Active Learning for Conditional Generative Compressed Sensing

Problem Statement
Generative compressed sensing (GCS) recovers structured signals from undersampled measurements by assuming the signal lies near the range of a pretrained generative model $G$. While existing theory establishes recovery guarantees for unconditional models under Gaussian measurements, practical applications often involve structured, subsampled unitary transforms (e.g., Fourier measurements in MRI) and possess auxiliary information available at acquisition or inference time. This paper addresses the gap in conditional generative compressed sensing, where the generative model is conditioned on prompts (e.g., text descriptions, metadata). The core challenge is that conditioning introduces a tripartite interaction: the prompt defining the true signal ($c^*$), the prompt used to design the sampling distribution ($c_s$), and the prompt used to define the recovery model ($c_r$). Mismatches between these prompts can lead to suboptimal sampling strategies or biased reconstructions, yet existing frameworks largely treat conditioning as a static prior rather than a dynamic design variable for both sensing and recovery.

Methodology
The authors formulate a framework for prompt-conditioned active learning under subsampled Fourier measurements, adapting the Christoffel Sampling for Machine Learning (CS4ML) approach.

1. Prompt-Conditioned Model Classes: The signal space is defined by a conditional generator $G(z, c)$, where $z$ is a latent vector and $c$ is a prompt. The authors distinguish three prompts:

   • $c^*$: The true-signal prompt (if $f^* \in F_{c^*}$).
   • $c_r$: The recovery prompt defining the candidate set $F_{c_r}$.
   • $c_s$: The sampling prompt used to construct the sampling distribution $\mu_{c_s}$.

2. Prompt Compatibility Factor ($\Lambda$): To quantify the interaction between these prompts, the authors define a Prompt Compatibility Factor:
   $$\Lambda(c_1, c_2, c_3) = \max_{i \in D} \frac{K(F_{c_1} - F_{c_2})(i)}{\mu_{c_3}(i)}$$
   where $K$ is the generalized Christoffel function measuring the "informativeness" of a Fourier coordinate for distinguishing signals in a class, and $\mu$ is the sampling distribution.

   • $\Lambda(c_r, c_r, c_s)$ controls the sample complexity required for stable recovery.
   • $\Lambda(c^*, c_r, c_s)$ controls the reconstruction error (approximation error) when the true signal does not match the recovery prompt.

3. Theoretical Analysis: The paper derives sample complexity bounds for two classes of conditional generators:

   • ReLU Networks: Assuming a bias-free, depth-$d$ feedforward ReLU network.
   • Lipschitz Generators: Assuming the generator is $L$-Lipschitz on the latent ball.
     In both cases, the bounds show that the number of measurements $m$ scales with $\Lambda$, the complexity of the generator (e.g., width/depth for ReLU, Lipschitz constant $L$), and the minimum sampling probability $\mu$.

4. Recovery Guarantees: Under the Set-Restricted Eigenvalue Condition (S-REC), the authors prove that the reconstruction error is bounded by the best approximation error in the recovery class, amplified by $\sqrt{\Lambda(c^*, c_r, c_s)}$, plus terms for noise and optimization error.

Key Contributions

• Formulation: A unified framework for prompt-conditioned GCS that explicitly separates the roles of prompts in sampling design and recovery modeling.
• Theoretical Metric: The introduction of the Prompt Compatibility Factor $\Lambda$, which unifies the analysis of sampling-recovery mismatch and signal-recovery mismatch into a single statistic governing both sample complexity and error bounds.
• Sample Complexity Bounds: Explicit bounds for ReLU and Lipschitz conditional generators, demonstrating that prompt mismatch incurs a penalty (increased sample complexity) proportional to the divergence between the sampling and recovery geometries.
• Empirical Validation: Experiments using Stable Diffusion 1.5 (SD15) demonstrate that:
  • Prompts meaningfully reshape Christoffel sampling distributions (concentrating mass on different Fourier frequencies).
  • Mismatches between sampling and recovery prompts ($c_s \neq c_r$) increase the compatibility factor $\Lambda$.
  • Reconstruction performance is sensitive to these mismatches, though the authors note that unconditional recovery ($c_r = \text{""}$) often performed robustly in their specific experimental setup, potentially due to the flexibility of the unconditional model or the specific optimization dynamics.

Results

• Theoretical: The paper proves that stable recovery is possible with high probability if the number of measurements scales with $\Lambda(c_r, c_r, c_s)$. If $c_s = c_r$, $\Lambda$ reduces to the standard Christoffel complexity constant. If $c_s \neq c_r$, $\Lambda$ can grow, reflecting the need for more samples to capture signal features relevant to $c_r$ that are underrepresented in the $c_s$-based sampling.
• Experimental:
  • Sampling Distributions: Different prompts (e.g., "sunset beach" vs. "cat") induce distinct sampling distributions, with matched pairs minimizing $\Lambda$.
  • Reconstruction: In "in-range" experiments (where the true signal is generated by the model), sampling with a semantically similar prompt ($c_s$ close to $c^*$) improved PSNR and SSIM compared to mismatched prompts.
  • CFG Ablation: The authors found that high Classifier-Free Guidance (CFG) values (e.g., 7.5) restricted the generator too strongly, leading to poorer reconstruction performance compared to lower CFG values (e.g., 1), suggesting that the "effective" recovery class is highly dependent on hyperparameters.

Significance and Claims
The paper claims to establish that prompts should be treated as design variables with distinct effects on sensing (sampling distribution), approximation (recovery class), and the interaction between them. The primary significance lies in extending GCS theory to conditional settings, showing that:

1. Active Learning is Prompt-Dependent: The optimal sampling strategy depends on the specific prompt used for recovery, not just the signal class.
2. Mismatch Penalties: There is an explicit theoretical cost (in terms of sample complexity and error amplification) when the sampling prompt does not align with the recovery prompt.
3. Practical Implications: While the theory assumes idealized generators, the experiments with Stable Diffusion suggest that prompt conditioning can substantially alter the induced sampling distributions and influence reconstruction quality, though the interplay with optimization hyperparameters (like CFG) is complex.

The authors modestly note that their theory applies to idealized generator classes (ReLU/Lipschitz) and that modern diffusion models are complex abstractions of these classes. They suggest future work should investigate divergence-based versions of $\Lambda$ and better characterize the effective Lipschitz constants of practical diffusion models under varying CFG scales.