Gaussian Random Hyperplanes

• Gaussian random hyperplanes are codimension-one linear subspaces drawn from a standard Gaussian distribution that encode geometry via one-bit comparisons.
• They underpin uniform tessellations and δ-isometric binary embeddings, supporting applications in compressed sensing and locality-sensitive hashing.
• The hyperplane count is governed by the Gaussian mean width, ensuring optimal scaling with error δ for robust high-dimensional signal recovery.

A Gaussian random hyperplane is a codimension-one linear subspace in ℝⁿ specified by an independent direction drawn from the standard Gaussian distribution. Collections of such hyperplanes are a central object in high-dimensional geometry, random matrix theory, geometric functional analysis, and discrete dimension reduction. Their utility arises from their rotational invariance, covering properties, and strong concentration phenomena. A principal application is the construction of binary embeddings or tessellations: the data is mapped by the signs of its inner products with Gaussian directions, encoding geometry via one-bit comparisons. The theory attaches critical importance to set complexity measures such as mean width or Gaussian complexity, which determine how many random hyperplanes are required for various uniform covering, dimension reduction, and recovery properties.

1. Uniform Tessellations and Isometric Binary Embeddings

The fundamental operation is the uniform tessellation of a bounded set K ⊆ Sⁿ⁻¹ via m independent Gaussian hyperplanes. Namely, for a random matrix A ∈ ℝ^{m×n} with rows gᵢ ~ N(0, Iₙ), the sign map

$f(x) = \operatorname{sign}(A x) \in \{-1, +1\}^m$

embeds K into the Hamming cube. The normalized Hamming distance d_H(f(x), f(y)) represents the proportion of hyperplanes separating x and y. Uniform tessellation holds if this fraction closely matches the geodesic distance d(x, y) up to a small error δ:

$$|d_A(x, y) - d(x, y)| \leq \delta \quad \forall x, y \in K.$$

Under this property, the embedding f is a δ-isometric embedding (Gromov–Hausdorff sense) of K into the Hamming cube. The method exploits the strong concentration of the empirical separation fractions around their expectation, inherited from Gaussian measure symmetries and anti-concentration properties (Plan et al., 2011).

2. Gaussian Mean Width and Complexity-Driven Hyperplane Counts

The number m of hyperplanes required for δ-uniform tessellation is governed by the Gaussian mean width

$$w(K) = \mathbb{E}_{g \sim N(0, I_n)} \sup_{x \in K} |\langle g, x\rangle|.$$

For many structured sets, w(K) ≪ \sqrt{n}, yielding an efficient reduction. The main result is that

$m \gtrsim \delta^{-6} w(K)^2$

hyperplanes suffice to δ-tessellate K with high probability. In the case of finite point sets, w(K) ≤ C\sqrt{\log|K|}, so m = O(\log|K|) achieves JL-type compression. This order is ensured to be sharp in many regimes via Sudakov's minoration and covering number bounds

$$\log_2 N(K, \delta) \leq m(K, \delta) \leq C \delta^{-6} w(K)^2,$$

where N(K, δ) is the minimal δ-covering number (Plan et al., 2011).

The hyperplane approach thus offers a discrete, one-bit alternative to the Johnson-Lindenstrauss embedding, with comparable complexity for many K and added efficiency for highly structured or sparse sets.

3. Embedding Quality, Comparison to Linear Maps, and Robustness

Unlike linear JL embeddings into $\mathbb{R}^m$, the hyperplane sign map is not Lipschitz, a reflection of the discontinuity of the $\operatorname{sign}$ function. As a result, the embedding is δ-isometric only in the Gromov–Hausdorff sense. Nevertheless, the local geometric properties (separation probabilities, minimal Hamming distortion) are preserved due to large deviation inequalities and high-probability uniform control over all pairs via covering arguments.

Classical JL-type linear projections require $m \sim \delta^{-2} \log |K|$ for finite sets, while the hyperplane method achieves comparable rates for finite K, but with distinct phase transitions in the error exponents when extended to general (non-finite) sets. Curvature arguments allow improvements in the dependence on δ in some geometric settings, lowering the exponent from δ⁻⁶ to δ⁻⁴ (Plan et al., 2011).

4. Applications: One-Bit Compressed Sensing, Locality-Sensitive Hashing, and Binary Representations

The binary embedding by Gaussian hyperplanes underpins one-bit compressed sensing: signal measurements are quantized to {–1, 1} by taking sign ⟨g, x⟩. Uniform tessellation with geometric control guarantees that distinct signals are not confounded by quantization, enabling recovery up to precision δ. In localities where K is compressible or lies on a low-dimensional manifold, mean width is small, enabling highly compressed, efficient, and robust measurement schemes.

The same machinery applies to locality-sensitive hashing for fast approximate nearest neighbor search; the sign pattern provides a Hamming code summarizing geometry so that nearby points on the sphere are mapped to similar codewords, controlling false positive and false negative rates under binary query (Plan et al., 2011).

5. Analytical Guarantees, Probabilistic Bounds, and Experimental Validation

The approach is underpinned by precise probabilistic guarantees: with high probability (at least $1-2\exp(-c\delta^2 m)$), uniform tessellation holds for all pairs. The analysis employs a combination of chaining, covering nets, small-ball estimates, and sharp concentration of empirical averages of Bernoulli random variables induced by the Gaussian hyperplane signs.

The method "softens" the discontinuity of the sign metric by controlling not just the hard Hamming distance but also its concentration over ε-nets of K. In finite K, the approach recovers JL bounds up to constants; in empirical settings (e.g., sparse signals), performance matches theoretical predictions, as confirmed by simulation and prior results in one-bit signal processing.

6. Extensions, Limitations, and Optimality

While the δ-exponent δ–⁶ is not always optimal, it cannot be universally improved for arbitrary sets due to lower bounds arising from covering entropy. In situations with favorable geometric properties, further refined analysis may lower the exponent, but the quadratic dependence on w(K)² is an optimal feature for a wide class of sets. Additionally, the randomized construction is robust to various forms of perturbations: using rows from Haar-uniformly random orthogonal ensembles in place of Gaussians is valid due to concentration of measure.

The sharpness of the mean width dependence, connection to Sudakov’s inequality, and competitive performance in the discrete binary regime make Gaussian random hyperplanes a foundational tool in contemporary high-dimensional discrete geometry, compression, and statistical signal processing (Plan et al., 2011).

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Summary Table: Key Parameters and Results for Gaussian Random Hyperplane Tessellations

Parameter	Definition / Value	Role in Embedding Quality
m	Number of random hyperplanes	Embedding dimension, controls distortion
w(K)	Gaussian mean width of set K	Governs required hyperplane count
δ	Target uniformity/additive error	Accuracy in preserving pairwise distances
f(x)	sign(Ax), A random Gaussian (m × n) matrix	Binary embedding map
d_H(f(x), f(y))	Fraction of separating hyperplanes (Hamming distance)	Approximates geodesic/Euclidean distance
Embedding bound	$m \gtrsim \delta^{-6} w(K)^2$	Sufficient #hyperplanes for δ-uniformity

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The paper of Gaussian random hyperplanes thus constitutes the rigorous theoretical underpinning for a variety of discrete dimension reduction, binary compression, and geometric embedding methods in contemporary mathematics and data science.