CNN/Neural Network: Can I still estimate 3 parameters if my input data has insufficient parameter labels?
Yes, this can be done.  Basically, you define a custom loss function, and then train the neural network to minimize this loss function.  In particular, you construct a single neural network that has 3 outputs; the first output is a measure of symmetry, the second of repetitiveness, and the third roundness.  The custom loss function should look at the output for which you have a known label and penalize the model according to how much that output is in error (it ignores the other 2 outputs).  For instance, if you are currently using mean-squared error, your custom loss function would be equal to the square of the difference between the known label and the corresponding output of the network.  This custom loss function is differentiable, so all the standard training methods (backpropagation, gradient descent, Adam solver, etc. etc.) work directly, with no modifications needed.

Many libraries for training neural networks will let you specify a custom loss function.