
Testing tensor products
A function f:[n]^d→F_2 is a direct sum if it is of the form f((a_1,...,...
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Quantum Fourier Transform Revisited
The fast Fourier transform (FFT) is one of the most successful numerical...
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Optimal Fast JohnsonLindenstrauss Embeddings for Large Data Sets
We introduce a new fast construction of a JohnsonLindenstrauss matrix b...
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Lower Memory Oblivious (Tensor) Subspace Embeddings with Fewer Random Bits: Modewise Methods for Least Squares
In this paper new general modewise JohnsonLindenstrauss (JL) subspace e...
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Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations
Fast linear transforms are ubiquitous in machine learning, including the...
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The Fast Cauchy Transform and Faster Robust Linear Regression
We provide fast algorithms for overconstrained ℓ_p regression and relate...
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Guarantees for the Kronecker Fast JohnsonLindenstrauss Transform Using a Coherence and Sampling Argument
In the recent paper [Jin, Kolda Ward, arXiv:1909.04801], it is prove...
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Faster JohnsonLindenstrauss Transforms via Kronecker Products
The Kronecker product is an important matrix operation with a wide range of applications in supporting fast linear transforms, including signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast JohnsonLindenstrauss projection for embedding vectors with Kronecker product structure, the Kronecker fast JohnsonLindenstrauss transform (KFJLT). The KFJLT drastically reduces the embedding cost to an exponential factor of the standard fast JohnsonLindenstrauss transform (FJLT)'s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. We prove that this computational gain comes with only a small price in embedding power: given N = ∏_k=1^d n_k, consider a finite set of p points in a tensor product of d constituent Euclidean spaces ⊗_k=d^1R^n_k⊂R^N. With high probability, a random KFJLT matrix of dimension N × m embeds the set of points up to multiplicative distortion (1±ε) provided by m ≳ε^2·log^2d  1 (p) ·log N. We conclude by describing a direct application of the KFJLT to the efficient solution of largescale Kroneckerstructured least squares problems for fitting the CP tensor decomposition.
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