In almost all practical uses of matrix multiplication, we have rounding errors. For example, in 3D it is hard to reverse exactly a rotation and get the exact initial position back.

I don't know what amount of losses we are talking about but in deep learning, several operations don't require a crazy level of compression, and it led to some lightweight float implementations (bfloat, on 16 bits, being the most common but there are also 8 bits floats for extreme cases)

If that's really a 10-100x speed increase at the cost of a bit of loss, I am sure machine learning will love it.