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jsteinhardt comments on A Fervent Defense of Frequentist Statistics - Less Wrong
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Further reading shows that http://en.wikipedia.org/wiki/Sparse_approximation is supposed to be NP-hard, so it can't be that the L1-norm minimum produces the "L0-norm" minimum every time.
http://statweb.stanford.edu/~donoho/Reports/2004/l1l0approx.pdf which is given as the Wikipedia reference for L1 producing L0 under certain conditions, only talks about near-solutions, not exact solutions.
Also Jacob originally specified that the coefficients were drawn from a Gaussian and nobody seems to be using that fact.
As long as the matrix formed by the x_i satisfies the "restricted isometry property", the optimization problem given by V_V above will recover the optimal solution. For m >> k*log(n), (where in this case m = 10,000, k = 100, and n = 10^6), a random Gaussian matrix will satisfy this property with overwhelmingly large probability. I should probably have checked that the particular numbers I gave will work out, although I'm pretty sure they will, and if not we can replace 10,000 with 20,000 and the point still stands.