Linear-Scaling Tensor Train Sketching
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arXiv
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| Format: | Preprint |
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2026
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| author | Cazeaux, Paul Dupuy, Mi-Song Justiniano, Rodrigo Figueroa |
| author_facet | Cazeaux, Paul Dupuy, Mi-Song Justiniano, Rodrigo Figueroa |
| contents | We introduce the TTStack sketch, a structured random projection tailored to the tensor train (TT) format that unifies existing TT-adapted sketching operators. By varying two integer parameters $P$ and $R$, TTStack interpolates between the Khatri-Rao sketch ($R=1$) and the Gaussian TT sketch ($P=1$). We prove that TTStack satisfies an oblivious subspace embedding (OSE) property with parameters $R = \mathcal{O}(d(r+\log 1/δ))$ and $P = \mathcal{O}(\varepsilon^{-2})$, and an oblivious subspace injection (OSI) property under the condition $R = \mathcal{O}(d)$ and $P = \mathcal{O}(\varepsilon^{-2}(r + \log r/δ))$. Both guarantees depend only linearly on the tensor order $d$ and on the subspace dimension $r$, in contrast to prior constructions that suffer from exponential scaling in $d$. As direct consequences, we derive quasi-optimal error bounds for the QB factorization and randomized TT rounding. The theoretical results are supported by numerical experiments on synthetic tensors, Hadamard products, and a quantum chemistry application. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11009 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear-Scaling Tensor Train Sketching Cazeaux, Paul Dupuy, Mi-Song Justiniano, Rodrigo Figueroa Numerical Analysis Data Structures and Algorithms 15A69, 65F55, 65F99, 65Y20, 68W20 We introduce the TTStack sketch, a structured random projection tailored to the tensor train (TT) format that unifies existing TT-adapted sketching operators. By varying two integer parameters $P$ and $R$, TTStack interpolates between the Khatri-Rao sketch ($R=1$) and the Gaussian TT sketch ($P=1$). We prove that TTStack satisfies an oblivious subspace embedding (OSE) property with parameters $R = \mathcal{O}(d(r+\log 1/δ))$ and $P = \mathcal{O}(\varepsilon^{-2})$, and an oblivious subspace injection (OSI) property under the condition $R = \mathcal{O}(d)$ and $P = \mathcal{O}(\varepsilon^{-2}(r + \log r/δ))$. Both guarantees depend only linearly on the tensor order $d$ and on the subspace dimension $r$, in contrast to prior constructions that suffer from exponential scaling in $d$. As direct consequences, we derive quasi-optimal error bounds for the QB factorization and randomized TT rounding. The theoretical results are supported by numerical experiments on synthetic tensors, Hadamard products, and a quantum chemistry application. |
| title | Linear-Scaling Tensor Train Sketching |
| topic | Numerical Analysis Data Structures and Algorithms 15A69, 65F55, 65F99, 65Y20, 68W20 |
| url | https://arxiv.org/abs/2603.11009 |