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Dimension reduction by random hyperplane tessellations

Yaniv Plan, Roman Vershynin

arXiv:1111.4452v2math.PRmath.FA

TL;DR

The paper asks how to obtain uniform hyperplane tessellations with as few hyperplanes as possible while preserving distances on K. It uses random hyperplanes and the associated sign map, obtaining near-optimal mean-width bounds and Hamming-cube embeddings, while the curvature argument has a stated limitation for uniform estimates.

  • Problem

    The paper seeks the smallest number of hyperplanes whose separation fractions approximate Euclidean distances uniformly over K.

  • Method

    The paper uses independent Haar-uniform random hyperplanes and maps each point to its vector of hyperplane signs.

  • Results

    With probability at least 1 − 2 exp(−cδ^2m), random hyperplanes yield a δ-uniform tessellation, and K embeds almost isometrically into a Hamming cube.

  • Takeaways & Limitations

    Uniform tessellations provide a discrete dimension-reduction mechanism by representing Euclidean subsets through Hamming-cube sign vectors.

  • Takeaways & Limitations

    The curvature argument cannot establish the two-sided estimate required by Theorem 1.2 because a uniform argument must handle conditional distributions depending on x and y.

Abstract

from arXiv · show

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map that sends x in K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R^n embeds into the Hamming cube {-1, 1}^m with a small distortion in the Gromov-Haussdorf metric. Since for many sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.

1. Introduction

The paper asks how few hyperplanes can uniformly tessellate a set while making separation fractions approximate Euclidean distances. Random hyperplanes achieve this with high probability, yielding embeddings into the Hamming cube through sign vectors.

  • The problem: The problem is to arrange the fewest affine hyperplanes so that separation fractions approximate Euclidean distances uniformly over K.A δ-uniform tessellation requires |dA(x, y) − d(x, y)| ≤ δ for all x, y ∈ K.
  • The main result: The main bound depends on K through its Gaussian mean width, with m = m(K, δ) hyperplanes needed for fixed accuracy δ.The paper identifies mean width as the global parameter controlling the estimate.
  • Random tessellations: With probability at least 1 − 2 exp(−cδ^2m), independent Haar-uniform random hyperplanes provide a δ-uniform tessellation of K.The result applies to subsets K ⊆ S^{n−1}.
  • Geometric interpretation: The resulting embedding perspective connects uniform tessellations with metric embeddings and the Gromov-Haussdorff distance between K and subsets of the Hamming cube.An ε-isometric embedding controls the corresponding Gromov-Haussdorff distance.
  • Hamming-cube embeddings: The sign map sends each point to its hyperplane-orientation vector, while normalized Hamming distance equals the fraction of separating hyperplanes.This identifies uniform tessellations with almost-isometric embeddings into {−1, 1}^m.

1.2. Almost isometry of K and the tessellation graph.

A uniform tessellation makes K almost isometric to its tessellation graph. The graph encodes cells as vertices and shared hyperplane boundaries as edges, with random tessellations requiring m proportional to δ^−6w(K)^2.

  • Almost-isometric graph representation: A δ-uniform tessellation makes K δ-isometric to its tessellation graph.The graph distance records the number of hyperplanes separating the corresponding cells.
  • Random construction: For random hyperplanes, the graph representation uses m ∼ δ^−6w(K)^2 hyperplanes.The mean width controls the required graph dimension.

1.3. Computing mean width.

Mean width provides a geometric measure governing tessellation complexity and can be interpreted through average directional widths. For finite sets, this yields logarithmic-size Hamming embeddings, while the construction differs from Lipschitz dimension reduction because Hamming distance is discontinuous.

  • Geometric interpretation: Mean width can be interpreted as a constant-factor multiple of spherical mean width, the average width of K over uniformly random directions.For origin-symmetric K, spherical mean width is twice the average directional width.
  • Geometric interpretation: w(K)^2 acts as an effective dimension that may be much smaller than the ambient algebraic dimension and remains robust under perturbations.This parallels the effective-rank concept from numerical linear algebra.
  • Finite sets: m ∼ log |K| suffices for finite K, yielding a δ-isometric embedding into {−1, 1}^m when m ≥ Cδ−6 log |K|.The bound follows from w(K) ≤ C√log |K|.
  • Finite sets: Unlike Johnson–Lindenstrauss embeddings into R^m, Lipschitz embeddings of arbitrarily close Euclidean points into the Hamming cube are generally impossible.The paper therefore uses δ-isometric rather than Lipschitz embeddings.
  • Proof considerations: For finite sets, concentration inequalities and a union bound over |K|^2 pairs prove the result directly, but this argument does not straightforwardly extend to general sets.The paper notes that discontinuity of Hamming distance complicates extending net-based estimates to nearby points.

1.5. Cells of uniform tessellations.

Uniform tessellations force every cell to have small diameter, and random hyperplanes achieve this property for spherical sets with a mean-width-dependent number of hyperplanes. The construction extends to bounded subsets of R^n through a spherical embedding, with affine hyperplanes and a scaling factor.

  • Spherical sets: Every cell of a δ-uniform tessellation has diameter at most δ.Points in the same cell are not separated, so the uniform-tessellation distance bound controls their diameter.
  • Spherical sets: m ≥ Cδ−6w(K)^2 random hyperplanes ensure all cells have diameter at most δ with probability at least 1 − exp(−cδ^2m).A direct curvature argument gives a slightly better dependence, m ∼ δ−4w(K)^2.
  • Extension to R^n: Bounded subsets of R^n can be handled by embedding them into a sphere in one higher dimension with small bi-Lipschitz distortion.This transfers the spherical tessellation results to affine hyperplanes in R^n.
  • Extension to R^n: w(K − K) captures the directional width of K more accurately than w(K), through distances between parallel supporting hyperplanes.The quantity is bounded by 2w(K).

1.7. Optimality.

The paper compares the minimal tessellation size with covering numbers and shows that random hyperplanes achieve an almost optimal decomposition governed by mean width. The dependence on distortion remains unoptimized, and the paper connects the result to prior normed-space embedding work and one-bit compressed sensing.

  • Covering-number bounds: m(K) is bounded below by log_2 N(K,δ) and above by a mean-width-based expression from the random tessellation theorem.The lower bound follows because a δ-uniform tessellation has at most 2^m cells, each of diameter at most δ.
  • Covering-number bounds: log N(K,δ) ≤ Cδ−2w(K)^2, but Sudakov’s inequality is not generally reversible, so the two bounds can have a gap.The inequalities are often close in settings where Sudakov’s bound is sharp, and up to logarithmic factors at some scale via Dudley’s inequality.
  • Optimality: m ∼ w(K)^2 yields an almost best decomposition of K by random hyperplanes, making mean width the principal complexity parameter for uniform tessellations.The lower bound applies more generally to any decomposition into 2^m pieces of diameter at most δ.
  • Limitations: The dependence of m(K,δ) on δ is not optimized and is related to the open question of optimal distortion dependence in Dvoretzky’s theorem.This is identified as an interesting open problem.
  • Connections: The work relates to prior embeddings into normed spaces and to one-bit compressed sensing, where A serves as a measurement map.The paper notes Gaussian and sub-Gaussian choices for the matrix A in related embedding results.

1.9. Related work: one-bit compressed sensing.

The paper situates uniform tessellations within one-bit compressed sensing and related dimension-reduction methods. Its approach embeds the set first into ℓ1 and then into the Hamming cube, using soft Hamming distance to overcome discontinuity barriers in general sets.

  • One-bit compressed sensing: One-bit compressed sensing recovers signals from the sign vector f(x) = sign Ax produced by m ≪ n linear measurements.Uniform tessellations connect this quantized recovery problem to preserving the geometry of sparse and compressible signal sets.
  • Prior results: Prior sparse-signal results established uniform tessellations and Hamming-cube embeddings with m ∼ δ−1s log(n/δ) or m ∼ δ−2s log(n/δ).For compressible signals, related bounds included m ∼ δ−6s log(n/s).
  • Scope: The paper does not address the algorithmic problem of efficiently recovering signals from their one-bit measurements.Its result concerns only the theoretical preservation of signal geometry.
  • Proof strategy: The proof first embeds K almost isometrically into ℓ1 and then re-embeds it into the Hamming cube.The intermediate ℓ1 embedding uses m ∼ δ−2w(K)^2, while a curvature argument gives a weaker route with m ∼ δ−4w(K)^2.
  • Proof strategy: The curvature argument cannot establish the main tessellation theorem, because extending estimates from a finite ε-net fails for discontinuous Hamming distance.This limitation motivates softening the Hamming distance, whose continuity supports the ε-net approximation step.

1.12. Notation.

The notation section defines support-based sparsity, standard ℓp spaces and Euclidean unit sets, and Gaussian distributions used throughout the paper.

  • Vector notation: For x ∈ R^n, ∥x∥0 denotes the cardinality of its support.Strictly speaking, ∥·∥0 is not a norm.
  • Norms: The spaces ℓp are represented as (R^n, ∥·∥p) for p ∈ {1, 2, ∞}.
  • Euclidean sets: The unit Euclidean ball is B^n_2 = {x ∈ R^n : ∥x∥2 ≤ 1}, while the unit sphere is S^{n−1} = {x ∈ R^n : ∥x∥2 = 1}.
  • Gaussian notation: N(0, 1) denotes the standard univariate normal distribution, and N(0, I_n) the centered multivariate normal with identity covariance.

2. Embedding into ℓ1

The paper constructs a random linear embedding of a bounded set K into ℓ1 using Gaussian measurements. Concentration controlled by Gaussian mean width yields small distortion with high probability and enables dimension reduction when w(K) ≪ n.

  • Concentration: E Z ≤ 4w(K)/√m bounds the expected deviation of the random Gaussian embedding.The proof combines symmetrization, contraction, rotational invariance, and Gaussian concentration.
  • Embedding theorem: The map embeds K linearly into ℓ1 with Gromov–Hausdorff distortion at most δ.
  • Embedding theorem: With probability at least 1 − 2 exp(−mδ^2/32), the normalized Gaussian map is a δ-isometry on K.The construction applies the concentration lemma to K − K.
  • Dimension reduction: When w(K) ≪ n, the embedding dimension satisfies m ≪ n, producing dimension reduction from n to m dimensions.
  • Dimension reduction: For finite K, the construction gives m ∼ log |K| and small Gromov–Hausdorff distortion, while general bounded sets need not admit lower-dimensional Lipschitz embeddings.The Gromov–Hausdorff embedding remains available for general sets through the stated result.

3. Proof of Corollary 1.9 by a curvature argument

The curvature argument proves a small-diameter result for cells whose points share all hyperplane signs, but it cannot establish the stronger uniform tessellation theorem when signs mismatch.

  • Curvature argument: With probability at least 1 − 2 exp(−cδ4m), every cell of the random tessellation has diameter at most δ.The hyperplanes are independent and Haar-uniform, represented by Gaussian normals.
  • Curvature argument: The proof uses midpoint convexity within a cell and sphere curvature to force the midpoint near the sphere, thereby making the endpoint distance small.For points in one cell, their midpoint remains in the same cell after normalization; Lemma 2.1 then gives ∥z∥2 ≈ 1.
  • Curvature argument: For points separated by a fraction d of hyperplanes, mismatched signs add an error term to the midpoint estimate.The argument estimates this term through the size of the mismatched-sign set T.
  • Limitations: If the error is approximately d, curvature yields only d(x, y) ≲ √d, weaker than the required d(x, y) ≲ d.The weakness is that the argument uses |T| but ignores the geometric nature of T.
  • Limitations: The paper therefore replaces the curvature route with a more conceptual argument to prove Theorem 1.2.The conditional distribution of hyperplane normals depends on x and y, making a uniform conditional-expectation argument difficult.
  • Broader connection: The cell-diameter bound’s δ−4 dependence is connected to the unresolved optimal distortion dependence in Dvoretzky’s theorem.For ℓ_m^1, the known construction has dimension n ∼ ε2m, while optimality of the ε2 dependence is unknown.

4. Toward Theorem 1.2: a soft Hamming distance

A standard net argument fails to extend hard Hamming-distance estimates because the distance is discontinuous. The paper introduces soft Hamming distance to recover a continuity mechanism and formulate a stronger theorem.

  • Motivation: The covering strategy controls distances for fixed pairs and finite nets, but extending the estimate from a net to all of K is obstructed by discontinuity.Some tessellations can be uniform on an ε-net while remaining highly non-uniform on the full set.
  • Soft Hamming distance: Soft Hamming distance replaces hard sign mismatches with thresholded separation events determined by a parameter t.For positive t it counts hyperplanes separating points sufficiently well; for negative t it also counts nearly separating hyperplanes.
  • Soft Hamming distance: The soft distance is non-increasing in t and becomes continuous under small perturbations when the transformed coordinates are bounded in L∞.The continuity lemma sandwiches perturbed separation events between thresholds t + ε and t − ε.
  • Main result: Theorem 4.4 states a stronger uniform tessellation result for the soft Hamming distance.Taking t = 0 recovers Theorem 1.2, while general t is useful because the proof uses multiple thresholds.

5. Proof of Theorem 4.4 on the soft Hamming distance

The proof of Theorem 4.4 combines concentration for fixed pairs, a mean-width-controlled net, tail control, and continuity under L1 perturbations to extend soft-distance estimates uniformly over K.

  • Fixed-pair concentration: The proof first establishes concentration of the soft Hamming distance for each fixed pair using its binomial distribution and Chernoff bounds.The number of counted hyperplanes has distribution Bin(m, p), with p equal to the expected soft distance.
  • Net argument: An ε-net of K can be chosen with log |Nε| ≤ Cε−2w(K)2, linking the covering size to Gaussian mean width.A union bound then controls all pairs in the net.
  • Tail control: Tail vectors in (K − K) ∩ εB_n^2 are controlled separately with probability at least 1 − 2 exp(−cm).The estimate uses Lemma 2.1 and the bound w(K − K) ≤ 2w(K).
  • Extension to K: Because tail perturbations are controlled in L1 rather than L∞, the proof establishes a relaxed continuity lemma with adjustable threshold M.Coordinates exceeding the threshold are bounded in number, yielding the comparison needed for perturbed pairs.
  • Extension to K: Combining net concentration, tail control, and L1 continuity completes Theorem 4.4 with a uniform estimate over all x, y ∈ K.The proof selects ε = δ2/100 and M = 10/δ before applying a union bound.

6. Proof of Theorem 1.10 on tessellations in Rn

The paper lifts a bounded subset of R^n into the sphere in R^{n+1}, applies the spherical tessellation theorem, and pulls the resulting separation estimates back to affine hyperplanes in R^n.

  • Lifting construction: Lifting x to x ⊕ t converts affine hyperplanes in R^n into linear hyperplanes in R^{n+1}.Separation is preserved because ⟨a_i ⊕ a, x ⊕ t⟩ = ⟨a_i, x⟩ + at.
  • Distance transfer: The lifted points are normalized by the spherical projection Q(u) = u/∥u∥2, and their geodesic distances are compared with distances in K.The lift uses a sufficiently large parameter t to control the normalization error.
  • Spherical theorem: Applying the spherical theorem gives |d_A(x′, y′) − d(x′, y′)| ≤ δ0 uniformly on the lifted set.The resulting bound is then combined with the distance comparison and the separation identity.
  • Distance transfer: The fraction of hyperplanes separating lifted points equals the fraction separating the original points: d_A(x′, y′) = d_A(x, y).This identity transfers the tessellation estimate between the lifted spherical set and the original bounded set.
  • Parameter choice: Choosing t := 2C1/δ and δ0 = δ2/(4πC1) yields the desired δ accuracy, under the stated mean-width-dependent lower bound on m.The proof notes that the required condition m ≥ Cδ0^−6w(K)2 follows from the assumed bound.
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