The PS3 was used a few time in clusters – some NN work was done on it back in the day. My understanding (somewhat echoed in TFA) is that when programming Cell, you really needed to think about communication patterns to avoid quickly running into memory bandwidth limitations, especially given memory hierarchy and bus quirks.
For it's day, it packed a lot of compute into cheap package, so long as you could do something useful with a data set that fit into 256kB, the size of the local memory buffer on each SPE. If you overflowed that, the anemic system bandwidth would make it suck. Protein folding was an example of a problem that back then used tons of compute but could be fit into small space.
It was the biggest contributor to folding @ home at one point. It came bundled with the PS3 and played relaxing music and showed a heat map of the world ps3 compute nodes as it went on. There was also https://en.wikipedia.org/wiki/PlayStation_3_cluster
With enough effort you could definitely do it. Just remember it is a device that came out in 2006 and it has 256MB of system RAM and 256MB of VRAM, at best you're running a quite small model after a lot work trying to port some inference code to CELL processors. Honestly it does sound a cool excuse to write code for the CELL processors, but don't expect amazing performance or anything.
It's a nearly 20 year old gaming console. Even if you could port a deep learning workload to run efficiently on the Cell architecture, it would be thoroughly outclassed by a modern cell phone (to say nothing of a desktop computer).
The PS3 only had 256mb of main memory so you'd be pretty limited there. Memory bandwidth, great at the time, is pretty poor by today's standards (25 gb/s)
https://open.clemson.edu/all_theses/629/