Dimension-Independent Classification

• Dimension-independent classification schemes are methods that avoid exponential performance degradation in high dimensions by exploiting regularity in function spaces or kernel constructions.
• They utilize neural network approximations and kernel-based designs that maintain polynomial risk bounds and convergence rates without being affected by the ambient dimension.
• The approaches rely on strict regularity conditions and coherent data geometry, making assumptions about boundary smoothness and independence critical for scalable high-dimensional performance.

A dimension-independent classification scheme consists of statistical or algorithmic procedures for supervised classification that maintain convergence rates, risk bounds, or operational performance that do not degrade exponentially with the ambient dimension $d$. These schemes are designed to circumvent the curse of dimensionality, either by exploiting function classes with special regularity (such as $RBV^2$), by using kernel or signal constructions dependent only on Euclidean distances, or by leveraging dimension-insensitive statistical functionals. Major approaches include regularity-constrained neural network classifiers, dimension-invariant kernel signal methods, distributional discriminants with stable asymptotics, and certain distance-based rules tailored to the high-dimensional regime.

1. Function Space Foundations and Decision Boundary Regularity

A central paradigm for dimension-independent classification leverages the regularity of decision boundaries via function spaces such as $RBV^2$. Let $B_1^d\subset\mathbb R^d$ denote the unit ball. A function $f:B_1^d\to\mathbb R$ belongs to $RBV^2$ if it admits an extension $f_{ext}\in RBV^2(\mathbb R^d)$ with finite norm,

$$\|f\|_{RBV^2} = RTV^2_{B_1^d}(f) + |f(0)| + \sum_{k=1}^d |f(e_k)-f(0)| < \infty,$$

where $RTV^2_{B_1^d}(f)$ denotes a Radon-domain total variation of the second derivative. Every such $f$ admits an integral-affine representation in terms of the ReLU function $\varrho(t) = \max\{0, t\}$, a finite signed measure $\mu$ on $\mathbb S^{d-1}\times [-1,1]$, and affine components: $$f(x) = \int_{\mathbb S^{d-1}\times[-1,1]} \varrho(w^\top x - b)\, d\mu(w,b) + c^\top x + c_0, \quad x\in B_1^d.$$ For binary classification, the decision function can be formulated as a “horizon function” $$h_f(x) = \mathbf 1_{\{x_d \le f(x_1,\ldots,x_{d-1})\}}$$, whose boundary $x_d = f(x_1,\ldots,x_{d-1})$ is $RBV^2$-regular (Lerma-Pineda et al., 2024).

2. Neural and Kernel-Based Approximation Frameworks

Dimension-independent approximation is obtained by constructing classifiers that exploit specific architectural or kernel properties:

• Shallow ReLU Networks: For each $N\in\mathbb N$ and $f\in RBV^2(B_1^d)$, there exists a shallow ReLU network with two layers and width $N+2$ that uniformly approximates $f$ over $B_1^d$ with error

$$\|f - f_N\|_{L^\infty(B_1^d)} \lesssim_d \|f\|_{RBV^2} N^{-(d+3)/(2d)},$$

with all weights bounded by $5\|f\|_{RBV^2}$ (Lerma-Pineda et al., 2024). Since the exponent is uniformly bounded away from zero as $d\to\infty$, these rates break the classical exponential dependence on $d$.

• Dimension-Independent Kernel Signals: For a data set $D = \{(x^i, y^i)\}$ in $\mathbb R^d\times\mathbb R^n$, a "data signal" $u_D$ is constructed as the unique minimizer of the regularized energy

$$E_\alpha(u) = \tfrac{\alpha}{2c_d}\|(1-\Delta)^{(d+1)/4}u\|_{L^2(\mathbb R^d)}^2 + \tfrac12\sum_{i=1}^m |u(x^i) - y^i|^2,$$

where the fundamental solution is the Laplace kernel $G(x-x') = \exp(-2\pi|x-x'|)$, with no explicit $d$-dependence outside the norm. The resulting linear system $(\alpha I + M)U = M Y$ with $M_{ij} = G(x^i-x^j)$ enables the classification of queries via competitive superlevel sets, yielding robust, continuous, and dimension-insensitive boundaries (Guidotti, 2022).

3. Dimension-Independence and Learning Rates

Dimension-independence is realized by ensuring convergence rates and risk bounds remain polynomial (not exponential) in $d$:

• Neural Risk Bounds: For $m$ samples and suitable statistical regularity (tube-compatible measure of order $\alpha$), empirical risk minimization over two-hidden-layer ReLU networks yields a classifier whose misclassification risk is

$$\mathbb E_S\;\mathbb P_{X\sim\mu}\{H(\Phi_{m,S}(X))\neq h_f(X)\} \lesssim m^{-(d+3)/(3d+3)+\kappa}$$

for any fixed $\kappa>0$. As $d\to\infty$, the exponent tends to $1/3$, demonstrating the absence of the curse of dimensionality for this regularity class (Lerma-Pineda et al., 2024).

• Trace-Based Discriminants: The trace-based (T-) criterion, which classifies based on simple Euclidean distances in $\mathbb R^p$, achieves vanishing misclassification error as $p\to\infty$ (assuming signal energy is distributed across coordinates):

$$P(\Pi_1\to\Pi_2) \to \Phi(-\alpha_2\|\delta\|^2/B_p)$$

with $B_p^2$ scaling as $O(p)$ and error $P(\cdot)\to0$ as $p\to\infty$ so long as variables are weakly correlated (Li et al., 2015). No matrix inversion is required, making it robust in “large $p$, small $n$” settings.

• HP Divergence Estimators: The minimum-weight cross-match statistic consistently estimates the Henze–Penrose divergence (HP-divergence), providing a dimension-independent bound on the Bayes error rate, as both expected value and variance of the statistic are dimension-free for the null case. In application, the convergence rate and bias remain stable as $d$ increases (Sekeh et al., 2018).

4. Methodological Instantiations

Dimension-independent classification is realized through diverse methodologies:

Scheme/Rule	Operational Principle	Limiting Conditions
$RBV^2$ neural classifiers	Shallow or two-layer ReLU networks, hinge ERM	$RBV^2$-regular boundaries, tube-compatibility
Data signals (Guidotti)	Laplace kernel signal, level-set competition	Data geometry must be coherent
HP-divergence matching	Optimal weighted matching, nonparametric bound	Large $N$ computationally challenging
Trace Rule (Li & Yao)	Independence rule—Euclidean distance in $\mathbb R^p$	Fails if variables correlated

These approaches differ in statistical assumptions (e.g., signal regularity, class structure), computational burden, and sensitivity to geometric properties of the data.

5. Empirical Results and Practical Considerations

Empirical evaluations on real and simulated data illustrate the operational value of these schemes:

• The $RBV^2$ classifier achieves polynomially decaying misclassification risk without exponential dependence on $d$, under sufficient regularity (Lerma-Pineda et al., 2024).
• The data signal approach yields 98.56% test accuracy on MNIST, well above nearest-neighbor methods, provided local geometric coherence is preserved (Guidotti, 2022).
• The minimum-weight matching estimator provides accurate, stable Bayes error bounds on multiple UCI datasets, with no hyperparameter tuning (Sekeh et al., 2018).
• The trace-based rule achieves error rates that decrease with the number of variables, attaining zero training error and competitive test error (two errors out of 34 in leukemia gene-expression data with $p=7129$ genes) (Li et al., 2015).

Computationally, both network-based and kernel-based schemes exhibit cubic scaling in sample size when using direct linear algebra or matching, although approximate or local methods can mitigate this cost in high-throughput settings. No feature selection or shrinkage is required for the trace-based or kernel data signal approaches.

6. Limitations and Assumptions

Dimension-independent classification depends critically on matched regularity between the model, loss, and data distribution:

• The neural-network-based scheme requires the true boundary to be $RBV^2$-regular and the sampling measure to be tube-compatible. Violation of these leads to loss of dimension-independence (Lerma-Pineda et al., 2024).
• Data signal approaches rely on the underlying class structure being connected via Euclidean (or kernel) geometry. Non-geometrically coherent classes yield degraded boundaries (Guidotti, 2022).
• The trace-based rule is efficient only in near-independence; strong correlation across variables invalidates its assumptions (Li et al., 2015).
• The optimal weighted matching estimator is computationally heavy at large $N$ and inapplicable to multiclass settings without adaptation (Sekeh et al., 2018).

Regularization parameters (for example, $\alpha$ in kernel signal methods) require tuning to balance smoothness and fit, but excessive smoothing can erase class distinctions while under-smoothing may overfit noise.

7. Synthesis and Theoretical Significance

All dimension-independent classification schemes share a key theoretical feature: the curse of dimensionality is broken either by leveraging function/class regularity (e.g., Radon-domain TV for $RBV^2$), by exploiting kernel shapes with no explicit $d$-scaling, or by harnessing statistical functionals that are invariant/covariant under change in $d$. These frameworks guarantee that, under clear explicit conditions on regularity and data geometry, the error bounds and computational controls of the classifier are not subject to exponential growth in $d$.

A plausible implication is that for data sets where class separation is reflected in geometric or analytic regularity—rather than complex anisotropic or sparse directions—dimension-independent schemes provide scalable, theoretically sound alternatives to traditional high-dimensional discriminants. However, the precise domain of applicability must be carefully matched to the assumptions of each methodology (Lerma-Pineda et al., 2024, Guidotti, 2022, Sekeh et al., 2018, Li et al., 2015).