New and Improved Johnson–Lindenstrauss Embeddings via the Restricted Isometry Property

Krahmer, Felix; Ward, Rachel

doi:10.1137/100810447

Cited by 238 publications

(289 citation statements)

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“…Random matrices known to have this property include matrices with independent subgaussian entries (such as Gaussian or Bernoulli matrices), see for example [DG03]. Moreover, it is shown in [KW11] that any matrix that satisfies the classical RIP will satisfy the Johnson-Lindenstrauss lemma and thus the D-RIP with high probability after randomizing the signs of the columns. The latter construction allows for structured random matrices with fast multiplication properties such as randomly subsampled Fourier matrices (in combination with the results from [RV08]) and matrices representing subsampled random convolutions (in combination with the results from [RRT12, KMR14]); in both cases, however, again with randomized column signs.…”

Section: Theorem 22 ([Cenr10])mentioning

confidence: 99%

“…This can be interpreted as a dictionary version of the universality property discussed above. It has now been shown that several classes of random matrices satisfy this property as well as subsampled structured matrices, after applying random column signs [DG03,KW11]. The matrices in both of these cases are motivated by application scenarios, but typically in applications they appear without the randomized column signs.…”

mentioning

confidence: 99%

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Compressive Sensing with Redundant Dictionaries and Structured Measurements

Krahmer¹,

Needell²,

Ward³

2015

SIAM J. Math. Anal.

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ABSTRACT. Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary D. This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary, if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the D-restricted isometry property (D-RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in D. While many classes of random matrices are known to satisfy such conditions, such matrices are not representative of the structural constraints imposed by practical sensing systems. We close this gap in the theory by demonstrating that one can subsample a fixed orthogonal matrix in such a way that the D-RIP will hold, provided this basis is sufficiently incoherent with the sparsifying dictionary D. We also extend this analysis to allow for weighted sparse expansions. Consequently, we arrive at compressive sensing recovery guarantees for structured measurements and redundant dictionaries, opening the door to a wide array of practical applications.

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Section: Theorem 22 ([Cenr10])mentioning

confidence: 99%

mentioning

confidence: 99%

Compressive Sensing with Redundant Dictionaries and Structured Measurements

Krahmer¹,

Needell²,

Ward³

2015

SIAM J. Math. Anal.

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“…it is more efficient to use a fast random projection [1,9] which results in a projection cost of O(nd log d). Right protection involves multiplying each entry of the projection data matrix of size k × n with a scalar, with a cost of O(kn).…”

Section: Complexitymentioning

confidence: 99%

Private and Right-Protected Big Data Publication: An Analysis

Heckel

Vlachos²

2017

Proceedings of the 2017 SIAM International Conference on Data Mining

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The ease of digital data dissemination has spurred an amplified interest in technologies related to data privacy and right protection. We examine how both goals can be achieved simultaneously by constructing modified data instances that are both differentially private and right protected. The proposed method first produces a sketch of the dataset via random projection and then perturbs the sketch just enough to ensure privacy. The right-protection mechanism inserts small noise in the dataset which subsequently can be used to verify ownership. We provide analytical privacy, right-protection, and utility guarantees. Our utility guarantees ensure approximate preservation of pairwise distances, thus mining operations such as search, classification, and clustering can be performed on the differentially private and right protected dataset.Extended version with proofs:

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“…This notion, due to Candès and Tao [11], was intensively studied during the last decade and found various applications and connections to several areas of theoretical computer science, including sparse recovery [8,20,27], coding theory [14], norm embeddings [6,23], and computational complexity [4,31,25]. The original motivation for the restricted isometry property comes from the area of compressed sensing.…”

Section: Introductionmentioning

confidence: 99%