Generative Inversion Methods for Ill-Posed Problems

• Generative inversion methods are algorithmic frameworks that leverage pretrained deep generative models to constrain solutions in ill-posed inverse problems.
• They employ non-convex optimization via projected-gradient descent, achieving linear convergence under restricted strong convexity and smoothness conditions.
• Extensions incorporating block-coordinate descent address model mismatch by recovering sparse innovations alongside signals from the generative prior.

A generative inversion method is an algorithmic framework that exploits learned generative network priors—typically deep neural networks such as GANs—for solving ill-posed inverse problems, wherein the objective is to recover a latent or structured representation from observed data that is incomplete, noisy, or otherwise insufficient for direct inversion. By constraining candidate solutions to lie within the range of a pretrained generator, the method leverages the expressive power of the generative model to regularize and structure the search space, leading to improved reconstructions, enhanced convergence properties, and robustness to model mismatch. This approach encompasses projected-gradient methods, theoretical recovery guarantees under network-specific conditions, and principled extensions for situations where the generative prior does not perfectly capture the true signal.

1. Inverse Problem Formulation and Generative Priors

A standard inverse problem is cast as the recovery of an unknown signal $x^*\in\mathbb R^n$ from partial and possibly noisy observations $y = \mathcal{A}(x^*) + e \in \mathbb R^m$, where %%%%2%%%% is a potentially nonlinear measurement operator and $e$ is an additive noise vector. For $m < n$, the inverse mapping is ill-posed. To regularize the solution, generative inversion assumes that valid $x$ reside in the range of a pretrained generator $G:\mathbb R^k \to \mathbb R^n$; that is, $$x \in \mathrm{Range}(G) = \{G(z): z\in\mathbb R^k\}$$ with $k\ll n$. GANs and other rich neural network priors replace traditional handcrafted regularizers (e.g. sparsity) by constraining solutions to this generative manifold (Hegde, 2018).

2. Non-Convex Optimization Objectives and Constrained Search

The inversion objective is formulated as minimization of a fidelity loss subject to the generative constraint: $$\min_{z\in\mathbb R^k}\; F(z) \equiv \ell(\mathcal{A}(G(z)), y) + R(z),$$ where $\ell$ is the data fidelity term (typically $\ell(u,v) = \|u-v\|_2^2$), and $R(z)$ is an optional regularization over the latent space (often set to zero in theory). Equivalently, one may pose

$$\min_{x\in\mathbb R^n} \; \ell(\mathcal{A}(x), y) \quad \text{subject to} \quad x \in \mathrm{Range}(G).$$

This constrained minimization is typically nonconvex due to the complicated geometry of $\mathrm{Range}(G)$, prompting algorithmic innovations to derive provable and efficient solution methods (Hegde, 2018).

3. Projected-Gradient Algorithms and Theoretical Guarantees

A core technique is the $\varepsilon$-projected-gradient descent ($\varepsilon$-PGD), which alternates standard gradient descent steps in data space with projections back onto the generator range via

$$\begin{aligned} &\text{Initialize } x_0 = 0 \ &\text{For } t = 0..T-1: \ &\quad g_t \gets \nabla_x \ell(\mathcal{A}(x_t), y) \ &\quad z_t \gets x_t - \eta g_t \ &\quad x_{t+1} \gets P_G(z_t), \ &\text{End} \end{aligned}$$

where $P_G(v)$ approximately solves $\arg\min_{x\in\mathrm{Range}(G)} \|v-x\|_2^2$ up to accuracy $\varepsilon$. This outer loop converges linearly to an $O(\varepsilon)$ neighborhood of the optimum under restricted strong convexity/smoothness (RSC/RSS) conditions on $F(x)$ and mild assumptions on the geometry of $\mathrm{Range}(G)$ (Hegde, 2018).

More precisely, if $F$ is RSC/RSS with parameters $\alpha > 0$ and $\beta > \alpha$, and $0 < 2 - \beta/\alpha < 1$, then with step size $\eta = 1/\beta$, the error contracts as

$$F(x_{t+1}) - F(x^*) \leq (2-\beta/\alpha)[F(x_t)-F(x^*)] + C\varepsilon,$$

for constant $C$, guaranteeing linear convergence to $F(x^*) + O(\varepsilon)$ after $O(\log(1/\varepsilon))$ iterations (Hegde, 2018).

4. Extensions for Model Mismatch and Block-Coordinate Descent

In practice, the true signal $x^*$ may not lie exactly in $\mathrm{Range}(G)$, resulting in model-mismatch. The generative inversion framework accommodates this by decomposing $x = u + v$, with $u = G(z)$ and $v$ being $s$-sparse in a known basis $B$. The inversion objective becomes

$$\min_{u\in\mathrm{Range}(G),\, v:\|B^Tv\|_0 \leq s} F(u+v).$$

The algorithm proceeds via a block-coordinate version of $\varepsilon$-PGD: $$\begin{aligned} &u_{t+1} \gets P_G(u_t - \eta g_t) \ &v_{t+1} \gets \text{HardThreshold}_{B,s}(v_t - \eta g_t) \ &x_{t+1} \gets u_{t+1} + v_{t+1} \end{aligned}$$ where $\text{HardThreshold}_{B,s}$ zeroes all but the $s$ largest coefficients of $B^Tw$. Under RSC/RSS on $\mathrm{Range}(G) \oplus \mathrm{Span}(B)$ and an incoherence assumption between $\mathrm{Range}(G)$ and $\mathrm{Span}(B)$, the error contracts similarly, ensuring robust recovery in the presence of innovations outside the generator range (Hegde, 2018).

5. Empirical Performance and Comparison to Latent-Space Gradient Descent

Empirical evaluations instantiate $\varepsilon$-PGD on compressive-sensing inverse problems using GAN priors (e.g. MNIST, CelebA). Compared against "latent-space back-propagation"—direct gradient descent on $z$ to minimize $\|\mathcal{A}(G(z)) - y\|^2$—the projected-gradient method achieves near-zero reconstruction error in $30$–$50$ iterations (vs. $>200$ for back-prop), $2$–$5\times$ lower final MSE, and total runtime speedup by $4$–$5\times$ due to similar per-iteration costs but faster convergence (Hegde, 2018). This motivates the use of principled iterative algorithms over naïve gradient schemes for generative inversion.

6. Significance, Limitations, and Future Directions

Generative inversion methods provide a rigorous mechanism for exploiting high-dimensional deep network priors in inverse problems, offering linear convergence under appropriate problem and network conditions, and transparent extensions handling model-mismatch via block-coordinate strategies. Theoretical analyses highlight the importance of convexity and smoothness properties restricted to the generator range, with practical success contingent upon reliable projection or approximation oracles $P_G$.

Key limitations include dependence on the expressiveness and coverage of the generative model; if physically realizable signals depart significantly from the generator manifold, recovery may stall or accuracy degrade. The model-mismatch extension partially alleviates this by capturing sparse innovations.

Potential future directions involve developing more efficient inner-projection routines $P_G$, advancing theory to encompass broader classes of generative architectures, and integrating learned priors with adaptive sparsity or structured innovations. There is active interest in extending these techniques to settings with nontrivial forward operators, stochastic noise, and in adapting $\varepsilon$-PGD variants for large-scale, high-dimensional data domains (Hegde, 2018).