The Self-Organizing Map (SOM), an unsupervised learning neural network, is employed to extract patterns evinced by the Loop Current (LC) system and to identify regions of sea surface height (SSH) variability in the eastern Gulf of Mexico (GoM) from 23 years (1993-2015) of altimetry data. Spatial patterns are characterized as different LC extensions and different stages in the process of LC eddy shedding. The temporal evolutions and the frequency of occurrences of these patterns are obtained, and the typical trajectories of the LC system progression on the SOM grid are investigated. For an elongated, northwest-extended, or west-positioned LC, it is common for the LC anticyclonic eddy (LCE) to separate and propagate into the western GoM, while an initially separated LCE in close proximity to the west Florida continental slope often reattaches to the LC and develops into an elongated LC, or reduces intensity locally before moving westward as a smaller eddy. Regions of differing SSH variations are also identified using the joint SOM-wavelet analysis. Along the general axis of the LC, SSH exhibits strong variability on time scales of 3 months to 2 years, also with energetic intraseasonal variations, which is consistent with the joint Empirical Orthogonal Function (EOF)-wavelet analysis. In the more peripheral regions, the SSH has a dominant seasonal variation that also projects across the coastal ocean. The SOM, when applied to both space and time domains of the same data, provides a powerful tool for diagnosing ocean processes from such different perspectives.