CNNs). We call our network the ManifoldNet. As in vector spaces where convolutions are equivalent to computing the weighted mean of functions, an analogous definition for manifold-valued data can be constructed involving the computation of the weighted Fr*{e}chet Mean (wFM). To this end, we present a provably convergent recursive computation of the wFM of the given data, where the weights makeup the convolution mask, to be learned. Further, we prove that the proposed wFM layer achieves a contraction mapping and hence the ManifoldNet does not need the additional non-linear ReLU unit used in standard CNNs. Operations such as pooling in traditional CNN are no longer necessary in this setting since wFM is already a pooling type operation. Analogous to the equivariance of convolution in Euclidean space to translations, we prove that the wFM is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside. This equivariance property facilitates weight sharing within the network. We present experiments, using the ManifoldNet framework, to achieve video classification and image reconstruction using an auto-encoder+decoder setting. Experimental results demonstrate the efficacy of ManifoldNet in the context of classification and reconstruction accuracy.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



 This paper proposes a method to use weighted Frechet Mean (wFM) for the operation on Manifold valued data for CNN. The novel point is to view wFM as a convolutional layer. Overall, this paper is mathematically well written, however, how each theory improves CNN and the model used in experiments are not clear enough. Pros + The use of wFM instead of a convolutional layer is an interesting idea. + This paper is mathematically well written. Cos - It is hard to understand how each theory presented in this paper helps to improve CNN. For example, the invariance to group operation. Some experimental results would help to understand the advantage of the group invariance. - It is also unclear why the authors constructed the invariant last layer although the inputs of the last layer are invariant under group operations. - In the introduction section, the authors raised the omnidirectional camera, diffusion magnetic resonance imaging, elastography as examples of manifold-valued data. However, experiments are limited to standard video sequences. - It is unclear how to obtain the weights {w_i} of wFM by backpropagation. - Since the contribution of this paper is to to use wFM instead of a convolutional layer, it is more interesting to visualize the weights {w_i}. - More explanation needs for the model used for experiments. Especially in dimensional reduction experiments, I could not understand how each subspace is obtained and averaged. If each frame is a subspace, by averaging frames, the reconstruction would be blurred.
assistant
 This paper proposes a method to use weighted Frechet Mean (wFM) for the operation on Manifold valued data for CNN. The novel point is to view wFM as a convolutional layer. Overall, this paper is mathematically well written, however, how each theory improves CNN and the model used in experiments are not clear enough. Pros + The use of wFM instead of a convolutional layer is an interesting idea. + This paper is mathematically well written. Cos - It is hard to understand how each theory presented in this paper helps to improve CNN. For example, the invariance to group operation. Some experimental results would help to understand the advantage of the group invariance. - It is also unclear why the authors constructed the invariant last layer although the inputs of the last layer are invariant under group operations. - In the introduction section, the authors raised the omnidirectional camera, diffusion magnetic resonance imaging, elastography as examples of manifold-valued data. However, experiments are limited to standard video sequences. - It is unclear how to obtain the weights {w_i} of wFM by backpropagation. - Since the contribution of this paper is to to use wFM instead of a convolutional layer, it is more interesting to visualize the weights {w_i}. - More explanation needs for the model used for experiments. Especially in dimensional reduction experiments, I could not understand how each subspace is obtained and averaged. If each frame is a subspace, by averaging frames, the reconstruction would be blurred.