High-Dimensional Privacy Characterizations

• High-dimensional privacy characterizations are rigorous frameworks that define how privacy can be quantified and preserved when working with large, complex data spaces.
• They employ techniques such as differential privacy, local DP, and divergence-based methods, alongside dimensionality reduction strategies like DP-PCA to mitigate the curse of dimensionality.
• These methods inform privacy–utility trade-offs, optimizing sample complexities and robust statistical estimation in machine learning, generative modeling, and high-dimensional inference.

High-dimensional privacy characterizations encompass mathematical principles, algorithmic techniques, and rigorous performance analyses that describe how privacy can be preserved or quantified when data and statistical tasks inhabit spaces of very large dimension. This area is pivotal for statistical inference, machine learning, generative modeling, interactive data collection, and other applications where both privacy risk and statistical utility are strongly dimension-dependent. The modern literature develops a spectrum of characterizations, ranging from differential privacy for high-dimensional statistics and sparse bandits, to privacy for text and manifold data, to topological and geometric perspectives.

1. Fundamental Privacy Definitions in High Dimensions

High-dimensional privacy mechanisms formalize guarantees either through classical differential privacy (DP), metric extensions, or more advanced divergence-based relaxations. Key formulations include:

• $(\varepsilon,\delta)$-Differential Privacy: For data matrices $\mathbf X, \mathbf X' \in \mathbb{R}^{n \times d}$ differing in one row, a mechanism $\mathcal{M}$ is $(\varepsilon,\delta)$-DP if, for all measurable $S$,

$$\Pr[\mathcal{M}(\mathbf X) \in S] \le e^\varepsilon \Pr[\mathcal{M}(\mathbf X') \in S] + \delta.$$

• Local Differential Privacy (LDP) and Sensitivity: For high-dimensional queries $f: \mathcal{X} \to \mathbb{R}^d$, the required noise scale is dictated by the $\ell_1$ or $\ell_2$ sensitivity, which can scale with $\sqrt{d}$ or $d$, underlying the "curse of dimensionality" (Mansbridge et al., 2020).
• $d_X$-Privacy and Geo-Privacy: For data in a metric space $(\mathcal{D}, d)$, mechanisms satisfy $\varepsilon\,d$-privacy if

$$\Pr[M(x)\in S] \leq e^{\varepsilon d(x,y)} \Pr[M(y)\in S].$$

Concentrated Geo-Privacy (CGP) (Liang et al., 2023) and concentrated differential privacy (CDP) generalize these via Rényi divergences.

• Rényi and Gaussian DP (GDP): These divergences enable sharp high-dimensional compositions and facilitate privacy accounting for mechanisms like the exponential mechanism or iterative gradient methods (Yun et al., 10 Nov 2025).

2. Curse of Dimensionality and Sensitivity Reduction

High dimensionality fundamentally amplifies the requisite noise for privacy unless additional structure is exploited:

• Curse in Anonymization: For $k$-anonymity and its variants, the minimum required generalization and resulting information loss grow rapidly with $d$ (the dimension of quasi-identifiers), leading to nearly complete utility loss in the worst case (Zakerzadeh et al., 2014).
• Sensitivity in High-Dimensional Queries: For principal component analysis (PCA) or linear regression, the global sensitivity of spectral or regression queries grows with $d$, making naive privatization infeasible (Yun et al., 10 Nov 2025, Sang et al., 3 Jun 2025).
• Recursive Preconditioning: For high-dimensional learning (e.g., multivariate Gaussian estimation), recursive preconditioning successively reduces the condition number of covariance matrices via private weak estimators, enabling efficient privatization at the cost determined by "average" rather than "worst-case" sensitivity (Kamath et al., 2018).
• Dimensionality Reduction via DP-PCA: In generative modeling, reducing the data to a lower-dimensional subspace using a private PCA (with Gaussian mechanism noise on the covariance) confines the privacy cost to the reduced dimension, rather than $d$ (Takagi et al., 2020).

3. High-Dimensional Algorithms and Architectures

Modern high-dimensional privacy mechanisms combine structural exploitation and modular privacy design:

Algorithm or Framework	Key Components	Dimensionality Handling
P3GM (Phased Generative) (Takagi et al., 2020)	Phase I: DP-PCA + DP-EM; Phase II: DP-SGD	Noise and parameter count confined to $d' \ll d$
HPTR (High-dim PTR) (Liu et al., 2021)	Propose-Test-Release, Exponential Mech, Resilience	Sensitivity reduced to robust 1D stats
Vertical Fragmentation (Zakerzadeh et al., 2014)	MI-based fragmentation, standalone anonymization	Splits $d$ into fragments of small effective dim
FLIPHAT / PrivateLASSO (Bandits) (Chakraborty et al., 22 May 2024, Shukla, 6 Feb 2024)	Sparse private regression (N-IHT), support recovery	Privacy cost scales with sparsity $s^* \ll d$
Representation LDP (Mansbridge et al., 2020)	Learned representation + Laplace on code	Manifold dimension $k \ll d$ sets noise scale
Differentially Private PCA (Yun et al., 10 Nov 2025)	Exponential mechanism, spectral analysis	Sharp privacy tuned to true spectral geometry

These mechanisms often use modular approaches, e.g., post-processing invariance for privatized eigenvalues or compositional accounting by Rényi DP.

4. Privacy–Utility Trade-Offs and Sample Complexities

Sample complexity and statistical efficiency hinge on the structure exploited to reduce the effective noise per query:

• Gaussian and Product Distribution Estimation: The sample complexity for DP learning of Gaussians scales as $\tilde O(d^2/\alpha^2 + d^{3/2}/\epsilon)$, nearly matching the non-private rate up to lower order (Kamath et al., 2018).
• Regression, Covariance, PCA: With robust, trimmed estimators and Propose-Test-Release, mean and regression tasks achieve $\tilde O(d/\xi^2 + d/(\xi\epsilon))$ and covariance estimation $\tilde O(d^2/\xi^2 + d^2/(\xi\epsilon))$, both matching private lower bounds (Liu et al., 2021).
• Generative Modeling: P3GM achieves total-variation distance $$D_{\rm TV}(\widehat p,p^*) \leq \sqrt{\tfrac12 \mathrm{KL}}$$ vanishing as $n \gg d'/\epsilon$ or $n \gg d'/\epsilon^2$, confining privacy cost to reduced $d'$ (Takagi et al., 2020).
• Bandits and Online Learning: In sparse linear bandits under joint DP, the minimal regret grows as $$\Omega(\max\{s^*\sqrt{T\log (d/s^*)},\,\sqrt{s^*}\log(d/s^*)/\epsilon\})$$; FLIPHAT matches this up to logs (Chakraborty et al., 22 May 2024).
• Text and Embeddings: For $d_X$-privacy on word embeddings, the typical Laplace noise required in high dimensions is so large relative to word similarity gaps that only very dissimilar outputs are likely, unless further semantic postprocessing is applied (Asghar et al., 21 Nov 2024, Feyisetan et al., 2019).

5. Statistical and Geometric Characterizations

Structural analysis underpins the sharpness and feasibility of high-dimensional privacy:

• Resilience and Robustness: If the data distribution is resilient (robust to small, localized corruption), then 1D robust statistics (trimmed means/variances) have low local sensitivity, and combine with PTR to yield optimal DP mechanisms (Liu et al., 2021).
• Geometric Tools: Steiner-point stability (Ben-Eliezer et al., 2022), private projection oracles, and convex floating bodies yield tight, robust quantile estimation under minimal assumptions. For $d$-dimensional context, the error scales polynomially with $d$, in contrast to exponential blow-up in worst-case settings.
• Spectral and Contiguity Arguments: In high-dimensional DP PCA, sharp privacy–utility trade-offs are established using spectral gap, Hilbert transform, and Le Cam’s contiguity, identifying exactly when privacy loss achieves a Gaussian limit—finer than worst-case bounds suggest (Yun et al., 10 Nov 2025).
• Topological Approaches: Lattice and simplicial complex methods (Dowker complexes and Galois lattices) formalize privacy as the absence of “free faces” and identify settings where “holes” or high-dimensional Betti numbers delay or even prevent exact victim identification (Erdmann, 2017).

6. Practical Implications and High-Dimensional Phenomena

• Dimension Reduction is Essential: Private learning and inference in high-dimensional spaces is viable only if one can exploit concentration, sparsity, manifold or spectral structure, or effective low-dimensional summaries.
• Advanced Composition: Concentrated privacy notions (CDP, CGP) yield improved error scaling (noise $\sim \sqrt{d}$, error $\sim \sqrt{k}$ under $k$ queries), with full support for advanced and adaptive composition (Liang et al., 2023).
• Empirical Performance: P3GM achieves $>90\%$ of non-private accuracy under tight $(\varepsilon,\delta)$ budget in $d\geq 784$; representation LDP mechanisms achieve $4$–$7\times$ boosts in classification accuracy on vision/text benchmarks over classical (uncorrelated) Laplace (Takagi et al., 2020, Mansbridge et al., 2020). In unsupervised settings, vertical fragmentation reduces information loss by over $70\%$ for $d\sim 40$ (Zakerzadeh et al., 2014).

7. Limitations, Phase Transitions, and Extensions

• Pathologies: If the data exhibit no correlation or manifold structure, or have heavy tails, private estimation reverts to the worst-case, with effective sample size and noise growing with $d$.
• Phase Transitions: For bandits and online learning, there is a regime ($\epsilon$ small) where privacy cost dominates, but as $\epsilon$ increases, non-private rates are recovered (Chakraborty et al., 22 May 2024).
• Extensibility: Modern approaches (e.g., HPTR, spectral contiguity) are adaptable to a wide class of statistical models, including GLMs, robust M-estimation, and complex latent variable models. The geometric and topological tools generalize to privacy in relational, graph-structured, and functional data.

References to Key Papers

• "Testing for large-dimensional covariance matrix under differential privacy" (Sang et al., 3 Jun 2025)
• "High-Dimensional Privacy-Utility Dynamics of Noisy Stochastic Gradient Descent on Least Squares" (Lin et al., 19 Oct 2025)
• "High-Dimensional Asymptotics of Differentially Private PCA" (Yun et al., 10 Nov 2025)
• "Concentrated Geo-Privacy" (Liang et al., 2023)
• "Towards Breaking the Curse of Dimensionality for High-Dimensional Privacy: An Extended Version" (Zakerzadeh et al., 2014)
• "FLIPHAT: Joint Differential Privacy for High Dimensional Sparse Linear Bandits" (Chakraborty et al., 22 May 2024)
• "$d_X$-Privacy for Text and the Curse of Dimensionality" (Asghar et al., 21 Nov 2024)
• "Leveraging Hierarchical Representations for Preserving Privacy and Utility in Text" (Feyisetan et al., 2019)
• "Privately Learning High-Dimensional Distributions" (Kamath et al., 2018)
• "Representation Learning for High-Dimensional Data Collection under Local Differential Privacy" (Mansbridge et al., 2020)
• "Differential privacy and robust statistics in high dimensions" (Liu et al., 2021)
• "Archimedes Meets Privacy: On Privately Estimating Quantiles in High Dimensions Under Minimal Assumptions" (Ben-Eliezer et al., 2022)
• "Topology of Privacy: Lattice Structures and Information Bubbles for Inference and Obfuscation" (Erdmann, 2017)

High-dimensional privacy characterizations, as synthesized above, enable rigorous and sharp analysis of privacy risk and data utility—while clarifying the necessary and sufficient structures that make strong privacy feasible in modern, complex data regimes.