**Derivation.** Starting with Bayes' theorem for the posterior:

$$p(Z_T | Z_C, \eta) = \frac{p(Z_C, \eta | Z_T) p(Z_T)}{p(Z_C, \eta)}.$$

Substituting the conditional independence assumption from Eq. (56):

$$p(Z_T | Z_C, \eta) = \frac{p(Z_C | Z_T) p(\eta | Z_T) p(Z_T)}{p(Z_C, \eta)}.$$

Next, we invert the individual likelihood terms using Bayes' theorem to express them in terms of the predictive distributions (the "experts"):

$$p(Z_C | Z_T) = \frac{p(Z_T | Z_C) p(Z_C)}{p(Z_T)} \quad \text{and} \quad p(\eta | Z_T) = \frac{p(Z_T | \eta) p(\eta)}{p(Z_T)}.$$

Substituting these back into the posterior equation:

$$p(Z_T | Z_C, \eta) = \frac{\left( \frac{p(Z_T|Z_C)p(Z_C)}{p(Z_T)} \right) \left( \frac{p(Z_T|\eta)p(\eta)}{p(Z_T)} \right) p(Z_T)}{p(Z_C, \eta)}.$$

Simplifying the terms:

$$p(Z_T | Z_C, \eta) = \frac{p(Z_T | Z_C) p(Z_T | \eta) p(Z_C) p(\eta)}{p(Z_T) p(Z_C, \eta)}.$$

Since $Z_C$ and $\eta$ are observed inputs (constants with respect to the optimization of $Z_T$), the terms $p(Z_C)$, $p(\eta)$, and $p(Z_C, \eta)$ can be absorbed into a normalization constant $\mathcal{Z}$. The relation reduces to:

$$p(Z_T | Z_C, \eta) \propto \frac{p(Z_T | Z_C) p(Z_T | \eta)}{p(Z_T)}.$$

In the BJEPA formulation, we approximate the marginal prior over targets $p(Z_T)$ as uniform (or absorbed into the learned bias of the experts), yielding the standard Product of Experts form:

$$p(Z_T | Z_C, \eta) \propto p(Z_T | Z_C) p(Z_T | \eta).$$

# J Toy Experiment Details

Here we provide the specific implementation details, hyperparameters, and architectural and training specifications used to generate the results in Section 9.

## J.1 Environment and Data Generation

The synthetic environment is a Linear-Gaussian system designed to simulate the "Noisy TV" scenario.

* Dimensions: Observation dimension $D_x = 20$, Signal dimension $D_s = 4$, Distractor dimension $D_d = 4$.

* Dynamics:
- Signal: $s_{t+1} = A_{\text{rot}}s_t + w_t$, where $A_{\text{rot}}$ is an orthogonal matrix generated via QR decomposition of a random Gaussian matrix. Process noise $w_t \sim \mathcal{N}(0, 0.1^2I)$.

- Distractor: $d_{t+1} = 0.9d_t + v_t$. Process noise $v_t \sim \mathcal{N}(0, 0.3^2I)$.

* **Observations:** $x_t = C_s t + D(\sigma d_t) + \epsilon_t$. Matrices $C$ and $D$ are random matrices with column-wise unit norm normalization. Sensor noise $\epsilon_t \sim \mathcal{N}(0, 0.01^2I)$.

* **Dataset:** We generate a continuous training trajectory of $T_{\text{train}} = 6000$ steps and a testing trajectory of $T_{\text{test}} = 2000$ steps.

* **Noise Scales:** The scaling factor $\sigma$ is swept from 0.0 to 8.0 in 9 equidistant steps.

## J.2 Model Architectures and Training Strategy

We implement five models to map the current observation $x_t \in \mathbb{R}^{D_x}$ to a latent representation $z_t \in \mathbb{R}^{D_z}$. To impose a hard trade-off between signal and distractor encoding, we set the latent dimension equal to the true signal dimension ($D_z = D_s = 4$). All models utilize strictly linear transformations ($f(x) = Wx$) without bias terms ('bias=False'), matching the linear nature of the data generation process.

We compare two modeling paradigms: **Generative Reconstruction** (VAE, AR) operating in pixel space, and **Joint-Embedding Prediction** (JEPA, VJEPA, BJEPA) operating via latent-space autoregression.

### J.2.1 Generative Baselines (Pixel Reconstruction)

These models must reconstruct the full observation $x$, including high-variance distractors.

**Linear Variational Autoencoder (VAE).** The VAE treats the problem as a static modeling task. It maps $x_t$ to a latent distribution $q(z_t|x_t)$ and reconstructs $x_t$.

* **Architecture:** A probabilistic encoder with two linear heads $\mu_\phi$, $\log \sigma_\phi^2$ mapping $D_x \to D_z$, and a deterministic decoder $\hat{x}_t = W_{\text{dec}}z_t$.

* **Objective:** The Evidence Lower Bound (ELBO)$^{27}$:

$$ \mathcal{L}_{\text{VAE}} = \|x_t - \hat{x}_t\|^2 + \beta D_{KL}(q(z_t|x_t)\|\mathcal{N}(0, I)). $$

* **Hypothesis:** To minimize reconstruction error, the VAE must prioritize features with the highest variance. We predict it will encode distractors and ignore the signal.

**Linear Pixel-Autoregressive (AR).** The AR model is a predictive autoencoder operating in pixel space. It compresses $x_t$ into a bottleneck to predict the next observation $x_{t+1}$.

* **Architecture:** A deterministic encoder $z_t = W_{\text{enc}}x_t$ mapping $D_x \to D_z$ and a predictive decoder $\hat{x}_{t+1} = W_{\text{pred}}z_t$ mapping $D_z \to D_x$.

* **Objective:** Mean Squared Error in pixel space: $\mathcal{L}_{\text{AR}} = \|x_{t+1} - \hat{x}_{t+1}\|^2$.

* **Hypothesis:** Since distractors are predictable ($d_{t+1} \approx 0.9d_t$), the AR model must track them to minimize prediction error, wasting capacity on distractor texture.

<footnote>$^{27}$Under the assumption of a Gaussian likelihood with identity covariance, $p(x_t|z_t) = \mathcal{N}(\hat{x}_t, I)$, minimizing the negative log-likelihood is equivalent to minimizing the squared Euclidean distance $\|x_t - \hat{x}_t\|^2$ plus a constant.</footnote>
## J.2.2 Joint-Embedding Architectures (Latent Autoregression)

These models are time-indexed, order-1 Markovian in latent space, trained to predict $z_{t+1}$ without reconstructing pixels.

**Linear JEPA (Deterministic).** JEPA aligns the predicted next state with the encoded future state.

* **Architecture:**

1. Context Encoder: $z_t = W_{enc}x_t$.

2. Predictor: A linear layer $\hat{z}_{t+1} = W_{pred}z_t$ mapping $D_z \to D_z$.

3. Target Encoder (EMA): Same architecture as Context Encoder.

* **Objective:** VICReg loss, minimizing prediction error $\|\hat{z}_{t+1} - z'_{t+1}\|^2$ with variance/covariance regularization to prevent collapse.

**Linear Probabilistic VJEPA.** VJEPA extends JEPA by treating the prediction as a probabilistic distribution.

* **Architecture:**

1. Deterministic Encoder: $z_t = W_{enc}x_t$.

2. Probabilistic Predictor: Two linear heads outputting $\mu_{pred}$, $\log \sigma_{pred}^2 = f_\phi(z_t)$ mapping $D_z \to D_z$.

3. Target Encoder (EMA): Two heads outputting $\mu_{\theta'}$, $\log \sigma_{\theta'}^2$ mapping $D_x \to D_z$.

* **Objective:** The regularized negative log-likelihood (Eq. 11):

$$ \mathcal{L}_{\text{VJEPA}} = \mathbb{E}_{\tilde{z}_{t+1} \sim q_{\theta'}} [-\log p_{\phi}(\tilde{z}_{t+1} | z_t)] + \beta D_{KL}(q_{\theta'}(z'_{t+1} | x_{t+1}) \| \mathcal{N}(0, I)). $$

**Linear Bayesian JEPA (BJEPA).** BJEPA incorporates a static Bayesian prior to filter out unpredictable dynamics by fusing a Predictive Likelihood (Dynamics Expert) with a Latent-Space Prior (Static Expert).

* **Architecture:** Identical to VJEPA, but adds learnable static parameters $\mu_{prior}$, $\log \sigma_{prior}^2$ (initialized to zero) to define a preferred latent manifold.

* **Inference:** BJEPA performs hard fusion via a Product of Experts (PoE):

$$ \mu_{post}, \log \sigma_{post}^2 = \text{PoE}((\mu_{dyn}, \sigma_{dyn}^2), (\mu_{prior}, \sigma_{prior}^2)). $$

* **Objective:** $\mathcal{L}_{\text{BJEPA}} = \mathcal{L}_{\text{VJEPA}} + \gamma D_{KL}(p_{\text{like}}(z_{t+1} | z_t) \| p_{\text{prior}}(z_{t+1})).$
**Stochastic Training Strategy.** In our implementation, we parameterize the context encoder $f_{\theta}(x_C)$ as a deterministic map, delegating uncertainty modeling entirely to the predictive transition $p_{\phi}(Z_T | Z_C)$ and the target inference $q_{\theta'}(Z_T | x_T)$. For both VJEPA and BJEPA, optimization of the variational objective (Eq. 11 and Eq. 33) requires estimating an expectation over the target distribution $E_{Z_T \sim q_{\theta'}}$. Following standard practice in stochastic gradient variational inference (SGVI), we approximate this expectation using a single Monte Carlo sample ($K = 1$) drawn from the target encoder's distribution $q_{\theta'}(Z_T | x_T)$ via the reparameterization trick at each training step. While this single-sample estimate is noisy for any individual data point, the stochasticity of mini-batch gradient descent (SGD) smooths the objective over the course of training, providing an unbiased gradient estimator.

Note that the sampling operation is applied only to the target encoder $q_{\theta'}$ to obtain a concrete regression target $Z_T$. We do not sample from the predictor $p_{\phi}$. Instead, the predictor outputs the parameters of its belief distribution (e.g. $\mu_{\phi}, \sigma_{\phi}^2$), and we analytically evaluate the negative log-likelihood density (Gaussian in our implementation) of the sampled target $Z_T$ under these parameters. Mathematically, this evaluates how well the predictor's belief explains the specific realization drawn from the target encoder's belief.

**Necessity of Target Sampling.** Sampling $Z_T$ from the target distribution $q_{\theta'}$ (rather than simply using its mean $\mu_{\theta'}$) is strictly necessary to prevent *variance collapse* in the predictor. Consider the Gaussian negative log-likelihood term minimized by the predictor:

$$ \mathcal{L}_{\text{pred}} \approx \frac{(Z_T - \mu_{\phi})^2}{2\sigma_{\phi}^2} + \frac{1}{2}\log(\sigma_{\phi}^2). $$

If we were to use the target mean as the training target (i.e. set $Z_T = \mu_{\theta'}$), the predictor could learn to output $\mu_{\phi} \approx \mu_{\theta'}$ perfectly, driving the squared error term to zero. This would encourage the variance $\sigma_{\phi}^2$ to collapse to zero (or $\log \sigma_{\phi}^2 \to -\infty$) to minimize the second term, resulting in a deterministic, overconfident model. By sampling $Z_T \sim N(\mu_{\theta'}, \sigma_{\theta'}^2)$, the target becomes a “moving target” that fluctuates around the mean according to the target’s uncertainty. The squared error term $(Z_T - \mu_{\phi})^2$ remains non-zero, forcing the predictor to maintain a non-zero variance $\sigma_{\phi}^2$ to cover the distribution of potential futures.

**Implementation and Limitations.** In our experimental implementation, we parameterize both the target encoder $q_{\theta'}$ and the predictor $p_{\phi}$ as independent diagonal Gaussian distributions:

$$ q_{\theta'}(Z | x_T) = N(Z; \mu_{\theta'}(x_T), \text{diag}(\sigma_{\theta'}^2(x_T))), \quad p_{\phi}(Z | Z_C) = N(Z; \mu_{\phi}(Z_C), \text{diag}(\sigma_{\phi}^2(Z_C))). $$

While this choice simplifies optimization and admits closed-form KL terms, the unimodal Gaussian assumption represents a limitation. In environments with complex, multi-modal bifurcations (e.g. distinct “go left” vs. “go right” futures), a single Gaussian may average the modes, resulting in a high-variance belief centered on an invalid state. Future work could address this by implementing more expressive heads, such as Gaussian Mixture Models or normalizing flows.

### J.2.3 Temporal Alignment in Training and Evaluation

All models receive observation pairs $(x_t, x_{t+1})$ during training, where $x_t$ denotes the current observation and $x_{t+1}$ denotes the next-step observation. The temporal alignment differs between model families based on their predictive structure:
## Training Alignment.

* VAE: trained on single observations $x_t$ with reconstruction target $x_t$ (no temporal prediction).

* AR: trained on pairs $(x_t, x_{t+1})$ with pixel-space prediction target $x_{t+1}$.

* JEPA/VJEPA/BJEPA: trained on pairs $(x_t, x_{t+1})$ with latent-space prediction target $z_{t+1} = f_{\theta'}(x_{t+1})$.

**Evaluation Alignment.** As predictive models (AR, JEPA, VJEPA, BJEPA) are trained to predict future states, their latent representations should be evaluated against the *future* ground-truth signal $s_{t+1}$, not the current signal $s_t$. Concretely:

* VAE: The encoder output $z_t = \mu_\phi(x_t)$ represents the current state and is evaluated against $s_t$.

* AR: The encoder output $z_t = W_{enc}x_t$ is designed to predict $x_{t+1}$, hence is evaluated against $s_{t+1}$.

* JEPA: The predictor output $\hat{z}_{t+1} = W_{pred}z_t$ explicitly predicts the next latent state and is evaluated against $s_{t+1}$.

* VJEPA: The predictor mean $\mu_{pred}(z_t)$ represents the expected next latent state and is evaluated against $s_{t+1}$.

* BJEPA: At inference, the hard-fused posterior mean $\mu_{post} = \text{PoE}(\mu_{dyn}, \mu_{prior})$ represents the constrained prediction of the next latent state and is evaluated against $s_{t+1}$.

This temporal alignment ensures that each model's representation is evaluated against the appropriate ground-truth signal consistent with its predictive objective.

## J.2.4 Linear Probe Evaluation Protocol

To assess how well each model's latent representation captures the true underlying signal $s_t$ (while ignoring the distractor $d_t$), we employ a linear probe methodology. For each model, we extract latent representations and fit a linear regression to predict the ground-truth signal, then measure the coefficient of determination ($R^2$) on held-out test data.

**Latent Extraction.** The latent representation used for probing differs by model, reflecting their distinct architectures:

* VAE: $z_t^{\text{probe}} = \mu_\phi(x_t)$ — the encoder mean, representing the current state.

* AR: $z_t^{\text{probe}} = W_{enc}x_t$ — the encoder output, which implicitly predicts the next state.

* JEPA: $z_t^{\text{probe}} = W_{pred}(W_{enc}x_t) = \hat{z}_{t+1}$ — the predictor output, explicitly predicting the next latent state.

* VJEPA: $z_t^{\text{probe}} = \mu_{pred}(W_{enc}x_t)$ — the predictor mean, representing the expected next latent state.
* BJEPA: $z_t^{\text{probe}} = \mu_{\text{post}}$ — the hard-fused posterior mean computed via Product of Experts:

$$ \mu_{\text{post}} = \frac{\sigma_{\text{prior}}^2 \cdot \mu_{\text{dyn}} + \sigma_{\text{dyn}}^2 \cdot \mu_{\text{prior}}}{\sigma_{\text{dyn}}^2 + \sigma_{\text{prior}}^2}, $$

where $(\mu_{\text{dyn}}, \sigma_{\text{dyn}}^2)$ are the dynamics predictor parameters and $(\mu_{\text{prior}}, \sigma_{\text{prior}}^2)$ are the learned static prior parameters.

**Probe Fitting and Evaluation.** Given extracted latents $\{z_i^{\text{probe}}\}_{i=1}^{N_{\text{train}}}$ from the training set and corresponding ground-truth signals $\{s_i^{\text{target}}\}_{i=1}^{N_{\text{train}}}$, we fit a linear regression:

$$ \hat{s} = W_{\text{probe}} z^{\text{probe}} + b_{\text{probe}}, $$

where $(W_{\text{probe}}, b_{\text{probe}})$ are learned via ordinary least squares. The target signal $s^{\text{target}}$ is chosen according to the temporal alignment described above: $s_t$ for VAE, and $s_{t+1}$ for all predictive models (AR, JEPA, VJEPA, BJEPA).

We then evaluate on the test set by computing the $R^2$ score:

$$ R^2 = 1 - \frac{\sum_i \|s_i^{\text{target}} - \hat{s}_i\|^2}{\sum_i \|s_i^{\text{target}} - \bar{s}\|^2}, $$

where $\bar{s}$ is the mean of the test targets. An $R^2$ close to 1 indicates that the latent representation successfully captures the true signal, while an $R^2$ close to 0 indicates that the representation has failed to disentangle signal from distractor.

**Interpretation.** This evaluation protocol directly tests the core hypothesis: predictive models operating in latent space (JEPA, VJEPA, BJEPA) should learn representations that capture the predictable signal while discarding unpredictable distractors, whereas generative models (VAE, AR) operating in pixel space are forced to allocate capacity to high-variance distractors regardless of their task relevance. The use of predictor outputs (rather than encoder outputs) for JEPA-family models ensures we evaluate the representation that the model is actually trained to produce, i.e. the predicted future state.

## J.3 Optimization and Hyperparameters

All models are trained using the *Adam* optimizer [38]. We use full-batch training (processing the continuous trajectory as a single batch) for stability in this linear setting.
<table><caption>Table 6: Optimization settings and loss function hyperparameters.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Description</th></tr></thead><tbody><tr><td colspan="3"><em>Global Optimization</em></td></tr><tr><td>Random Seed</td><td>111</td><td>For reproducibility</td></tr><tr><td>Optimizer</td><td>Adam [38]</td><td>Standard implementation</td></tr><tr><td>Learning Rate</td><td>1 × 10<sup>-3</sup></td><td>Constant throughout training</td></tr><tr><td>Training Steps</td><td>6000</td><td>Sufficient for convergence</td></tr><tr><td colspan="3"><em>JEPA Family Common Settings</em></td></tr><tr><td>EMA Decay (τ)</td><td>0.99</td><td>Target encoder update rate</td></tr><tr><td colspan="3"><em>Model-Specific Coefficients</em></td></tr><tr><td><strong>JEPA</strong> (VICReg Loss)</td><td></td><td></td></tr><tr><td>Invariance Coeff</td><td>25.0</td><td>MSE term weight</td></tr><tr><td>Variance Coeff</td><td>25.0</td><td>Hinge loss on standard deviation</td></tr><tr><td>Covariance Coeff</td><td>1.0</td><td>Off-diagonal decorrelation weight</td></tr><tr><td><strong>VJEPA</strong></td><td></td><td></td></tr><tr><td>β (Eq. 11)</td><td>0.01</td><td>Target encoder KL regularization weight</td></tr><tr><td><strong>BJEPA</strong></td><td></td><td></td></tr><tr><td>β (Eq. 11)</td><td>0.01</td><td>Target encoder KL regularization weight</td></tr><tr><td>γ (Eq. 33)</td><td>0.1</td><td><strong>Structural Prior</strong> KL weight</td></tr></tbody></table>

**Note on BJEPA Priors.** In the Python code implementation for the “Noisy TV” experiment, the Structural Prior used for the $\gamma$ term is *static* (time-invariant) but *learnable*. It is parameterized as a Gaussian with parameters $\mu_{\text{prior}}$, $\sigma_{\text{prior}}^2$ that are optimized alongside the model weights to find the stable manifold of the signal. The reference prior used for the $\beta$ term (target regularization) is a fixed unit Gaussian $N(0, I)$.

## J.4 Results

We measure performance using the coefficient of determination ($R^2$) between the learned latent $Z_t$ and the true signal $s_t$ via a linear probe. The quantitative results<sup>28</sup> are summarized in Table.4 and visualized across all 9 noise scales in Fig.3. Additionally, Fig.4 provides a qualitative view of the signal reconstruction at low, medium, and high noise levels.

**Failure of Generative Models.** As shown in Table.7 and Fig.3, the reconstructive baselines (VAE, AR) suffer catastrophic performance degradation as the distractor noise scale increases. At the highest noise scale ($\sigma = 8.0$), VAE and AR signal recovery drops to $R^2 \approx 0.50$ and $0.58$ respectively, while their noise recovery remains high ($R^2 = 0.62$ and $R^2 = 0.45$). This confirms that likelihood-based objectives force the models to encode high-variance distractors, leading to representation collapse on the target task.

<footnote n="28"><sup>28</sup>We do not compare training loss values across methods because their objectives define optimality in fundamentally different metric spaces. Generative models (VAE, AR) minimize pixel-level reconstruction error, whereas JEPA-based models minimize feature distance or maximize likelihood in an abstract latent space with arbitrary scaling. Consequently, the raw loss magnitudes are not comparable.</footnote>
**Robustness of Joint-Embedding Architectures.** In contrast, all joint-embedding architectures (JEPA, VJEPA, BJEPA) demonstrate robustness to the “Noisy TV” distractor. They maintain high signal recovery ($R^2 > 0.85$) even at $\sigma = 8.0$, effectively filtering out the distractor dynamics. In this specific trial, the deterministic JEPA achieved the highest final accuracy ($R^2 = 0.93$), though it exhibited higher variance in training stability compared to the probabilistic variants<fnref n="29" />.

**Stability and Optimization.** The high-variance noise makes the target embeddings ($z_{t+1}$) jittery and unpredictable. As standard JEPA is deterministic, it struggles to average out this jitter, leading to potential training instability. Even in this successful run, JEPA showed a momentary instability at Noise Scale 3.0, dropping to 0.841 (Test) while VJEPA/BJEPA remained at > 0.90. This collapse at Scale 3.0 likely indicates optimization difficulties where the variance regularization term fought against the predictive loss. This hints at the optimization difficulty deterministic models face when the noise magnitude creates complex gradients, even if the model recovered at Scale 4.0.

<footnote n="29">²⁹While the deterministic JEPA achieved high performance in the reported run (Seed 111), we observed in auxiliary experiments with different random seeds that it is prone to occasional training collapse (sudden performance drops to $R^2 < 0.5$) at intermediate noise scales. We attribute this to the difficulty of optimizing a deterministic predictor on highly stochastic targets. The probabilistic methods (VJEPA and BJEPA) consistently exhibited greater training stability across seeds by accounting for aleatoric uncertainty via their variance outputs.</footnote>
<table><caption>Table 7: Performance metrics ($R^2$) across models at all noise scales. Tr denotes training set, Te denotes test set. Note the robustness of JEPA family models at high noise levels (e.g. Scale 8.0) compared to generative baselines.</caption><thead><tr><th>Scale (SNR)</th><th>Model</th><th>↑Signal R<sup>2</sup> (Tr/Te)</th><th>↓Noise R<sup>2</sup> (Tr/Te)</th><th>↓Time (Tr/Te)</th></tr></thead><tbody><tr><td>0.0 (inf dB)</td><td>VAE</td><td><u>1.000</u> / <u>1.000</u></td><td>NA / NA</td><td>12.6s / 0.01s</td></tr><tr><td>0.0 (inf dB)</td><td>AR</td><td>0.999 / 0.999</td><td>NA / NA</td><td><u>6.4s</u> / 0.01s</td></tr><tr><td>0.0 (inf dB)</td><td>JEPA</td><td>0.947 / 0.930</td><td>NA / NA</td><td>16.9s / 0.01s</td></tr><tr><td>0.0 (inf dB)</td><td>VJEPA</td><td>0.999 / 0.999</td><td>NA / NA</td><td>13.9s / 0.01s</td></tr><tr><td>0.0 (inf dB)</td><td>BJEPA</td><td>0.987 / 0.981</td><td>NA / NA</td><td>23.5s / 0.01s</td></tr><tr><td>1.0 (15.8 dB)</td><td>VAE</td><td>0.978 / 0.973</td><td>0.065 / 0.002</td><td>12.6s / 0.01s</td></tr><tr><td>1.0 (15.8 dB)</td><td>AR</td><td>0.983 / 0.980</td><td>0.041 / -0.007</td><td><u>6.6s</u> / 0.01s</td></tr><tr><td>1.0 (15.8 dB)</td><td>JEPA</td><td>0.947 / 0.930</td><td>0.230 / 0.178</td><td>16.9s / 0.01s</td></tr><tr><td>1.0 (15.8 dB)</td><td>VJEPA</td><td><u>0.999</u> / <u>0.998</u></td><td><u>0.006</u> / <u>-0.012</u></td><td>13.8s / 0.01s</td></tr><tr><td>1.0 (15.8 dB)</td><td>BJEPA</td><td>0.975 / 0.976</td><td>0.029 / -0.003</td><td>23.5s / 0.01s</td></tr><tr><td>2.0 (9.8 dB)</td><td>VAE</td><td>0.903 / 0.853</td><td>0.283 / 0.194</td><td>12.8s / 0.01s</td></tr><tr><td>2.0 (9.8 dB)</td><td>AR</td><td>0.923 / 0.889</td><td>0.179 / 0.094</td><td><u>6.4s</u> / 0.01s</td></tr><tr><td>2.0 (9.8 dB)</td><td>JEPA</td><td>0.947 / 0.930</td><td>0.213 / 0.172</td><td>17.0s / 0.01s</td></tr><tr><td>2.0 (9.8 dB)</td><td>VJEPA</td><td><u>0.996</u> / <u>0.995</u></td><td><u>0.020</u> / -0.012</td><td>14.2s / 0.01s</td></tr><tr><td>2.0 (9.8 dB)</td><td>BJEPA</td><td>0.966 / 0.967</td><td>0.042 / <u>-0.023</u></td><td>23.3s / 0.01s</td></tr><tr><td>3.0 (6.3 dB)</td><td>VAE</td><td>0.852 / 0.770</td><td>0.458 / 0.393</td><td>12.5s / 0.01s</td></tr><tr><td>3.0 (6.3 dB)</td><td>AR</td><td>0.870 / 0.801</td><td>0.328 / 0.256</td><td><u>6.5s</u> / 0.01s</td></tr><tr><td>3.0 (6.3 dB)</td><td>JEPA</td><td>0.916 / 0.841</td><td><u>0.000</u> / <u>-0.003</u></td><td>16.8s / 0.01s</td></tr><tr><td>3.0 (6.3 dB)</td><td>VJEPA</td><td><u>0.947</u> / <u>0.930</u></td><td>0.224 / 0.176</td><td>13.5s / 0.01s</td></tr><tr><td>3.0 (6.3 dB)</td><td>BJEPA</td><td>0.946 / 0.919</td><td>0.106 / 0.078</td><td>23.0s / 0.01s</td></tr><tr><td>4.0 (3.8 dB)</td><td>VAE</td><td>0.822 / 0.730</td><td>0.512 / 0.458</td><td>12.5s / 0.01s</td></tr><tr><td>4.0 (3.8 dB)</td><td>AR</td><td>0.839 / 0.756</td><td>0.394 / 0.338</td><td><u>6.5s</u> / 0.01s</td></tr><tr><td>4.0 (3.8 dB)</td><td>JEPA</td><td><u>0.999</u> / <u>0.999</u></td><td><u>0.004</u> / <u>-0.010</u></td><td>16.6s / 0.02s</td></tr><tr><td>4.0 (3.8 dB)</td><td>VJEPA</td><td>0.994 / 0.993</td><td>0.025 / -0.007</td><td>13.8s / 0.01s</td></tr><tr><td>4.0 (3.8 dB)</td><td>BJEPA</td><td>0.920 / 0.899</td><td>0.213 / 0.156</td><td>23.2s / 0.01s</td></tr><tr><td>5.0 (1.8 dB)</td><td>VAE</td><td>0.777 / 0.681</td><td>0.543 / 0.492</td><td>12.5s / 0.01s</td></tr><tr><td>5.0 (1.8 dB)</td><td>AR</td><td>0.806 / 0.717</td><td>0.418 / 0.364</td><td><u>6.4s</u> / 0.01s</td></tr><tr><td>5.0 (1.8 dB)</td><td>JEPA</td><td><u>0.947</u> / <u>0.930</u></td><td><u>0.200</u> / <u>0.183</u></td><td>16.5s / 0.01s</td></tr><tr><td>5.0 (1.8 dB)</td><td>VJEPA</td><td>0.945 / 0.928</td><td>0.227 / 0.190</td><td>14.1s / 0.01s</td></tr><tr><td>5.0 (1.8 dB)</td><td>BJEPA</td><td>0.884 / 0.841</td><td>0.317 / 0.240</td><td>22.3s / 0.01s</td></tr><tr><td>6.0 (0.3 dB)</td><td>VAE</td><td>0.729 / 0.627</td><td>0.577 / 0.531</td><td>12.3s / 0.01s</td></tr><tr><td>6.0 (0.3 dB)</td><td>AR</td><td>0.770 / 0.677</td><td>0.437 / 0.386</td><td><u>7.1s</u> / 0.01s</td></tr><tr><td>6.0 (0.3 dB)</td><td>JEPA</td><td><u>0.999</u> / <u>0.999</u></td><td><u>0.007</u> / <u>-0.011</u></td><td>16.0s / 0.01s</td></tr><tr><td>6.0 (0.3 dB)</td><td>VJEPA</td><td>0.947 / 0.930</td><td>0.197 / 0.202</td><td>13.7s / 0.01s</td></tr><tr><td>6.0 (0.3 dB)</td><td>BJEPA</td><td>0.935 / 0.914</td><td>0.205 / 0.171</td><td>22.8s / 0.01s</td></tr><tr><td>7.0 (-1.1 dB)</td><td>VAE</td><td>0.675 / 0.566</td><td>0.615 / 0.574</td><td>12.7s / 0.01s</td></tr><tr><td>7.0 (-1.1 dB)</td><td>AR</td><td>0.729 / 0.631</td><td>0.462 / 0.416</td><td><u>7.1s</u> / 0.01s</td></tr><tr><td>7.0 (-1.1 dB)</td><td>JEPA</td><td><u>0.946</u> / <u>0.929</u></td><td><u>0.209</u> / 0.191</td><td>16.1s / 0.01s</td></tr><tr><td>7.0 (-1.1 dB)</td><td>VJEPA</td><td>0.939 / 0.918</td><td>0.264 / <u>0.182</u></td><td>13.4s / 0.01s</td></tr><tr><td>7.0 (-1.1 dB)</td><td>BJEPA</td><td>0.894 / 0.851</td><td>0.291 / 0.237</td><td>23.1s / 0.01s</td></tr><tr><td>8.0 (-2.2 dB)</td><td>VAE</td><td>0.613 / 0.499</td><td>0.656 / 0.620</td><td>12.3s / 0.01s</td></tr><tr><td>8.0 (-2.2 dB)</td><td>AR</td><td>0.680 / 0.578</td><td>0.491 / 0.449</td><td><u>7.1s</u> / 0.01s</td></tr><tr><td>8.0 (-2.2 dB)</td><td>JEPA</td><td><u>0.947</u> / <u>0.930</u></td><td><u>0.226</u> / <u>0.183</u></td><td>16.1s / 0.01s</td></tr><tr><td>8.0 (-2.2 dB)</td><td>VJEPA</td><td>0.905 / 0.870</td><td>0.299 / 0.251</td><td>13.4s / 0.01s</td></tr><tr><td>8.0 (-2.2 dB)</td><td>BJEPA</td><td>0.896 / 0.841</td><td>0.292 / 0.238</td><td>23.0s / 0.01s</td></tr></tbody></table>
## J.5 Computational Resources

All experiments were conducted using the standard CPU runtime provided by Google Colab [22]. The key hardware specifications are summarized below:

* **Platform:** Google Colab (Linux x86_64)

* **CPU:** Intel(R) Xeon(R) CPU @ 2.20GHz

* **CPU Topology:** 2 Logical CPUs (1 Physical Core, 2 Threads per core)

* **Memory (RAM):** 13.61 GB

* **Disk Space:** 242.49 GB

* **Instruction Set:** x86_64 with AVX2 and FMA support

Given the lightweight nature of the linear toy experiment (latent dimension $D_z = 4$), training was performed entirely on the CPU, with individual model runs completing in approximately 10-35 seconds (see Table 7).

## J.6 Code Availability

The complete source code and data generation procedures used for the experiments in this paper are available at: https://github.com/YongchaoHuang/VJEPA.