Dimension-Insensitive Sample Complexity

• Dimension-insensitive sample complexity is defined as a regime where the number of samples required depends primarily on intrinsic properties like sparsity or low rank, rather than the ambient dimension.
• It leverages combinatorial and statistical mechanics frameworks, yielding bounds that remain nearly invariant to the number of extraneous features.
• Applications in dictionary learning, sparse recovery, and private learning illustrate phase transitions and robust efficiency in high-dimensional settings.

Dimension-insensitive sample complexity refers to regimes in statistical learning, optimization, or data analysis where the number of samples required for a desired accuracy depends minimally—or not at all—on the ambient dimension of the data. This stands in contrast to classical “curse of dimensionality” phenomena, where sample complexity grows polynomially or exponentially with the number of features or parameters. Dimension-insensitive bounds emerge when complexity is controlled by intrinsic properties such as sparsity, low effective rank, covering numbers, or combinatorial measures that do not scale with extraneous coordinates. Various lines of work rigorously characterize, achieve, or exploit such bounds in both theory and algorithm design across disparate domains.

1. Foundational Principles and Definitions

Dimension-insensitive sample complexity formalizes settings where the minimum number of examples or queries required to recover underlying structure, learn accurate models, or test properties is independent of the ambient dimension $d$, or at most only weakly (e.g., polylogarithmically) dependent. Formally, consider learning or recovery mappings $\mathcal{L}$ from data $Y \in \mathbb{R}^{M \times P}$, generated by a structured latent process (such as $Y = (1/\sqrt{N}) D X$, with sparse $X$ and dictionary $D \in \mathbb{R}^{M\times N}$), or evaluating risk of estimators over a high-dimensional simplex or ball.

A prototypical definition involves specifying the sample complexity $n^*(\epsilon, d, \ldots)$ required to achieve error $\leq \epsilon$ as

$$n^*(\epsilon, d, \ldots) = \mathcal{O}(f(\epsilon, \text{intrinsic parameters})) + g(d)$$

where $g(d)$ is subdominant, for example $g(d) = \log d$ or vanishes entirely. The primary complexity driver is then not dimensionality but a problem-specific combinatorial or geometric parameter, such as sparsity $\rho$, intrinsic rank $k$, fat-shattering dimension, or metric entropy measured in a dual norm.

2. Bayesian Optimal Dictionary Learning: Replicated Statistical Mechanics Approach

A canonical instance is dictionary learning with observations $Y = (1/\sqrt{N}) D X$, where the goal is to identify both a planted dictionary $D$ and a sparse coefficient matrix $X$ generated independently from specified distributions. In this setting, the sample complexity $P_c$, defined as the critical number of training examples needed for perfect recovery (zero mean-squared error), is shown via the replica method of statistical mechanics to scale linearly with the dictionary size $N$ as long as the compression rate $\alpha = M/N$ exceeds the sparsity $\rho$: $P_c = N \cdot \frac{\alpha}{\alpha - \rho}$ This result ((Sakata et al., 2013) Eq. (2)) demonstrates that the number of samples for correct model identification is $O(N)$, i.e., dimension-insensitive with respect to other structural parameters, so long as the generative conditions are met. The analysis distinguishes between “success,” “failure,” and “middle” solutions in the space of posterior distributions, with the replica symmetric ansatz leading to an entropy collapse uniquely at the planted solution for $\alpha > \alpha_M(\rho)$.

The macroscopic order parameters (e.g., $$q_D = \frac{1}{MN} [\langle D \rangle_\theta \cdot \langle D \rangle_\theta]_Y$$) are computed via saddle-point equations arising from the statistical mechanics formulation, and the derivation provides phase diagrams identifying condensed posterior phases conducive to recovery by belief propagation algorithms—whose computational complexity is polynomial in $N$ and thus compatible with high-dimensional scaling.

3. Combinatorial and Information-Theoretic Frameworks

Dimension-insensitive complexity can often be attributed to surrogate combinatorial measures or intrinsic structures:

• Probabilistic Representation Dimension (RepDim): In private learning under differential privacy constraints, the sample complexity is determined not by the hypothesis class’s VC-dimension, but by the minimal size of a probabilistic representation (RepDim), which may be constant even when VC-dimension is large or unbounded (Beimel et al., 2014). This combinatorial quantity governs not only private learning, but also private data release and optimizations.
• Density Dimension in Semimetric Spaces: For classification in semimetric spaces where the triangle inequality fails, the sample complexity is dictated by the density dimension log₂μ(𝒳) rather than the doubling dimension. Efficient sample compression schemes based on nets (of cardinality $$k \approx (2 \cdot \text{rad}(S)/\gamma)^{O(\text{dens}(S))}$$) yield error guarantees in terms of $k$, enabling “dimension-insensitive” learning even when conventional geometrical arguments break down (Gottlieb et al., 2015).
• Fat-shattering Dimension and Metric Entropy: In property testing and outcome indistinguishability frameworks, characterizations are established in terms of fat-shattering dimension or metric entropy with respect to dual Minkowski norms. For instance, in outcome indistinguishability, sample complexity is completely characterized by the covering number $N_{\mu, \mathcal{D}}(\mathcal{P}, \epsilon)$, independent of the ambient input space, and duality relations further establish tight bounds (Hu et al., 2022).

4. Algorithms and Statistical Models Enabling Dimension-Insensitive Bounds

Several algorithmic and model-based strategies explicitly target or exploit dimension-insensitive sample complexity:

• Fourier and Sparse Recovery: Optimal sparse recovery in the Fourier domain can be achieved with $O(k \log N)$ samples in any constant dimension (Indyk et al., 2014), by leveraging hash-and-filter procedures that reuse permutations and exploit the underlying sparsity, thereby avoiding exponential scaling in $d$.
• Intrinsic/Regularized Metric Learning: In metric learning, naive uniform convergence bounds scale with representation dimension $D$, but by introducing regularization (e.g., penalizing the Frobenius norm of $M^\top M$), sample complexity can be tuned to the intrinsic complexity $d_M = \|M^\top M\|_F^2$, if $d_M \ll D$ (Verma et al., 2015). This enables robust generalization even with many uninformative features.
• Dimension-Insensitive Zeroth-Order Optimization: For stochastic zeroth-order convex optimization, sparsity-inducing projection steps combined with weak $\ell_1$-norm assumptions $\|x^*\|_1 \leq R$ yield query complexity $O((D_0 + R)^3 L \sigma^2 \log d / \epsilon^3)$, preserving dimension-insensitivity except for a logarithmic term, and crucially without requiring gradient sparsity (Liu et al., 2021).
• Private Learning of Axis-Aligned Rectangles: Algorithms such as RandMargins achieve sample complexity $O(d \cdot (\log^*|X|)^{1.5})$ for axis-aligned rectangle learning under differential privacy, via novel deletion strategies for “exposed” data points that circumvent composition cost accumulation (Sadigurschi et al., 2021).
• Learning High-Dimensional Simplices with Noise: Recent work shows that for signal-to-noise ratio SNR at least $\Omega(K^{1/2})$, learning a K-dimensional simplex subject to Gaussian noise is possible with $n \geq K^2/ \epsilon^2$ samples, matching the noiseless regime and confirming dimension-insensitive sample complexity in favorable SNR conditions (Saberi et al., 2022).

5. Regimes and Phase Transitions in Sample Complexity Scaling

Several lines of work identify critical thresholds or “phase transitions” delineating dimension-insensitive from dimension-dependent regimes:

• Lossy Population Recovery: The sample complexity of estimating a high-dimensional binary distribution from erasure-corrupted samples exhibits two distinct regimes. For erasure $\epsilon \leq 1/2$, sample complexity is $\Theta(1/\delta^2)$ independent of $d$. When $\epsilon > 1/2$, the problem becomes nonparametric and complexity increases superpolynomially with $1/\delta$, but even then remains only logarithmically dependent on $d$ (Polyanskiy et al., 2017).
• Empirical Risk Minimization in Convex Optimization: In classical (Lipschitz) stochastic convex optimization in the $\ell_2$ setting, ERMs require $n = \tilde{O}(d/\epsilon + 1/\epsilon^2)$ samples, establishing an inescapable linear term in $d$, but separating the dimension-dependent (realizable) and dimension-insensitive (agnostic/statistic-limited) regimes (Carmon et al., 2023).

6. Practical Implications, Applications, and Limitations

Dimension-insensitive sample complexity results have substantial impact across areas:

• Private and Robust Learning: They provide the foundation for efficient private algorithms, multi-group learning, and robust optimization—ensuring feasible sample sizes in high-dimensional regimes for practical tasks such as fairness, federated learning, and data release under strong privacy constraints (Ghazi et al., 2020, Peng, 2023).
• Signal Processing, Quantum State Testing: In sparse Fourier recovery and quantum property testing, optimal bounds eliminate ambient dimension as a bottleneck, using tailored estimators and symmetry-exploiting frameworks (Indyk et al., 2014, Fanizza et al., 2021).
• Geometric Learning and Coreset Construction: The sensitivity sampling framework for classification coresets can yield subset selection for loss approximation with coreset size independent of dimension (Alishahi et al., 7 Feb 2024), further reinforcing algorithmic tractability in high-dimensional settings.

Limitations persist: all favorable results rely critically on structural assumptions—sparsity, bounded variation, regularity, or strong prior knowledge. In agnostic or adversarial settings absent such structure, dimensionality often cannot be avoided. Additionally, practical performance depends on algorithmic constants and computational cost, which may still scale with dimension even if sample complexity does not.

7. Connections to Broader Theory and Future Directions

The paper of dimension-insensitive complexity intersects information theory, combinatorics, and geometry. Metric entropy duality (Hu et al., 2022) links sample complexity of learning and distinguishing models to convex geometric conjectures. Ongoing research probes sharper characterizations in private learning, optimization under weaker assumptions, and algorithmic frameworks for further reducing or exploiting dimension dependence.

Open problems include tightening polynomial factors in “almost” dimension-insensitive bounds, developing more universal regularizers that induce intrinsic complexity adaptation, and extending results to non-linear, non-i.i.d., or interactive settings. There is active interest in fully delineating the trade-offs between stability, capacity control, and computational efficiency needed to achieve dimension-insensitive statistical guarantees.