75 research outputs found

    Monotone Maps, Sphericity and Bounded Second Eigenvalue

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    We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on nn points can be embedded into l2nl_2^n, while, (in a sense to be made precise later), for almost every nn-point metric space, every monotone map must be into a space of dimension Ω(n)\Omega(n). It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on {\em sphericity} of graphs, which suggest one example of such a metric space - that defined by a complete bipartitegraph. We prove that an δn\delta n-regular graph of order nn, with bounded diameter has sphericity Ω(n/(λ2+1))\Omega(n/(\lambda_2+1)), where λ2\lambda_2 is the second largest eigenvalue of the adjacency matrix of the graph, and 0 < \delta \leq \half is constant. We also show that while random graphs have linear sphericity, there are {\em quasi-random} graphs of logarithmic sphericity. For the above bound to be linear, λ2\lambda_2 must be constant. We show that if the second eigenvalue of an n/2n/2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n2)o(n^2) entries. Furthermore, for any 0 < \delta < \half, and λ2\lambda_2, there are only finitely many δn\delta n-regular graphs with second eigenvalue at most λ2\lambda_2

    Sparser Johnson-Lindenstrauss Transforms

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    We give two different and simple constructions for dimensionality reduction in 2\ell_2 via linear mappings that are sparse: only an O(ε)O(\varepsilon)-fraction of entries in each column of our embedding matrices are non-zero to achieve distortion 1+ε1+\varepsilon with high probability, while still achieving the asymptotically optimal number of rows. These are the first constructions to provide subconstant sparsity for all values of parameters, improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar, and Sarl\'{o}s (STOC 2010). Such distributions can be used to speed up applications where 2\ell_2 dimensionality reduction is used.Comment: v6: journal version, minor changes, added Remark 23; v5: modified abstract, fixed typos, added open problem section; v4: simplified section 4 by giving 1 analysis that covers both constructions; v3: proof of Theorem 25 in v2 was written incorrectly, now fixed; v2: Added another construction achieving same upper bound, and added proof of near-tight lower bound for DKS schem

    Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform

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    The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from probability in Banach spaces that were successfully used in the context of sparse reconstruction to advance on an open problem in random pojection. In particular, we generalize and use an intricate result by Rudelson and Vershynin for sparse reconstruction which uses Dudley's theorem for bounding Gaussian processes. Our main result states that any set of N=exp(O~(n))N = \exp(\tilde{O}(n)) real vectors in nn dimensional space can be linearly mapped to a space of dimension k=O(\log N\polylog(n)), while (1) preserving the pairwise distances among the vectors to within any constant distortion and (2) being able to apply the transformation in time O(nlogn)O(n\log n) on each vector. This improves on the best known N=exp(O~(n1/2))N = \exp(\tilde{O}(n^{1/2})) achieved by Ailon and Liberty and N=exp(O~(n1/3))N = \exp(\tilde{O}(n^{1/3})) by Ailon and Chazelle. The dependence in the distortion constant however is believed to be suboptimal and subject to further investigation. For constant distortion, this settles the open question posed by these authors up to a \polylog(n) factor while considerably simplifying their constructions

    Isometric sketching of any set via the Restricted Isometry Property

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    In this paper we show that for the purposes of dimensionality reduction certain class of structured random matrices behave similarly to random Gaussian matrices. This class includes several matrices for which matrix-vector multiply can be computed in log-linear time, providing efficient dimensionality reduction of general sets. In particular, we show that using such matrices any set from high dimensions can be embedded into lower dimensions with near optimal distortion. We obtain our results by connecting dimensionality reduction of any set to dimensionality reduction of sparse vectors via a chaining argument.Comment: 17 page
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