Effective Encoding Dimension (EED)

• EED is a metric that quantifies the number of effective directions in encoded representations using spectral measures, rank statistics, and empirical thresholds.
• It is computed by analyzing the eigenspectrum of covariance matrices in models like Vision Transformers, hyperdimensional computing, and statistical frameworks.
• EED informs model design by identifying bottlenecks and optimizing the trade-off between dimensionality reduction and computational efficiency.

Effective Encoding Dimension (EED) is a mathematically formalized concept quantifying the number of degrees of freedom or “useful” directions present in an encoded representation, feature set, or parameter space, conditional on model, task, data, and algorithmic specifics. EED provides a principled, data-driven measure that generalizes across contexts: neural networks (especially Vision Transformers), hyperdimensional computing, dimensionality reduction frameworks, and statistical models. It is operationalized via spectral statistics (entropy, PCA, Fisher information), optimization criteria, or empirical accuracy thresholds, capturing the “true” representational or modeling capacity required for effective learning or inference.

1. Mathematical Definitions and General Formulations

EED is consistently grounded in spectral and rank-based measures:

• Spectral Entropy Definition (ViT context):

Given $H^{(l)} \in \mathbb{R}^{N \times D}$ (token embeddings at layer $l$), the feature covariance $\Sigma^{(l)} = \frac{1}{N}(H^{(l)})^T H^{(l)}$. The spectrum $\{\lambda_1, ..., \lambda_D\}$ of $\Sigma^{(l)}$ is normalized, and the spectral entropy computed:

$$S(\Sigma^{(l)}) = -\sum_{k=1}^D p_k^{(l)} \log p_k^{(l)}, \quad p_k^{(l)} = \frac{\lambda_k^{(l)}}{\sum_{j=1}^D \lambda_j^{(l)}}$$

Effective encoding dimension:

$N_{\text{eff}}^{(l)} = \exp[S(\Sigma^{(l)})]$

Normalized EED:

$$\text{EED}\%^{(l)} = \frac{N_{\text{eff}}^{(l)}}{D} \times 100\%$$

If the spectrum is flat, $N_{\text{eff}} \approx D$; for a collapsed spectrum, $N_{\text{eff}} \approx 1$ (Awadhiya, 8 Dec 2025).

• Fisher Information Definition (Statistical models):

For model family $$\mathcal{M} = \{P(x|\theta): \theta \in \Theta \subset \mathbb{R}^d\}$$ with Fisher information $F_{ij}(\theta)$ and scale resolution $\epsilon = 1/\sqrt{n}$,

$$\dim_{\text{eff}}(n; \mathcal{M}) = 2 \frac{\log\left(\frac{1}{V_\Theta} \int_\Theta \sqrt{\det(\mathrm{Id}_d + \frac{n}{2\pi} F(\theta))} d\theta \right)}{\log\left(\frac{n}{2\pi}\right)}$$

The EED interpolates between the count of “strong” directions and the nominal dimension $d$, depending on eigenvalue dispersion and sample size $n$ (Berezniuk et al., 2020).

• Encoding Map Definition (Linear algebra, dimension reduction):

With sample-encoding $\alpha: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and feature-encoding $\beta: \mathbb{R}^p \rightarrow \mathbb{R}^r$, the respective EEDs are

$$\text{EED}_\text{s} = \operatorname{rank}(\alpha) = m, \qquad \text{EED}_\text{f} = \operatorname{rank}(\beta) = r$$

If nonlinear reductions, $m = n^\alpha$, $r = p^\beta$, where $\alpha, \beta \in (0,1]$ (Banh et al., 2022).

2. Algorithmic Procedures for Computing EED

The computation of EED varies by application pattern:

• Vision Transformers (ViT, self-supervised):

For each layer $l$: 1. Gather token embeddings $H^{(l)}$. 2. Compute layer covariance $\Sigma^{(l)}$. 3. Perform eigendecomposition to extract $\lambda_k$. 4. Normalize spectrum and compute spectral entropy. 5. Exponentiate to obtain $N_{\text{eff}}^{(l)}$. 6. Normalize and repeat across all layers to yield the EED profile (Awadhiya, 8 Dec 2025).

• Hyperdimensional Computing (DistHD):

1. Encode each sample into D-dimensional hypervector.
2. Identify misleading dimensions using top-2 class scores, calculate global distance statistics per dimension.
3. Regenerate (replace) bases of top $R \cdot D$ misleading dimensions.
4. Repeat until model accuracy plateaus; the smallest $D$ achieving target accuracy is defined as the EED (Wang et al., 2023).

• Statistical/Linear Models:

1. Apply projection $\alpha$ (samples) or $\beta$ (features) to original data.
2. Induce encoded space; rank of encoding map is the EED.
3. Alternatively, in scale-space analysis, calculate the covering number of $\Theta$ under local Fisher metric, then log-normalize for EED (Berezniuk et al., 2020, Banh et al., 2022).

• Intrinsic Dimension Estimation (Autoencoder):

1. Normalize data.
2. For candidate dimension $d$, project onto $d$ PCA components.
3. Train a bottleneck autoencoder on the residual.
4. Compute reconstruction error and select $d^*$ at the “knee point” (Δ(MRSE) below threshold); return $d^*$ as EED (Kärkkäinen et al., 2022).

3. Empirical Observations Across Domains

Distinct empirical phenomena manifest in EED analyses:

• Vision Transformers:

Object-centric datasets (TinyImageNet, CIFAR-100) show a pronounced U-shaped EED profile: high EED% in early layers, low mid-layer bottleneck (min EED% ≈23–31%), and strong re-expansion before the head; texture-centric datasets (UC Merced) maintain high EED% throughout (≈95%), with no bottleneck (Awadhiya, 8 Dec 2025).

Dataset	Compositional Type	Min EED% (mid-layers)
CIFAR-100	Object-centric (high)	≈23%
TinyImageNet	Object-centric (med)	≈30.5%
UC Merced	Texture-centric	≈95% (no bottleneck)

• Hyperdimensional Classification:

Dynamic encoding (DistHD) reduces physical dimension $D$ required for target accuracy by up to 8× relative to static HDC; misleading dimensions are iteratively regenerated, converging typically in 5–10 iterations (Wang et al., 2023).

• Statistical Models:

EED tracks only directions with Fisher eigenvalues above noise threshold ($1/n$); dimensionality converges to ambient $d$ only for very large $n$ (slow in models with highly non-uniform Fisher spectra) (Berezniuk et al., 2020).

• Autoencoder-Based Estimation:

Shallow autoencoders suffice to detect the “knee point” in MRSE curves; deep architectures further reduce error but do not alter EED estimates (Kärkkäinen et al., 2022).

4. Interpretations and Theoretical Implications

EED encapsulates several functional roles:

• Information Bottlenecks:

EED quantifies information-theoretic bottlenecks, e.g. in ViTs, the mid-layer compression acts to isolate semantic features, modulating $I(T;X)$ and tightening generalization bounds according to $$\epsilon_{\text{gen}} \propto \sqrt{N_{\text{eff}}/M}$$ (Awadhiya, 8 Dec 2025).

• Model Complexity and Compression:

EED determines the description length for encoding parameters at given resolution or sample size, sharpening model complexity bounds and rationalizing overparameterization effects (Berezniuk et al., 2020).

• Algorithmic Design:

In HDC, EED motivates dynamic dimension adaptation via error-driven detection and replacement of misleading components, directly optimizing accuracy-to-dimension trade-offs (Wang et al., 2023).

• Dimensionality Reduction:

Effective rank reduction via encoding maps or SVD decompositions enables cubic-time computational savings with controlled approximation error, with EED as the quantifier of retained representational capacity (Banh et al., 2022).

5. Domain-Specific Applications

• Vision Transformers:

EED profiles diagnose emergent representational hierarchies, guide architectural choices (e.g., redundancy of explicit bottleneck stages), and inform training strategies for dense vs. semantic tasks (Awadhiya, 8 Dec 2025).

• Hyperdimensional Computing:

EED under DistHD offers an adaptive criterion for minimal dimension needed for desired classification accuracy, yielding substantial compute/memory savings and robustness to distributional shifts (Wang et al., 2023).

• Statistical and Linear Models:

EED-driven subspace selection accelerates linear mixed model inference (e.g., heritability estimation) and mixture model clustering, with empirically validated trade-offs between runtime and estimation error (Banh et al., 2022).

• Dimension Estimation with Autoencoders:

Additive pipelines combining PCA and autoencoders implement scalable EED estimation for arbitrary datasets; the minimal dimension $d^*$ yields a direct estimate of intrinsic complexity (Kärkkäinen et al., 2022).

6. Practical Guidelines and Selection Criteria

Selection of EED is governed by the balance between approximation fidelity and computational efficiency:

• Begin with moderate reduction exponents ($\gamma \approx 0.7$–$0.8$); empirically validate fit-loss.
• In HDC, select initial $D$ conservatively, run iterative regeneration to plateau accuracy, increase $D$ if target not met; intersection size $|U|$ of regenerated dims signals proximity to true EED (Wang et al., 2023).
• For mixed models, encode down to $m \approx n^{0.5}$–$n^{0.8}$, $r \approx p^{0.7}$; monitor fit by cross-validation or specific metrics (e.g., BIC, clustering accuracy) (Banh et al., 2022).
• Autoencoder pipelines require tuning of bottleneck dimension $d$ to thresholded MRSE improvement $\tau$ (typically $3 \times 10^{-3}$–$4 \times 10^{-3}$); shallow architectures suffice for robust EED detection (Kärkkäinen et al., 2022).

7. Conceptual Extensions and Research Directions

Recent studies propose:

• Use of spectral pruning or staged compression as inductive bias during network training.
• Extending EED analysis to large-scale models, dense prediction tasks, and causal interventions on the bottleneck structure.
• Adopting dynamic EED-attainment cycles for evolving data and shifting distributions, particularly for memory-constrained or real-time learning systems (Awadhiya, 8 Dec 2025, Wang et al., 2023).

EED thus unifies statistical, algorithmic, and representational perspectives—serving as a core metric for model reduction, adaptive encoding, and data-driven architectural analysis across contemporary machine learning domains.