Visual Equivalence Objective

• Visual Equivalence Objective is a framework that defines loss functions, evaluation protocols, and architectural criteria to align visual representations across different systems.
• It leverages linear, convolutional, and mutual information-based mappings to establish equivalence in applications such as tracking, semantic tokenization, and self-supervised learning.
• Empirical studies show improved performance in CNN accuracy, segmentation, and depth estimation in XR by applying reproducible and formal evaluation metrics.

The Visual Equivalence Objective defines a family of loss functions, evaluation protocols, and architectural criteria for visual representation learning that ensure two or more visual elements, systems, or transformations are judged “equivalent” according to formal, reproducible, and domain-specific objectives. This concept underpins modern work on representation matching, semantic tokenization, tracking, self-supervised learning, perception evaluation in extended reality (XR), and cross-modal alignment. Visual equivalence may be defined by exact loss minimization (as in MMSE objectives), by inclusion in an equivalence class via a simple transform (e.g., linear or convolutional), or by mutual information content, with each variant operationalized through domain-appropriate losses.

1. Formal Definitions and Mathematical Criteria

Visual equivalence asserts that representations, transformations, or system outputs are functionally interchangeable under a loss or mapping. In the context of image representations, this takes the form: given two representations $\phi(x)\in\mathbb R^d$ and $\phi'(x)\in\mathbb R^{d'}$, linear equivalence holds if a map $E: \mathbb R^d \to \mathbb R^{d'}$ exists such that

$$\phi'(x) \approx E\,\phi(x) \qquad \text{for all } x.$$

The empirical instantiation is the minimization of a regularized loss:

$$L(E, b) = \lambda R(E) + \frac{1}{n}\sum_{i=1}^n \ell(\phi'(x_i), E\,\phi(x_i) + b)$$

where $\ell$ is a regression or task-oriented loss and $R(E)$ a regularizer that promotes simplicity (Frobenius norm, channel sparsity) (Lenc et al., 2014). Visual equivalence may also refer directly to task losses: for tracking,

$$J_{\rm corr}(h) = \|y - \sum_{l=1}^d x^l \otimes h^l\|^2 + \lambda\sum_{l=1}^d \|h^l\|^2$$

and

$$J_{\rm conv}(h') = \|y - \sum_{l=1}^d x^l * h^l\|^2 + \lambda\sum_{l=1}^d \|h^l\|^2$$

where the solutions $h^*$ and $h'^*$ yield exactly the same minimum-mean-squared error and differ only by index reversals or trivial conjugation, provided the ideal response $y$ is a centrosymmetric Gaussian (Li et al., 2021).

In self-supervised geometric learning, visual equivalence is imposed by equivariance and geodesic preservation:

$$\phi(T \cdot I_{\rm src}) = h_T(\phi(I_{\rm src})),$$

with loss

$$\mathcal{L}_{\rm equiv} = \|\phi(I_{\rm goal}) - h_T(\phi(I_{\rm src}))\|_2^2$$

+

$$\mathcal{L}_{\rm geo} = \left|\|\phi(I_{\rm src}) - h_T(\phi(I_{\rm src}))\|_2 - c\|p\|_2\right|$$

quantifying the faithfulness of feature displacement to $SE(3)$ geodesic distance (Huh et al., 2022).

2. Applications in Deep Visual Representation Learning

Representation Equivalence in CNNs

Visual equivalence has been rigorously established for early convolutional representations in ConvNets through the introduction of “stitching layers”—simple (linear, convolutional) transformations inserted between the activations of different network instances. When trained with the objective above, these layers can align a wide family of histograms and deep networks, as measured by the preservation of downstream classification accuracy (Lenc et al., 2014). Equivalence holds robustly across architectures and datasets in early layers, but degrades in highly task-specific late layers, indicating generic edge and texture features are widely shared while higher semantic abstractions are network-specific.

Equivalence in Tracking: Correlation vs. Convolution Filters

The Visual Equivalence Objective reveals a formal interchangeability between correlation and convolution-based filters for tracking when the ideal response is a centrosymmetric Gaussian and normal equations are invertible. The two formulations yield conjugate solutions in the Fourier domain, identical MMSE, and spatial filters differing only by index flips. This refutes the view that “template matching” (correlation) and “filter response” (convolution) define essential functionally different trackers, enabling both to be deployed interchangeably in optimization and inference (Li et al., 2021).

Semantic Equivalence in Multimodal and Object-centric Visual Tokenization

Modern vision–language systems require that visual tokens correspond closely to meaningful semantic units. Methods such as SeTok dynamically merge pixels or patches into object-like clusters, enforcing equivalence by both semantic reconstruction (U-Net) and mask consistency with ground-truth or high-quality masks. The associated losses include both reconstruction error and regularized mask alignment:

$$L_{\mathrm{unet}} = \mathbb{E}\|\epsilon - \epsilon_\theta(z_t, t, \{u_n\})\|_2^2,\ L_{\mathrm{mask}} = \mathrm{KL}(M\,\|\,\pi) + L_{\rm bce}(M, \pi) + L_{\rm dice}(M, \pi)$$

This design preserves both low- and high-frequency information and empirically yields superior downstream performance in VQA, image captioning, generation, and segmentation (Wu et al., 7 Jun 2024, Zhong et al., 7 Oct 2025).

In object-centric masked image modeling (MIM), objects serve as the “visual equivalence” of language “words.” The objective targets pixel-level and object-balanced reconstruction loss over coarse object masks, suppressing shortcut learning (e.g., pixel inpainting via interpolation) and fostering globally consistent semantic reasoning:

$$L_{\mathrm{OBJ\text{-}MIM}} = L_{\mathrm{MIM}} + \lambda_1 L_{\mathrm{obj}}$$

where $L_{\mathrm{obj}}$ is a balanced-object term modulated by mask size (Zhong et al., 7 Oct 2025).

3. Geometric, Psychophysical, and Perceptual Instantiations

In extended reality, the Visual Equivalence Objective operationalizes “objective depth” by mapping biological measurements (e.g., gaze-measured vergence angle, GVA) to ground-truth spatial metrics. Empirically, the GVA, once baseline-corrected, provides nearly veridical depth estimation (within 5%) across physical and virtual environments, outperforming subjective reports by a wide margin. The analysis establishes that GVA–depth mapping is stable across experimental manipulations, movement directionality, and inter-individual variability, advocating for equivalence-based metrics in system calibration and performance evaluation (Arefin et al., 2023).

4. Architectural and Algorithmic Implementations

Implementation of visual equivalence commonly employs linear/convolutional “stitching layers” between representations, density-peaks clustering for semantic token extraction, and mask-consistency or reconstruction losses. Loss-architecture pairings enforce either direct feature regression, equivariance under group action, or mutual information preservation. In tracking and regression, block-diagonal or block-conjugate data matrices are solved in the Fourier domain. In geometric learning, shared-weight Siamese networks coordinate transformations and features, with iterative inference over parameterized $SE(3)$ topologies (Li et al., 2021, Huh et al., 2022, Wu et al., 7 Jun 2024, Zhong et al., 7 Oct 2025, Lenc et al., 2014).

Table: Representative Implementations and their Criteria

Domain	Equivalence Mechanism	Loss/Objectives
CNNs	Linear/convolutional mapping	$L(E, b)$ with activation or task loss
Tracking	Filter index flip/conjugation	MMSE loss; spatial filter equivalence
Multimodal token	Clustering + mask alignment	$L_{\mathrm{unet}},\,L_{\mathrm{mask}}$
XR perceptual	GVA–depth matching (baseline norm)	MSE between estimated and true depth
Geometric RL	Equivariance + geodesic matching	$\mathcal{L}_{\rm equiv},\,\mathcal{L}_{\rm geo}$

5. Empirical Findings and Impact

• Equivalence objectives allow rigorous and scalable transfer of learned representations across architectures, initialization seeds, and even domain boundaries at the early feature stage (Lenc et al., 2014).
• In object-centric modeling, the approach yields empirically higher segmentation, detection, VQA, and inpainting performance, with up to 3% accuracy improvement in VQA and 93% context recovery in toy context datasets (Zhong et al., 7 Oct 2025).
• Semantic tokenizers leveraging visual equivalence objectives achieve superior vision–language alignment, as evidenced by consistent gains across multiple MLLM and segmentation benchmarks (Wu et al., 7 Jun 2024).
• In visual servoing, enforcing 3D equivariance and geodesic proportionality yields >35% reduction in average pose error and >90% success rate in tight-tolerance tasks without ground-truth supervision (Huh et al., 2022).
• For XR and depth perception, GVA-based measures outperform subjective judgments by a factor of 3–8× in bias, providing a robust calibration metric (Arefin et al., 2023).

6. Limitations and Future Research

• Linear or convolutional equivalence captures only first-order or simple spatial transformations; non-linear or topologically disparate architectures require more powerful mapping functions (Lenc et al., 2014).
• Per-participant baselines and hardware calibration variability introduce shifts in empirical perceptual equivalence metrics (e.g., GVA offsets in XR), motivating hardware refinement and statistical control (Arefin et al., 2023).
• Mask or cluster-based tokenization is sensitive to segmentation errors and may not generalize to highly non-object-centric or amorphous scenes (Zhong et al., 7 Oct 2025, Wu et al., 7 Jun 2024).
• Partial equivalence degrades with task specificity, and full visual equivalence between abstract or highly compositional vision–language representations remains unsolved.

Prospective work may systematize non-linear visual equivalence for larger architectures, unify psychophysical and computational equivalence standards, and further harness visual equivalence objectives for robust, context-aware multimodal and interactive agents.