The curse of dimensionality is real and problematic, but it only applies to the real dimension of the problem, not the measured/perceived one.

Let's say you have samples with 5 dimensions. That's still not (usually) considered within the realm of the curse.

Now, let's assume that because of the way they are measured, you actually have 10,000,000 measured dimensions for each sample. Now, it starts to be a problem - especially if you don't know how that expansion happens -- it might be linear, nonlinear, with noise, etc.

Even though we are now working in 10,000,000 space, and it looks like the curse of dimensionality applies, it does not. You just have to find a reasonable way to compact everything back down to the real problem dimension, or somewhere close enough to it.

The most popular tool for this is called Random Projections, that follows from the work of Lindenstrauss and Johnson in 94. Compressive Sensing is a complementary field.