Differentially Private Model-X Knockoffs via Johnson-Lindenstrauss Transform: what it means for business leaders

Introduces a Johnson–Lindenstrauss-based pipeline for differentially private Model-X knockoffs, preserving geometry for stable selection while controlling false discoveries—letting regulated teams perform high-dimensional, explainable modeling without crippling utility or messy noise injection.

1. What the method is

A Johnson–Lindenstrauss (JL) random projection is used to privatize the design matrix before feature selection, so distances and second-order structure are largely preserved under (ε,δ)-differential privacy. On this privatized representation, the workflow constructs Model-X knockoffs and computes knockoff statistics using a debiased private Lasso, then applies the standard data-dependent threshold to select variables with false discovery rate (FDR) control. Because privacy is enforced at the input level, anything done afterward is “free” by DP post-processing immunity: analysts can re-fit models or re-run knockoffs without spending more privacy budget. The JL step also yields a positive semidefinite surrogate covariance, avoiding numerical pathologies from naively noising moments. In short, the method produces a reusable private data product that enables high-dimensional, interpretable selection with far less utility loss than classic output-noise mechanisms, while keeping the knockoff symmetry properties needed for valid FDR guarantees.

2. Why the method was developed

Many organizations must comply with strict privacy rules yet still explain “which variables matter.” Existing differentially private approaches often add noise to sufficient statistics or model outputs, which can break exchangeability, yield non-PSD covariances, and crater statistical power—especially in sparse, high-dimensional regimes. Model-X knockoffs offer rigorous FDR control, but naive privacy layers disrupt the geometric assumptions they rely on. The authors sought a design that preserves the essential geometry for knockoffs while meeting DP guarantees and remaining practical to deploy. By privatizing inputs via JL projections rather than noising outputs, they stabilize optimization, retain more signal, and let teams reuse a single privatized matrix across multiple downstream analyses. The goal is dependable, audit-friendly selection that satisfies privacy budgets without resorting to brittle engineering workarounds or accepting large drops in discovery power.

3. Who should care

Heads of data science in regulated sectors; biostatistics groups running genome-wide or claims analyses; risk and fraud teams building sparse, explainable models; and platform owners operating privacy gateways for sensitive telemetry. Product and compliance leaders gain clearer, attestable guarantees around both privacy and FDR, which simplifies reviews and stakeholder communication. MLOps teams benefit from the architecture: a one-time privatized projection can be shared as a sanctioned data asset, enabling repeated model fitting and variable selection without additional privacy accounting. Researchers building causal screens or feature ranking tools can adopt the pipeline to keep selection stable under tight privacy budgets, reducing the need for ad-hoc tuning while maintaining interpretability for decision makers and regulators.

4. How the method works

The raw design matrix is left-multiplied by a Gaussian projection R sized to deliver (ε,δ)-DP and preserve pairwise geometry with high probability. The projected second moment (RA)^⊤RA is PSD and well-conditioned for optimization. Model-X knockoffs are then built on the privatized inputs, and feature statistics are computed via a debiased private Lasso so that knockoff symmetry holds. Signed statistics W_j feed the knockoff filter, which selects variables at a data-dependent threshold controlling FDR at target level q. Theory characterizes power/FDR as functions of projection dimension, sparsity, SNR, samples, and privacy parameters; guidance is provided for choosing R and accounting for privacy once, after which all downstream steps are post-processing. In practice, the pipeline plugs into standard sparse-regression tooling with minimal code changes and predictable privacy-utility trade-offs.

5. How it was evaluated

The paper combines formal analysis with simulations and an applied case study. Asymptotic results for the debiased private Lasso under JL projections yield explicit predictions for selection power and FDR. Simulations sweep privacy budgets, projection sizes, sparsity levels, and signal strengths to compare JL-privatized knockoffs against a Gaussian-mechanism baseline that perturbs second moments directly. Robustness checks examine numerical stability (PSD preservation), selection reliability, and sensitivity to mis-specification. A real-data example illustrates end-to-end practicality, documenting privacy accounting, model fitting on the privatized matrix, and knockoff filtering outcomes, with emphasis on how results change as the projection dimension varies relative to samples and features.

6. How it performed

Empirically, the JL-based approach maintained target FDR and delivered higher discovery power than moment-noising baselines across representative (ε,δ) budgets and projection ratios. Positive-semidefinite surrogates improved optimization stability, producing cleaner knockoff statistics and more reproducible selections. Theory aligned with experiments: as projection dimension and sample size grow, power recovers toward non-private performance and, under stated conditions, can approach one. Operationally, privatizing once and reusing the projected data reduced privacy-accounting overhead and simplified pipelines. For leaders, the takeaway is a viable route to privacy-respecting, explainable feature selection that preserves statistical muscle and implementation simplicity. (Source: arXiv 2508.04800, 2025)