Vivek Gopalakrishnan

# A Quick Formulation of Compressed Sensing

2021-12-27 | A very, very mininal derivation of the optimization

Compressed sensing is a singal processing method for reconstructing a signal from an undersampled set of measurements. Here, we will derive the optimization problem at the core of compressed sensing.

• Let $x$ be the true signal.

• Let $x \stackrel{\mathcal U}{\mapsto} s$ be the representation of $x$ in a functional basis space.

• Let $x = \Psi s$ for some universal transform basis $\Psi$ (e.g., Fourier, wavelet, etc.).

• Let $y = Cx$ be the measured signal, where $C$ is the measurement matrix.

Our objective is to obtain a subsampled version of $s$ from $y$, and then recover $s \stackrel{\mathcal U^{-1}}{\mapsto} x$, i.e.,

$\min_s \frac{1}{2} ||y - C\Psi s||_2^2 \,.$

As stated, this optimization is a underspecified / indeterminate problem (i.e., there are infinitely many solutions for $s$), so we constrain the solutions to be sparse:

$\min_s \frac{1}{2} ||y - C\Psi s||_2^2 + \lambda||s||_0 \,.$

However, minimizing $l_0$ norms is NP hard, so we relax the problem with the $l_1$ norm:

$\min_s \frac{1}{2} ||y - C\Psi s||_2^2 + \lambda||s||_1 \,.$

  Because it requires combinatorial optimization.
This is a well-defined problem, and is solvable with linear programming (e.g., basis pursuit, denoising/matching). Additionally, Equation 3 is equivalent to Equation 2 under certain conditions (e.g., $\nu$-incoherence) and constraints on $C\Psi$. Then, given a noisy, downsampled measurement $y$, solve Equation 3 to get $\hat s$, and predict the original signal to be $\hat x = \Psi \hat s$ which is equivalent to $\hat s \stackrel{\mathcal U^{-1}}{\mapsto} \hat x$.