Signal reconstruction notes

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Problem

Start with a known sensing matrix $A \in \R^{m \times n}$ with $m \ll n$ , and noisy observations

y = A x^\star + \varepsilon, \qquad \varepsilon \sim \mathcal{N}(0, \sigma^2 I). \tag{1}

The unknown $x^\star$ is assumed sparse. A first estimator is basis-pursuit denoise,

\hat x = \argmin_{x} \tfrac{1}{2}\norm{y - Ax}^2 + \lambda \norm{x}_1. \tag{2}

Each numbered display is its own $$ block with one \label{...}. In prose, \eqref{eq:obs} and \eqref{eq:lasso} become links to those tags: the recovery in (2) is written directly in terms of the measurements from (1). Use \ref{eq:lasso} if you want the bare number 2 without parentheses.

Approach

ISTA applies a gradient step on the least-squares term in (2), then a soft-threshold. The two updates are one numbered block (aligned inside $$), still a single \label:

\begin{aligned} z &\leftarrow x - t A^\top(Ax - y) \\ x &\leftarrow \operatorname{prox}_{t\lambda\lVert\cdot\rVert_1}(z). \end{aligned} \tag{3}

Relative reconstruction error against the $x^\star$ that generated (1) is

e(x) = \frac{\norm{x - x^\star}}{\norm{x^\star}}. \tag{4}

The implementation splits into three stages:

Generate a sparse ground-truth vector and a Gaussian sensing matrix, i.e. a draw from (1).
Recover with the ISTA map (3), then with a short unrolled network when the same operator is reused.
Report (4) and support Hamming distance against $x^\star$ .

def ista_step(x, y, A, lam, t):
    grad = A.T @ (A @ x - y)
    z = x - t * grad
    return np.sign(z) * np.maximum(np.abs(z) - t * lam, 0.0)

The unrolled variant keeps $A$ fixed and learns step sizes. That is useful when the same operator is applied to many independent observations, and it stays honest about what is learned: a few scalars, not a new sensing model.

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