Generative Priors: Theory & Applications

• Generative priors are constraints imposed by deep generative models that encode the statistical structure of data, enabling realistic reconstructions.
• They offer global optimization guarantees for nonconvex inverse problems, ensuring descent directions exist outside minor exceptional regions.
• Applications span compressed sensing, image reconstruction, phase retrieval, and blind deconvolution, backed by rigorous theoretical analysis.

A generative prior is a constraint or regularization imposed on a solution space by the range of a generative model—typically a deep neural network—that has been trained to represent a data distribution of interest. Unlike classical hand-crafted or analytic priors (e.g., sparsity, smoothness, or total variation), generative priors reflect the statistical structure learned from large datasets, making them powerful tools for solving ill-posed inverse problems, enforcing plausible reconstructions, and enabling efficient dimension reduction. This entry addresses the theoretical underpinnings, algorithmic formulations, empirical findings, and broad implications of generative priors, with an emphasis on rigorous results emerging from the analysis of deep architectures.

1. Theoretical Foundations and Global Properties

The central theoretical advance in the paper of generative priors is the establishment of global optimization guarantees for inverse problems regularized by deep generative neural networks. When a signal $y_0$ is known to be in the range of a generator $G$—that is, $y_0 = G(x_0)$ for some latent vector $x_0$—recovery can be formulated as solving

$f(x) = \frac{1}{2}\|A G(x) - A y_0\|_2^2$

where $A$ is a measurement matrix (or the identity for direct inversion). For generators $G$ realized by deep, expansive, piecewise linear (typically ReLU-activated) networks with random Gaussian weights, it has been shown that the empirical risk minimization objective $f(x)$ is highly nonconvex yet globally benign under suitable conditions: except for small neighborhoods around the true $x_0$ and a negative scalar multiple $-\rho_d x_0$ (with $\rho_d \rightarrow 1$ as depth $d \rightarrow \infty$), there are no spurious stationary points, and descent directions exist everywhere else (Hand et al., 2017). This result provides a theoretical basis for the empirical observation that gradient-based optimization reliably finds the desired solution, even though $f(x)$ is nonconvex.

Key conditions required for such guarantees include:

• Expansivity: Layer widths satisfy $n_i \geq c \, n_{i-1} \log n_{i-1}$ for some $c > 0$.
• Randomness: Network weights and measurement matrices are i.i.d. Gaussians (or possess similar concentration properties).
• Sufficient measurement: $m = \Omega(k d \log(n_1 \dots n_d))$, where $k$ is latent dimension.

These results provide explicit global guarantees; outside the two small exceptional regions, descent-based algorithms cannot be trapped by spurious local minima or saddles.

2. Model Formulations and Optimization Strategies

Generative priors are enforced by restricting solutions to the range of a generative network. Two canonical problem settings are prominent:

Problem Setting	Objective	Comments
Inverting a Generative Network (full output accessible)	$\min_x \frac{1}{2} \|G(x) - y_0\|_2^2$	$A = I$
Inversion with Compressive Linear Observations	$\min_x \frac{1}{2} \|A G(x) - A G(x_0)\|_2^2$	$A$ random, $m \ll n$

The generator is typically of the form $$G(x) = \mathrm{ReLU}(W_d \cdots \mathrm{ReLU}(W_1 x)\cdots)$$. The strong nonlinearity and nonconvexity introduced by the composition of ReLUs is handled by leveraging concentration of measure and union bounds over the exponential number of activation patterns.

The optimization is conducted over the latent space (usually low-dimensional), employing gradient descent or variants thereof. These algorithms benefit from a global landscape that avoids non-global local minima under the stated conditions (Hand et al., 2017).

3. Statistical and Computational Guarantees

A distinctive property of generative priors is their ability to provide optimal sample complexity and reliable recovery in a variety of inverse problems, even in the presence of strong nonlinearity. Specifically:

• Statistical optimality: The number of measurements required for accurate recovery scales linearly with the dimension of the latent space $k$, not with the ambient dimension $n$ or sparsity level $s$ as in classical priors (Huang et al., 2018). For instance, $m = \Omega(k d \log(n_1\dots n_d))$ measurements suffice for successful inversion.
• Benign optimization geometry: Outside small neighborhoods of $x_0$ and $-\rho_d x_0$, every point offers a descent direction for the empirical risk—there are no spurious local minima or saddles. The origin itself is a local maximizer. The result is robust even for compressive observations.
• Algorithmic reliability: Simple gradient descent, with proper initialization and step size, is sufficient for convergence to the true code in these settings.

These properties bridge a significant gap between the empirical performance of deep generative priors and their theoretical justification in nonconvex, nonlinear inverse problems. Notably, the empirical success of recovery algorithms for restricted classes of signals is now matched by rigorous, non-asymptotic analysis.

4. Applications and Impact

Generative priors have broad applicability across the spectrum of inverse problems, with several notable areas:

• Compressed Sensing: Substituting hand-crafted sparsity priors with the learned range of a deep generator reduces both the required number of measurements and improves recovery fidelity, particularly for structured signals not adequately captured by classic priors.
• Image Reconstruction/Inpainting: The generative prior constrains the solution space to realistic outputs. Theoretical guarantees underpin successes in tasks such as image completion, super-resolution, and MRI reconstruction (Hand et al., 2017).
• Phase Retrieval and Blind Deconvolution: Generative priors permit low-sample-complexity and robust recovery in settings where traditional sparsity-based approaches fail due to the nonlinear nature of the measurement model.
• Robustness: Enforcing a generative prior, which constrains solutions to a low-complexity manifold of plausible images, has been shown to yield increased robustness to measurement noise, outliers, and adversarial perturbations.

5. Limitations, Extensions, and Future Directions

While the results provide foundational understanding and practical assurance, several limitations and directions for extension are identified:

• Model Mismatch: The theory assumes that the true signal lies exactly in the range of the generator; practical scenarios often present mismatch due to imperfect training, model bias, or noise. Extending guarantees to accommodate mismatch, noise, or outliers is an open challenge.
• Architectural Realism: The theoretical results currently apply to random, bias-free, fully connected networks. Analysis of architectures with biases, convolutions, pooling, or more general activation functions is needed for full relevance to state-of-the-art generative models.
• Measurement Operators: Results primarily address random Gaussian measurement matrices; many practical systems (e.g., Fourier or structured measurements in imaging and diffraction) require further theoretical investigation.
• Generalization: The extension of guarantees to settings with only approximate generative coverage or to cases involving model adaptation or transfer is a critical area for development.

Active research is addressing these challenges, including the analysis of robustness properties, generalization to broader neural network architectures, and empirical validation on more complex and realistic datasets and inverse problem settings.

6. Broader Implications

The theoretical results on generative priors have catalyzed a paradigm shift in the design and analysis of regularization techniques for inverse problems:

• From analytic to data-driven priors: Deep generative models provide high-capacity, data-driven characterizations of signal classes, significantly outperforming classical priors when the data distribution is complex or multimodal.
• Bridging practice and theory: Detailed geometric analysis validates the practical effectiveness observed across machine learning, signal processing, and imaging communities.
• Influence on algorithm design: The absence of spurious stationary points in generative-prior-constrained empirical risk leads to the development of direct, scalable solvers and underpins their reliability.

In summary, generative priors enforce low-dimensionality and learned structure through the range of a deep neural network. Under suitable regimes, they guarantee both favorable optimization geometry and sample-efficient, robust recovery, substantiating their rapidly growing role in computational imaging, inverse problems, and signal processing (Hand et al., 2017).