Problems such as those are closely related to "inverse problems", whereby one attempts to reconstruct a signal from an under-determined system of equations, (ill-posed problems) given some apriori information. As @Hilmar mentioned, you cannot reconstruct phase information from the absolute magnitude of a DFT, unless you have prior information that constrains you. For example, in the paper you mention, the author states:
Both the problem of phase
retrieval from two intensity measurements (in electron microscopy or
wave front sensing) and the problem of phase retrieval from a single
intensity measurement plus a non-negativity constraint (in astronomy)
are considered, with emphasis on the latter. It is shown [...]
The bold is mine. Even in the paper you allude to, there are constraints that then allow for a Maximum Likelihood framework to be applied, or one where sparsity is rewarded, etc.
Edit: The prior information I have is : Time-domain signal is
positive-valued. Time domain signal is zero for in some points, i.e.
signal has a support P⊂{0,1,2,⋯,N−1}
All your signal being positive valued is irrelevant to the phase information of the entire DFT spectrum. To see this, consider this following:
Note that having an all positive signal $x[n]$ means you are guaranteed to have a DC bias. This means that the first bin in the absolute magnitude of you DFT corresponding to $|X(0)|$ will be strictly positive. Let us split the input signal into its zero-mean partition, ($z[n]$) plus the mean value, $m$. Then:
$$x[n] = z[n] + m$$
The DFT of the result is, (exploiting the linearity property of the DFT):
$$
X[k] = \sum_{n=0}^{N-1} x[n] \ e^{-j \frac{ \Large 2 \pi n k }{N}} = \sum_{n=0}^{N-1} \Big[z[n] + m \Big] \ e^{-j \frac{ \Large 2 \pi n k }{N}} = \sum_{n=0}^{N-1} z[n] \ e^{-j \frac{ \Large 2 \pi n k }{N}} + \sum_{n=0}^{N-1} m \ e^{-j \frac{ \Large 2 \pi n k }{N}}
$$
Recall that the DFT is nothing but a projection of your signal $x[n]$ onto an orthogonal basis set composed of complex exponentials. Save for the DC component at $k=0$, all the complex sinusoids have mean $0$. Thus the second term $\sum_{n=0}^{N-1} m \ e^{-j \frac{ \Large 2 \pi n k }{N}}$ completely vanishes for all $k \ne 0$. Hence, we can now write the DFT of $x[n]$ as being:
$$
X[k] = \left\{
\begin{array}{l l}
\sum_{n=0}^{N-1} z[n] \ e^{-j \frac{ \Large 2 \pi n k }{N}} + mN = mN & ,k = 0\\
\sum_{n=0}^{N-1} z[n] \ e^{-j \frac{ \Large 2 \pi n k }{N}} & , k \ne 0
\end{array} \right.
$$
This then shows that your signal always being positive is completely irrelevant to the phase signal spectrum.
Hence, this prior information ($x[n] \ge 0, n \in 0, 1, ... N-1$) is irrelevant, and no recovery is possible given $|X[k]|$, with only this constraint.
You are right in that these types of problems have some solutions to them, but it is also important for you to also remember that you have to define your constraints before you can solve for your unknown entities. That is, you have to give prior information that you can utilize.
As a simple example, given the absolute magnitude vector $|X(f)|$, there is an infinite number of possible solutions to what values $\angle X(f)$ can take on. However, if you "know" before hand that, say, all the imaginary co-efficients of $X(f) \in \mathfrak{C} $ are zero, then we can easily recover what the real co-efficients are up to a sign ambiguity.
The take home message however, is what prior information do you have, and how is it related to the DFT. Hope this helped.
