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I tried to replicate the model architecture described in the paper. When training, I did not use batch norm. I stopped after 100 epochs due to convergence. The test accuracy of this model was 96.3%.
All of these instances were correctly predicted
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Chair
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Lamp
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Vase
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This chair was predicted as a lamp. In general, chairs are pretty easy to predict and this was one of the two failure casues. It is unsurprising because this barely has a seat, and only has two legs. This suggestss it is pretty out of distribution from the training examples.
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This is a vase that was predicted as a chair. My guess is the strong horizontal surface and rectangular structure were similar to the seat and sides of a chair, respectively. Furthermore, the plant, which could be a strong queue that it is a vase, is pretty small.
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This is a lamp that was predicted as a vase. I personally wouldn't know how to classify this. There's is nothing that looks like a lampshade, which is generally one of the stronger queues.
Again, I tried to replicate the architecture described in the paper, with the absense of batch norm. The test accuracy was 87.5%.
Groundtruths left, predictions right
The majority of the results match pretty well. The most common minor error I saw was that class boundaries, which are frequently annotated as straight lines, were frequently predicted as curves. This may stem from the fact that while each point has global context, it does not know the class predictions for its neighbors, so it cannot have globally consistent predictions.
Groundtruths left, predictions right
Many of the bad predictions were on couches, where the part boundaries were poorly defined, or rarely-seen types of chairs with complex geometries. Additionally, the model made mistakes on rare classes, such as the purple cushion and white footrest, since there were not many training examples. Another interesting observation is that in the last example, the model had trouble distinguishing the bar of the armrest from the seat. This makes sense because the network cannot easily reason about connectivity information.
As suggested, I tried two tasks. The first was randomly downsampling the mesh. The second was rotating the object about the x axis.
Classification performance degraded pretty severly until it was worse than chance with a rotation of π2. Interesting, some of the performance was regained after this point. This suggests that objects have some symetry top-to-bottom which can be exploited. 
I was pretty impressed with this result. The network was able to achieve very good results with as few a tenth of the points. Furthermore, with as few as 50 points, it still did far better than chance. This robustness is likely a function of the max pooling operation which already enforces that predictions are made based on a subset of the data.
For angles between 0 and π2, we see a trend which is similar to the classificaiton result. However, after this point the accuracy does not substantially rebound. This is likely because there is a strong bias toward classes appearing at a certain height within the scene. 
Segmentation is also quite robust downsampling. There appears to be slightly more of a perforamence penalty for segmentation versus classication. This is likely because segmentation requires more local context, which is lost more quickly than the global structure. 