The Restricted Isometry Property of Subsampled Fourier Matrices

Haviv, Ishay; Regev, Oded

doi:10.1137/1.9781611974331.ch22

Cited by 21 publications

(3 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…)) that have been shown in [11]. Namely, it is shown that m = (δ −2 (log 1 δ ) 2 k(log k δ ) 2 log(N )) implies the (k, δ)-RIP with probability arbitrarily close to 1 for sufficiently large N .…”

Section: Unrestricted Fast Johnson-lindenstrauss Embeddingsmentioning

confidence: 79%

Optimal fast Johnson–Lindenstrauss embeddings for large data sets

Bamberger

Krahmer

2021

Sampl. Theory Signal Process. Data Anal.

View full text Add to dashboard Cite

Johnson–Lindenstrauss embeddings are widely used to reduce the dimension and thus the processing time of data. To reduce the total complexity, also fast algorithms for applying these embeddings are necessary. To date, such fast algorithms are only available either for a non-optimal embedding dimension or up to a certain threshold on the number of data points. We address a variant of this problem where one aims to simultaneously embed larger subsets of the data set. Our method follows an approach by Nelson et al. (New constructions of RIP matrices with fast multiplication and fewer rows. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1515-1528, 2014): a subsampled Hadamard transform maps points into a space of lower, but not optimal dimension. Subsequently, a random matrix with independent entries projects to an optimal embedding dimension. For subsets whose size scales at least polynomially in the ambient dimension, the complexity of this method comes close to the number of operations just to read the data under mild assumptions on the size of the data set that are considerably less restrictive than in previous works. We also prove a lower bound showing that subsampled Hadamard matrices alone cannot reach an optimal embedding dimension. Hence, the second embedding cannot be omitted.

show abstract

Section: Unrestricted Fast Johnson-lindenstrauss Embeddingsmentioning

confidence: 79%

Optimal fast Johnson–Lindenstrauss embeddings for large data sets

Bamberger

Krahmer

2021

Sampl. Theory Signal Process. Data Anal.

View full text Add to dashboard Cite

show abstract

“…The problem of determining whether A has RIP for any accuracy is proven to be NPhard [40]. Constructing RIP matrices deterministically is a hard problem [41], but RIP matrices can be generated with high probability by simple random methods [42,43].…”

Section: Ripmentioning

confidence: 99%

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Swamy

Wang

et al. 2023

Mathematics

View full text Add to dashboard Cite

Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article.

show abstract

“…Also, just as the JL lemma can be so-used to obtain an RIP matrix, the reverse is also true: Krahmer and Ward [KW11] showed that any ( (log(1/ )), ( ))-RIP matrix Π gives rise to an ( , )-JL distribution defined by picking independent Rademachers 1 , … , and then producing the matrix Π⋅ ( ). The best known improvements of the FJLT combine this result with analyses of RIP for Π that support fast matrix-vector multiplication, such as sampling rows from the Discrete Fourier Transform [HR17]. 5.2.…”

Section: Quantitatively Improving Upon Previous Bounds Candèsmentioning

confidence: 99%

Dimensionality Reduction in Euclidean Space

Nelson¹

2020

Notices Amer. Math. Soc.

View full text Add to dashboard Cite

I begin with a description of what this article is not about. It is not about Principal Component Analysis (PCA), Kernel PCA, Multidimensional Scaling, ISOMAP, Hessian Eigenmaps, or other methods of dimensionality reduction primarily created to help understand high-dimensional datasets. Rather, this article focuses on high dimensionality as a barrier to algorithmic efficiency (i.e., low running time and/or memory consumption), and explores how dimension reduction can be used as an algorithmic tool to overcome this barrier. In fact, as we discuss more in length in Section 5.2, this view is not only different than

show abstract

The Restricted Isometry Property of Subsampled Fourier Matrices

Cited by 21 publications

References 28 publications

Optimal fast Johnson–Lindenstrauss embeddings for large data sets

Optimal fast Johnson–Lindenstrauss embeddings for large data sets

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Dimensionality Reduction in Euclidean Space

Contact Info

Product

Resources

About