What about tensors?

A fair question with an honest answer: no single tensor inverse is in common use. Several definitions exist — built on the Einstein product, or on the t-product for order-3 tensors — and they are active research.

In practice, in machine learning, you unfold the tensor into a matrix, use the matrix pseudoinverse, and fold the result back. That works because unfolding loses nothing.

T = np.random.randn(4, 3, 5)
M = unfold(T, 0)              # (4, 15)
M_plus = np.linalg.pinv(M)     # (15, 4)
np.allclose(M @ M_plus @ M, M) # True

A general lesson worth remembering: when a tensor problem is hard, unfold it to a matrix, solve it there, and fold back.