Machine learning for Point Clouds Lidar data
First, CNNs are great for image recognition, where you usually take sub sampled windows of about 80 by 80 pixels, 27,000 input neurons is too large and it will take you forever to train a CNN on that.

Furthermore, why did you choose CNN? Why don't you try some more down to earth algorithms fisrst? Like SVMs, or Logistic regressions.

4500 Data points and 27000 features seems unrealistic to me, and very prone to over fitting.

Check this first.

http://scikit-learn.org/stable/tutorial/machine_learning_map/