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New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property
Felix Krahmer, Rachel Ward
TL;DR
The paper addresses how to obtain Johnson–Lindenstrauss embeddings from RIP matrices while retaining near-optimal embedding dimensions. It randomizes column signs, proves uniform finite-set norm preservation, and applies the result to structured matrices and compressed sensing. The resulting JL bounds are optimal up to logarithmic factors in N, including an ε^-2 dependence for partial Fourier and Hadamard matrices.
Problem
Existing JL constructions seek structured, fast embeddings with optimal dimension, while the connection between JL concentration and RIP motivates a converse construction from RIP matrices.
Method
The paper randomizes the column signs of an RIP matrix and uses concentration and RIP-based estimates to control norm distortion uniformly over a finite set.
Results
RIP matrices with randomized column signs yield JL embeddings optimal up to logarithmic factors in N; partial Fourier and Hadamard bounds improve from ε^-4 to ε^-2 dependence.
Takeaways & Limitations
The construction supplies fast structured JL embeddings and has consequences for compressed sensing, including applications to redundant dictionaries.
Takeaways & Limitations
The embedding construction still requires N random bits, although this randomness is removed when the embedded vectors themselves have random signs.
Abstract
from arXiv · showhide
Consider an m by N matrix Phi with the Restricted Isometry Property of order k and level delta, that is, the norm of any k-sparse vector in R^N is preserved to within a multiplicative factor of 1 +- delta under application of Phi. We show that by randomizing the column signs of such a matrix Phi, the resulting map with high probability embeds any fixed set of p = O(e^k) points in R^N into R^m without distorting the norm of any point in the set by more than a factor of 1 +- delta. Consequently, matrices with the Restricted Isometry Property and with randomized column signs provide optimal Johnson-Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound m = O(delta^(-4) log(p) log^4(N)) appearing in Ailon and Liberty to m = O(delta^(-2) log(p) log^4(N)), which is optimal up to the logarithmic factors in N. Our results also have a direct application in the area of compressed sensing for redundant dictionaries.
1 Introduction
The introduction connects Johnson–Lindenstrauss embeddings with compressed sensing and the Restricted Isometry Property, then presents randomized column signs as a converse construction with near-optimal dimension. The approach improves structured-matrix bounds, especially for partial Fourier and Hadamard matrices.
- Johnson–Lindenstrauss embeddings: The JL Lemma embeds p points into O(ε^-2 log(p)) dimensions while preserving pairwise distances within factors 1−ε and 1+ε.The embedding-dimension bound is tight up to an O(log(1/ε)) factor.
- Johnson–Lindenstrauss embeddings: Linear JL embeddings reduce distance preservation to norm preservation for the finite difference set E = {x_i − x_j}.For random matrices, a concentration inequality for each fixed vector combines with a union bound over E.
- Compressed sensing and RIP: RIP is a sufficient condition for minimal ℓ1-norm recovery to coincide with sparsest recovery in many underdetermined systems Φx = y.Gaussian and Bernoulli matrices achieve RIP with m ≳ δ^-2 k log(N/k), while structured matrices typically incur additional logarithmic factors in N.
- Contribution of this work: RIP matrices with randomized column signs provide JL embeddings that are optimal up to logarithmic factors in the ambient dimension N.Without randomization, null-space vectors of the fixed parent matrix prevent such a converse.
- Contribution of this work: For partial Fourier and partial Hadamard matrices, the embedding bound improves from m ≍ ε^-4 log(p/η) log^4(N) to m ≍ ε^-2 log(p/η) log^4(N).These constructions support O(N log(N)) matrix-vector multiplication, compared with O(Nm) for purely random matrices.
2 Notation
This section establishes notation for norms, asymptotic comparisons, block decompositions, matrix restrictions, and diagonal operators used in the analysis.
- Norms and matrix norms: The ℓp norm is defined for vectors in R^N, while the operator norm and Frobenius norm are defined for matrices.The operator norm is the supremum of ||Φx||2 over unit Euclidean-norm vectors.
- Asymptotic notation: The notation f ≳ g means f is bounded below by a positive constant multiple of g, and f ≍ g means both comparisons hold.These comparisons are uniform over the stated domain S.
- Block decompositions: Vectors are arranged by decreasing absolute value and decomposed into blocks of size s = k/2, with a potentially smaller final block.The section also defines coarse decompositions, block-index relations, and restricted column matrices.
- Matrix restrictions and diagonal operators: For a vector x, D_x denotes the diagonal matrix whose diagonal entries equal the coordinates of x.The notation supports matrix expressions involving coordinatewise scaling.
3 The main results
The main results show that randomized column signs convert RIP matrices into high-probability embeddings for finite point sets, while also supplying a converse to the JL-to-RIP implication.
- Theorem 3.1: For a finite set E of p points, the theorem assumes an RIP matrix of order k with k at least proportional to log(4p/η) and random Rademacher column signs.The randomized matrix is ΦD_ξ, where ξ is uniformly distributed over {−1, 1}^N.
- Theorem 3.1: With probability exceeding 1−η, the randomized RIP map preserves the relevant norm uniformly for every x ∈ E.This establishes the finite-set embedding guarantee from the RIP hypothesis.
- Converse result: The result provides a direct converse to the implication that JL-type concentration yields the Restricted Isometry Property.The converse requires randomized column signs and is formulated through RIP matrices satisfying an admissibility condition.
- Converse result: For admissible pairs (k, m), randomized signs make ΦD_ξ satisfy the concentration inequality with c0 proportional to log^-1(N).The admissibility condition is k ≤ c3δ^2m / log(N/k).
4 Concrete examples and applications
The section applies the RIP-to-JL result to structured matrices and compressed sensing, improving bounds for several fast matrix constructions and extending applicability to redundant dictionaries and cross validation.
- Deterministic constructions: Theorem 3.1 converts RIP bounds into JL bounds for multiple structured and deterministic constructions, with improvements recorded against prior JL estimates.The comparison lists RIP dimension, previous JL dimension, and the resulting JL dimension.
- Matrices arising from bounded orthonormal systems: m ≳ ε−2 log(p) log(N) suffices for bounded orthonormal-system embeddings with randomized column signs, up to logarithmic factors in N.The underlying RIP condition requires m ≳ δ−2k log4(N).
- Matrices arising from bounded orthonormal systems: ε−4 becomes ε−2 for randomized partial Fourier and partial Hadamard matrices, improving the prior Ailon–Liberty bound.These constructions retain fast matrix-vector multiplication through their orthogonal structure.
- Partial circulant matrices: m ≳ ε−2 log2(p) was previously sufficient for randomized-sign partial circulant matrices, while Theorem 3.1 improves the bound for large p.Partial circulant matrices also support fast multiplication through convolution and Fourier-domain multiplication.
- Compressed sensing applications: The results extend compressed sensing from orthonormal bases to redundant dictionaries, including tight frames, for essentially all random RIP-based matrix constructions.They also permit RIP matrices to support cross validation after randomizing column signs.
- Optimal asymptotics: m ≍ k log(N/k) and m ≍ δ−2 are identified as optimal asymptotics for RIP dimension in N, k, and δ under the stated regimes.Combined with JL optimality, the results establish near-optimal embedding dimensions up to logarithmic factors in N.
5 Proof Ingredients
The proof combines concentration inequalities for random signs with norm estimates derived from the RIP, using block structure and standard inequalities to control the resulting random quantities.
- Proof Ingredients: The proof uses concentration inequalities for Rademacher sequences and explicit RIP-based norm estimates.These ingredients provide the probabilistic and geometric parts of the argument.
- Proof Ingredients: Hoeffding’s inequality supplies the first concentration estimate for the randomized-sign analysis.It is introduced as a classical concentration result.
- Proof Ingredients: A deviation bound for Rademacher chaos supplies the second concentration-of-measure ingredient.The cited bound is presented with explicit constants and a simplified proof.
- Proof Ingredients: The RIP estimate applies to disjoint subsets of size at most s when the matrix has the (2s, δ)-RIP.This proposition is used in the norm estimate for RIP matrices.
- Proof Ingredients: For decreasingly arranged vectors with unit ℓ2 norm, later blocks have controlled ℓ∞ norm, supporting the blockwise norm estimate.The argument also uses the arithmetic–geometric mean inequality and the RIP estimate.
- Proof Ingredients: The block decomposition partitions coordinates into R = ⌈N/s⌉ groups and applies a symmetric-matrix estimate before combining it with Hoeffding’s inequality.This links the RIP norm estimate to the final concentration step.
6 Proof of the main results
The proof analyzes the randomized-sign embedding for fixed vectors, bounds its component terms using RIP and concentration inequalities, and then applies union bounds. The resulting parameter conditions establish the theorem and yield near-optimal embedding dimensions for matrices with optimal-order RIP.
- Proof of the main results: The proof fixes a normalized vector, reorders its entries, and uses column-permutation invariance of RIP before estimating the embedding error.The analysis assumes ||x||_2 = 1 and eventually takes a union bound over the finite set.
- Proof of the main results: RIP makes each restricted matrix Φ(J) an almost isometry, controlling the first term in the decomposition.The proof uses ||D x(J) ξ(J)||_2 = ||x(J)||_2 when applying the RIP estimate.
- Proof of the main results: Hoeffding’s inequality and a Rademacher-chaos deviation bound provide concentration estimates for the remaining terms.The proof combines Hoeffding’s inequality with an RIP-based norm estimate and separately bounds the decomposed terms.
- Proof of the main results: k = 2s ≥ 40 log(4p/η) ensures the stated probability requirement, while the resulting bounds combine to prove the theorem.With τ = .55 and γ = .1, the second and last terms are bounded by .2δ and .55δ, respectively.
- Proof of the main results: For RIP matrices satisfying k ≤ c3δ^2m / log(N/k), the theorem gives a JL embedding whose dimension is optimal up to a single logarithmic factor in N.The argument also applies in the complex setting, with x ∈ C^N and Φ ∈ C^{m×N}.