Differentiable Perceptual Proxies Overview

• Differentiable perceptual proxies are continuous, neural-network-based surrogates designed to approximate non-differentiable human perceptual metrics, enabling end-to-end gradient optimization.
• They employ both analytical formulations like MS-SSIM and learned approaches using deep feature losses to more accurately reflect human judgments in tasks such as image and audio processing.
• Their integration into applications—from generative modeling to metric learning—results in measurable improvements in reconstruction quality, compression, and overall perceptual alignment.

Differentiable perceptual proxies are continuous, parameterized functions—typically neural networks or analytically specified differentiable operators—constructed to serve as surrogates for human perceptual judgments, or as stand-ins for non-differentiable black-box perceptual metrics, in machine learning pipelines. Their differentiability enables direct integration into gradient-based optimization, empowering applications ranging from generative modeling and compression to inverse procedural modeling and deep metric learning. Unlike heuristic or hand-crafted metrics, differentiable perceptual proxies are carefully designed or trained—often on human judgments or by mimicking high-level perceptual features—to yield gradients that more faithfully reflect human notions of similarity, quality, or recognition, thus reconciling optimization with perceptual objectives.

1. Foundations of Differentiable Perceptual Proxies

The core motivation for differentiable perceptual proxies arises from the inadequacy of classical loss functions (such as pixel-wise $\ell_1$ or $\ell_2$ norms) in aligning with human perception. While these losses are mathematically convenient and differentiable, they typically fail to capture image structures or audio artifacts salient to observers (Snell et al., 2015). To overcome this, perceptual metrics grounded in structural similarity, multiscale analysis, or deep feature embeddings—such as MS-SSIM, NLP, LPIPS, or learned discriminators—have been introduced as differentiable proxies for perceptual quality.

A differentiable perceptual proxy is defined such that, for input signals $x$ and $y$, a function $d_{\text{proxy}}(x,y)$ produces a real-valued score, and the entire computation graph from $x$ and $y$ through $d_{\text{proxy}}$ is differentiable with respect to its inputs and, if applicable, its own parameters. This property permits the proxy metric to be used as a loss in backpropagation.

The term “proxy” can be used in two senses:

• As a differentiable surrogate for a complex, perhaps non-differentiable perceptual function or black-box (e.g., SSIM, VMAF) (Chen et al., 2019, Hu et al., 2022).
• As a compact embedding or parameter vector that stands in for an object, class, or perceptual construct (e.g., class proxies in deep metric learning (Saberi-Movahed et al., 2023)).

2. Mathematical Formulations and Differentiable Metrics

Several differentiable perceptual proxies are prominent in literature:

A. Multiscale Structural Similarity (MS-SSIM):

MS-SSIM is formulated via patch-based luminance, contrast, and structure comparisons across scales: $$\operatorname{MS\text{-}SSIM}(x,y) = I_M(x,y)^{\alpha_M} \prod_{j=1}^M [C_j(x,y)^{\beta_j} \cdot S_j(x,y)^{\gamma_j}]$$ where operations $I, C, S$ are differentiable (Snell et al., 2015).

B. Perceptual Loss via Deep Features:

For feature maps $f^l$ from layer $l$: $$d(x, x_0) = \sum_l \frac{1}{H_l W_l} \sum_{h,w} \| w_l \odot (f^l_{hw} - f^l_{0hw}) \|_2^2$$ where $w_l$ are learned or fixed channel weights (Huang et al., 14 Jan 2024).

C. Learned Proxies in Deep Metric Learning:

Class proxies $P \in \mathbb{R}^{d \times r}$ are regularized for soft orthogonality: $$\min_P \ \ell_\text{ProxyAnchor}(P) + \lambda \|P^T P - I_r\|_F^2$$ where $\ell_\text{ProxyAnchor}$ couples proxies to data samples by class, and the second term enforces near-orthogonality between proxies (Saberi-Movahed et al., 2023).

D. Linear Autoregressive Similarity Index (LASI):

Image embeddings $w_i$ are constructed for each pixel by solving, at inference time,

$$w_i([1,i)) = \arg\min_{w \in \mathbb{R}^N} \sum_{j < i} \omega^{\ell_{ij}} (n_j^T w - x_j)^2$$

leading to a similarity metric

$$d(x, y) = \frac{1}{k} \sum_{i = 1}^k \| w_i^{(x)} - w_i^{(y)} \|_2$$

which is fully differentiable and parameter-free (Severo et al., 2023).

3. Training and Construction Methodologies

Construction of differentiable perceptual proxies employs several strategies:

• Direct Analytical Differentiability: Metrics such as MS-SSIM or NLP are designed to be continuous and differentiable by construction, making them straightforward to integrate in loss functions (Snell et al., 2015, Ballé et al., 2016).
• Learning from Human Judgments: Proxies may be trained on labeled perceptual data, such as mean opinion scores (MOS), just-noticeable-difference (JND) pairings, or paired comparisons. For example, in perceptual audio assessment, a deep network is regressed on binary “same/different” labels near the JND threshold, yielding a metric D(x_ref, x_per) that is differentiable and empirically aligned with MOS and human preferences (Manocha et al., 2020).
• Proxy Networks for Non-differentiable Metrics: When the target perceptual metric is non-differentiable (e.g., VMAF), a proxy network is trained to regress these scores and used as a smooth substitute during training (e.g., ProxIQA (Chen et al., 2019)).
• Contrastive and Online Feature Disentanglement: Learning to disentangle perceptually-relevant dimensions from deep feature embeddings via contrastive or triplet loss strategies yields representations that are robust as perceptual proxies and more aligned with human similarity judgments (Mei et al., 2020).
• Proxy as Differentiable Approximators: Differentiable proxies may be trained to approximate the outputs of black-box procedural or graphics functions, enabling gradient-based optimization even when the original function is non-differentiable (Hu et al., 2022).

4. Applications in Image, Audio, and Structured Domains

The utility of differentiable perceptual proxies spans a wide array of domains:

• Image Generation and Reconstruction: Employing MS-SSIM or deep-feature loss as a differentiable proxy yields reconstructions that better preserve texture and structure, outperforming pixel-based losses as assessed by human observers (Snell et al., 2015).
• Compression and Super-Resolution: Perceptual metrics serve as rate-distortion objectives, guiding codecs to allocate bits where perceptual relevance is highest and achieving improved perceptual appearance often at lower bitrates, with quantifiable superiority over MSE-optimized baselines (Ballé et al., 2016, Chen et al., 2019).
• Audio Enhancement and Assessment: Learned proxies trained from crowdsourced JND datasets provide differentiable loss functions that correlate well with MOS and improve the subjective quality of denoised speech (Manocha et al., 2020).
• Inverse Procedural Modeling: Differentiable proxies approximate non-differentiable nodes in procedural graphs, enabling full gradient-driven optimization of both structure and style in photorealistic material synthesis (Hu et al., 2022).
• Metric Learning and Retrieval: In deep metric learning, class proxies are learned and regulated for orthogonality, structuring embedding spaces for high retrieval accuracy and efficient clustering (Saberi-Movahed et al., 2023).
• Probabilistic Inference and Visual Illusions: Differentiable probabilistic program interpreters enable visual puzzle synthesis by differentiating through Bayesian inference procedures, allowing adversarial optimization of inputs to induce desired perceptual errors under computational models of vision (Chandra et al., 2022).

5. Human Alignment, Evaluation, and Limitations

A central axis for evaluating differentiable perceptual proxies is alignment with human perceptual judgments:

• Empirical Preference Studies: Experiments consistently reveal human preference for outputs optimized under perceptual, differentiable losses over traditional pixel-based objectives across reconstructions and super-resolution outputs (Snell et al., 2015).
• Correlation with MOS and Quality Metrics: Learned proxies demonstrate high Spearman and Pearson correlations with mean opinion scores in both audio and image quality assessment (Manocha et al., 2020, Severo et al., 2023). Proxies trained on JND data most closely replicate human thresholds of perceptual change.
• Complementary Failure Modes: Maximum Differentiation (MAD) competitions between metrics such as LASI and LPIPS reveal that no single proxy universally dominates; each is sensitive to distinct artifact types, and combined approaches merit consideration (Severo et al., 2023).
• Proxy Calibration and Flexibility: Updating proxy networks concurrently with main system parameters, as in ProxIQA, maintains high metric fidelity and steers optimization to perceptual improvements while managing the risk of proxy misalignment or drift (Chen et al., 2019).
• Semantic Awareness via Auxiliary Tasks: Simultaneous training for both compression and classification fosters semantic-aware proxies, enhancing correlation with perceptual judgments without reliance on external networks such as VGG (Huang et al., 14 Jan 2024).

6. Architectural Integrations and Broader Paradigms

Differentiable perceptual proxies are deeply integrated into modern architectures and learning paradigms:

• End-to-End Approaches: Frameworks jointly optimize analysis and synthesis transforms, quantization, and perceptual objectives in an end-to-end differentiable fashion, as in nonlinear transform codecs for compression (Ballé et al., 2016).
• Contrastive Feature Disentanglement: Online construction and selection of contrastive triplets activate only the perception-relevant dimensions, yielding more effective perceptual losses (Mei et al., 2020)
• Attention and Transformer Backbones: Modern proxy-based metric learning leverages contextualized features from transformer architectures, e.g., DeiT, yielding higher discrimination in retrieval tasks and greater embedding robustness (Saberi-Movahed et al., 2023).
• Graph and Predictive Coding Models: In predictive coding, layers of differentiable, hierarchical, and dynamic state transitions are optimized with precision-weighted errors, providing rich internal proxies for both perception and planning, and modularity via Markov Blankets (Ofner et al., 2021).
• Optimization of Non-differentiable Black-Box Modules: By training differentiable neural proxies for procedural modules, full gradient backpropagation over entire computational graphs—including structured and procedural graphics pipelines—is enabled, supporting efficient material matching and inverse design (Hu et al., 2022).

7. Future Directions and Open Questions

The landscape of differentiable perceptual proxies is actively evolving:

• Hybrid and Adaptive Proxies: Multi-metric approaches, combining the strengths of deeply learned and data-free (analytical) proxies (e.g., LASI + LPIPS), offer improved robustness against metric-specific artifacts (Severo et al., 2023).
• Task-Specific and Semantic-Aware Proxies: Joint optimization with diverse auxiliary tasks and explicit semantic constraints can enhance the generality and discriminative utility of proxies, especially for applications in low-bandwidth and edge-device scenarios (Huang et al., 14 Jan 2024).
• Generalization Beyond Vision and Audio: Differentiable proxies have potential extensions in perceptual modeling for multimodal data, dynamic prediction, and even emergent properties in spatio-temporal graph structures, as suggested by recent work on graph hierarchies and Markov Blankets (Ofner et al., 2021).
• Efficient Integration in Procedural and Probabilistic Programming: Advances in differentiable probabilistic programming expose inference procedures to gradient-based optimization, enabling the synthesis of controlled perceptual effects and more transparent analysis of perceptual inference mechanisms (Chandra et al., 2022).
• Challenges in Proxy Calibration: Ensuring stability, overcoming overfitting, and calibrating proxies to accurately reflect human perception across diverse contexts—rather than merely reproducing dataset-specific statistics—remain ongoing challenges.

In summary, differentiable perceptual proxies constitute a foundational element in harmonizing machine optimization with sensory experience, offering mathematically tractable, gradient-compatible, and semantically rich surrogates for perceptual judgment across vision, audio, and structured domain applications. Their continuing evolution integrates metric design, proxy learning, architecture innovation, and biological inspiration, setting the agenda for perceptually driven optimization in machine intelligence.