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The JohnsonLindenstrauss lemma almost characterizes Hilbert space, but not quite
"... Let X be a normed space that satisfies the JohnsonLindenstrauss lemma (JL lemma, in short) in the sense that for any integer n and any x1,..., xn ∈ X there exists a linear mapping L: X → F, where F ⊆ X is a linear subspace of dimension O(log n), such that ‖xi − x j ‖ ≤ ‖L(xi) − L(x j) ‖ ≤ O(1) ..."
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Let X be a normed space that satisfies the JohnsonLindenstrauss lemma (JL lemma, in short) in the sense that for any integer n and any x1,..., xn ∈ X there exists a linear mapping L: X → F, where F ⊆ X is a linear subspace of dimension O(log n), such that ‖xi − x j ‖ ≤ ‖L(xi) − L(x j) ‖ ≤ O(1) · ‖xi − x j ‖ for all i, j ∈ {1,..., n}. We show that this implies that X is almost Euclidean in the following sense: Every ndimensional subspace of X embeds into Hilbert space with distortion 22O(log ∗ n). On the other hand, we show that there exists a normed space Y which satisfies the JL lemma, but for every n there exists an ndimensional subspace En ⊆ Y whose Euclidean distortion is at least 2Ω(α(n)) , where α is the inverse Ackermann function. 1
Almost Euclidean subspaces of ℓN 1 via expander codes
, 2009
"... We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ R N of dimension (1 − o(1))N such that for every x ∈ X, (log N) −O(log log log N) √ N ‖x‖2 � ‖x‖1 � √ N ‖x‖2. If we are allowed to use N 1 / log log N � N o(1) random bits and dim(X) � (1 − η)N for any ..."
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We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ R N of dimension (1 − o(1))N such that for every x ∈ X, (log N) −O(log log log N) √ N ‖x‖2 � ‖x‖1 � √ N ‖x‖2. If we are allowed to use N 1 / log log N � N o(1) random bits and dim(X) � (1 − η)N for any fixed constant η, the lower bound can be further improved to (log N) −O(1) √ N‖x‖2. Through known connections between such Euclidean sections of ℓ1 and compressed sensing matrices, our result also gives explicit compressed sensing matrices for low compression factors for which basis pursuit is guaranteed to recover sparse signals. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of errorcorrecting codes.
Ellipsoid Approximation Using Random Vectors
"... Abstract. We analyze the behavior of a random matrix with independent rows, each distributed according to the same probability measure on R n or on ℓ2. We investigate the spectrum of such a matrix and the way the ellipsoid generated by it approximates the covariance structure of the underlying measu ..."
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Abstract. We analyze the behavior of a random matrix with independent rows, each distributed according to the same probability measure on R n or on ℓ2. We investigate the spectrum of such a matrix and the way the ellipsoid generated by it approximates the covariance structure of the underlying measure. As an application, we provide estimates on the deviation of the spectrum of Gram matrices from the spectrum of the integral operator. 1
lattice problems such
, 2006
"... We present reductions from lattice problems in the ℓ2 norm to the corresponding problems in other norms such as ℓ1, ℓ ∞ (and in fact in any other ℓp norm where 1 ≤ p ≤ ∞). We consider ..."
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We present reductions from lattice problems in the ℓ2 norm to the corresponding problems in other norms such as ℓ1, ℓ ∞ (and in fact in any other ℓp norm where 1 ≤ p ≤ ∞). We consider
Uncertainty Principles, Extractors, and ExplicitEmbeddings of L2 into L1
"... ABSTRACT We give an explicit construction of a constant distortion embedding F of ln2 into lm1, with m = n1+o(1). As a bonus, our embedding also has good computational properties: for any input x, F x can be computed in n1+o(1) time. The previously known mappings required \Omega (n2) evaluation time ..."
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ABSTRACT We give an explicit construction of a constant distortion embedding F of ln2 into lm1, with m = n1+o(1). As a bonus, our embedding also has good computational properties: for any input x, F x can be computed in n1+o(1) time. The previously known mappings required \Omega (n2) evaluation time.
Electronic Colloquium on Computational Complexity, Report No. 126 (2006) Uncertainty Principles, Extractors, and Explicit Embeddings of L2 into
, 2006
"... The area of geometric functional analysis1 is concerned with studying the properties of geometric (normed) spaces. A typical question in the area is: for two spaces X and Y, equipped with norms ‖·‖X and ‖·‖Y, under which conditions is there an embedding F: X → Y such that for any p,q ∈ X, we have ‖p ..."
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The area of geometric functional analysis1 is concerned with studying the properties of geometric (normed) spaces. A typical question in the area is: for two spaces X and Y, equipped with norms ‖·‖X and ‖·‖Y, under which conditions is there an embedding F: X → Y such that for any p,q ∈ X, we have ‖p − q‖X ≤ ‖F(p) − F(q)‖Y ≤ C‖p − q‖Y for some constant2 C ≥ 1? A ubiquitous tool