In the paper *Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries* (page 2), the authors rewrite the minimization problem

\begin{align} \min_{\alpha \in \mathbb R^k} \| \alpha \|_0 && s.t. && \|D \alpha - y \|_2^2 \leq T, \end{align}

where $D \in \mathbb R^{n \times k}$ and $y \in \mathbb R^n$, into

\begin{align} \min_{\alpha \in \mathbb R^k} \| D \alpha - y \|_2^2 + \mu \| \alpha \|_0 \end{align}

and state that

for a proper choice of $\mu$, the two problems are equivalent.

There is no reference in the paper that explains why this would be true and in fact, I don't believe it is true, since $\| \cdot \|_0$ is not even convex. Does such a $\mu$ really exist?

I don't know if this is important, but the following assumptions were made: $y$ is a noisy version of $x$ with zero mean white noise of variance $\sigma^2$, and that there exists $x$ so that $\| D \alpha - x \|_2 \leq \epsilon$ with $\| \alpha \|_0 \leq L \ll n$. The $T$ from the first equation above is dictated by $\varepsilon$ and $\sigma$.

I already asked two professors who also don't believe the statement is true. However, it has to come from somewhere. Does it maybe work with $\| \cdot \|_1$? Or is this just an "analytic application" of a penalty method?