A Randomized Approximate Nearest Neighbors Algorithm

Jones, Peter W.; Osipov, Alexander V.; Rokhlin, Vladimir

doi:10.21236/ada555156

Cited by 26 publications

(28 citation statements)

References 9 publications

(21 reference statements)

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“…Second, we demonstrate the performance of the algorithm empirically, by running it on sets of points, generated according to the Gaussian distribution. The choice of uniform or Hamming distributions instead of Gaussian results in very similar performance (see [10] for results and details). The algorithm has been implemented in FORTRAN (Lahey 95 Linux version).…”

Section: Numerical Resultsmentioning

confidence: 94%

“…uniform distribution on the discrete set of the vertices of [0, 1] d ). For both uniform and Hamming distributions, the performance of the algorithm was very similar to that in the Gaussian case (see [10] for details).…”

mentioning

confidence: 70%

“…The following theorem provides an approximation to the expectation E [prop N ] (see (29)). The error of this approximation is verified via numerical experiments (see [10] for more details).…”

mentioning

confidence: 84%

“…In the rest of this subsection, we outline the results of the analysis (see [10] for details). We start with introducing some notation.…”

Section: Performance Analysismentioning

confidence: 99%

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A Randomized Approximate Nearest Neighbors Algorithm - A Short Version

Jones¹,

Osipov²,

Rokhlin³

2011

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We present a randomized algorithm for the approximate nearest neighbor problem in ddimensional Euclidean space. Given N points {x j } in R d , the algorithm attempts to find k nearest neighbors for each of x j , where k is a user-specified integer parameter. The algorithm is iterative, and its CPU time requirements are proportional to T • N • (d • (log d) + k • (d + log k) • (log N)) + N • k 2 • (d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N • (d + k). A byproduct of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {x j } for an arbitrary point x ∈ R d. The cost of each such query is proportional to T • (d • (log d) + log(N/k) • k • (d + log k)), and the memory requirements for the requisite data structure are of the order N • (d + k) + T • (d + N). The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme's behavior for certain types of distributions of {x j }, and illustrate its performance via several numerical examples.

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Section: Numerical Resultsmentioning

confidence: 94%

mentioning

confidence: 70%

mentioning

confidence: 84%

“…In the rest of this subsection, we outline the results of the analysis (see [10] for details). We start with introducing some notation.…”

Section: Performance Analysismentioning

confidence: 99%

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A Randomized Approximate Nearest Neighbors Algorithm - A Short Version

Jones¹,

Osipov²,

Rokhlin³

2011

Self Cite

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“…We note that the problem of selecting a unitary matrix uniformly at random finds application in machine learning (see [39] and the references therein). The algorithm developed by Can is similar to that developed by Jones, Osipov and Rokhlin [40] in that it alternates (partial) Hadamard matrices and diagonal matrices; the difference is that the unitary 3-design property of the Clifford group [11] provides randomness guarantees.…”

Section: Main Ideas and Discussionmentioning

confidence: 99%

Kerdock Codes Determine Unitary 2-Designs

Can

Rengaswamy

Calderbank

et al. 2020

IEEE Trans. Inform. Theory

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The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length N = 2 m over Z4. We show that exponentiating these Z4-valued codewords by ı √ −1 produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form N + 1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL(2, N). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary 2-design. The Kerdock design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary 2-designs on encoded qubits, i.e., to construct logical unitary 2-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to 16 qubits.

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Compressive mining: fast and optimal data mining in the compressed domain

2014

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Real-world data typically contain repeated and periodic patterns. This suggests that they can be effectively represented and compressed using only a few coefficients of an appropriate basis (e.g., Fourier, Wavelets, etc.). However, distance estimation when the data are represented using different sets of coefficients is still a largely unexplored area. This work studies the optimization problems related to obtaining the tightest lower/upper bound on Euclidean distances when each data object is potentially compressed using a different set of orthonormal coefficients. Our technique leads to tighter distance estimates, which translates into more accurate search, learning and mining operations directly in the compressed domain.We formulate the problem of estimating lower/upper distance bounds as an optimization problem. We establish the properties of optimal solutions, and leverage the theoretical analysis to develop a fast algorithm to obtain an exact solution to the problem. The suggested solution provides the tightest estimation of the L 2 -norm or the correlation. We show that typical data-analysis operations, such as k-NN search or k-Means clustering, can operate more accurately using the proposed compression and distance reconstruction technique. We compare it with many other prevalent compression and reconstruction techniques, including random projections and PCA-based techniques. We highlight a surprising result, namely that when the data are highly sparse in some basis, our technique may even outperform PCA-based compression.The contributions of this work are generic as our methodology is applicable to any sequential or highdimensional data as well as to any orthogonal data transformation used for the underlying data compression scheme.

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A Randomized Approximate Nearest Neighbors Algorithm

Cited by 26 publications

References 9 publications

A Randomized Approximate Nearest Neighbors Algorithm - A Short Version

A Randomized Approximate Nearest Neighbors Algorithm - A Short Version

Kerdock Codes Determine Unitary 2-Designs

Compressive mining: fast and optimal data mining in the compressed domain

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