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It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints.

The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the LS problem.

What they mean that the regular LS solution result doesn't necessarily yield a solution in the space of valid space of signals and even taking the closes one to the solution isn't optimal.

So we have the problem the constrained problem:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \text{subject to} \quad & s \in {QPSK}^{M} \end{align*}$$

Which its solution is optimal.

What they say isn't optimal is solving:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \end{align*}$$

And then find $ s \in {QPSK}^{M} $ as you'd do in MIMO.
Namely, the greedy method (Do LS while ignoring the constraint and then apply the constraint) isn't optimal.

Yet the problem is that there is no analytic or efficient way to solve the Constraint LS problem but doing brute force search which isn't feasible.

I agree the text isn't clear about making this observation.
Yet it well known in the optimization field that solving in this greedy 2 step method isn't guaranteed to be optimal.

Remark
I found a link to the PDF - Doron Ezri - MIMO OFDM Lecture Notes. What I talked about can be seen in part 5.4.

You must attention to the written text.
It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints.

The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the LS problem.

What they mean that the regular LS solution result doesn't necessarily yield a solution in the space of valid space of signals and even taking the closes one to the solution isn't optimal.

So we have the problem the constrained problem:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \text{subject to} \quad & s \in {QPSK}^{M} \end{align*}$$

Which its solution is optimal.

What they say isn't optimal is solving:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \end{align*}$$

And then find $ s \in {QPSK}^{M} $ as you'd do in MIMO.
Namely, the greedy method (Do LS while ignoring the constraint and then apply the constraint) isn't optimal.

Yet the problem is that there is no analytic or efficient way to solve the Constraint LS problem but doing brute force search which isn't feasible.

I agree the text isn't clear about making this observation.
Yet it well known in the optimization field that solving in this greedy 2 step method isn't guaranteed to be optimal.

You must attention to the written text.
It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints.

The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the LS problem.

What they mean that the regular LS solution result doesn't necessarily yield a solution in the space of valid space of signals and even taking the closes one to the solution isn't optimal.

So we have the problem the constrained problem:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \text{subject to} \quad & s \in {QPSK}^{M} \end{align*}$$

Which its solution is optimal.

What they say isn't optimal is solving:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \end{align*}$$

And then find $ s \in {QPSK}^{M} $ as you'd do in MIMO.
Namely, the greedy method (Do LS while ignoring the constraint and then apply the constraint) isn't optimal.

Yet the problem is that there is no analytic or efficient way to solve the Constraint LS problem but doing brute force search which isn't feasible.

I agree the text isn't clear about making this observation.
Yet it well known in the optimization field that solving in this greedy 2 step method isn't guaranteed to be optimal.

Remark
I found a link to the PDF - Doron Ezri - MIMO OFDM Lecture Notes. What I talked about can be seen in part 5.4.

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You must attention to the written text.
It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints.

The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the LS problem.

What they mean that the regular LS solution result doesn't necessarily yield a solution in the space of valid space of signals and even taking the closes one to the solution isn't optimal.

So we have the problem the constrained problem:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \text{subject to} \quad & s \in {QPSK}^{M} \end{align*}$$

Which its solution is optimal.

What they say isn't optimal is solving:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \end{align*}$$

And then find $ s \in {QPSK}^{M} $ as you'd do in MIMO.
Namely, the greedy method (Do LS while ignoring the constraint and then apply the constraint) isn't optimal.

Yet the problem is that there is no analytic or efficient way to solve the Constraint LS problem but doing brute force search which isn't feasible.

I agree the text isn't clear about making this observation.
Yet it well known in the optimization field that solving in this greedy 2 step method isn't guaranteed to be optimal.

You must attention to the written text.
It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints.

The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the LS problem.

What they mean that the regular LS solution result doesn't necessarily yield a solution in the space of valid space of signals and even taking the closes one to the solution isn't optimal.

You must attention to the written text.
It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints.

The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the LS problem.

What they mean that the regular LS solution result doesn't necessarily yield a solution in the space of valid space of signals and even taking the closes one to the solution isn't optimal.

So we have the problem the constrained problem:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \text{subject to} \quad & s \in {QPSK}^{M} \end{align*}$$

Which its solution is optimal.

What they say isn't optimal is solving:

$$\begin{align*} \arg \min_{s} \quad & \frac{1}{2} {\left\| H s - y \right\|}_{2}^{2} \\ \end{align*}$$

And then find $ s \in {QPSK}^{M} $ as you'd do in MIMO.
Namely, the greedy method (Do LS while ignoring the constraint and then apply the constraint) isn't optimal.

Yet the problem is that there is no analytic or efficient way to solve the Constraint LS problem but doing brute force search which isn't feasible.

I agree the text isn't clear about making this observation.
Yet it well known in the optimization field that solving in this greedy 2 step method isn't guaranteed to be optimal.

1
source | link

You must attention to the written text.
It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints.

The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the LS problem.

What they mean that the regular LS solution result doesn't necessarily yield a solution in the space of valid space of signals and even taking the closes one to the solution isn't optimal.