6 Search Results for "Christandl, Matthias"

Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2024.20
Discreteness of Asymptotic Tensor Ranks (Extended Abstract)

Authors: Jop Briët, Matthias Christandl, Itai Leigh, Amir Shpilka, and Jeroen Zuiddam

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points. Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank. Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank.
Cite as

Jop Briët, Matthias Christandl, Itai Leigh, Amir Shpilka, and Jeroen Zuiddam. Discreteness of Asymptotic Tensor Ranks (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2024.20,
  author =	{Bri\"{e}t, Jop and Christandl, Matthias and Leigh, Itai and Shpilka, Amir and Zuiddam, Jeroen},
  title =	{{Discreteness of Asymptotic Tensor Ranks}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.20},
  URN =		{urn:nbn:de:0030-drops-195483},
  doi =		{10.4230/LIPIcs.ITCS.2024.20},
  annote =	{Keywords: Tensors, Asymptotic rank, Subrank, Slice rank, Restriction, Degeneration, Diagonalization, SLOCC}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.12
On the Multilinear Complexity of Associative Algebras

Authors: Markus Bläser, Hendrik Mayer, and Devansh Shringi

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
Christandl and Zuiddam [Matthias Christandl and Jeroen Zuiddam, 2019] study the multilinear complexity of d-fold matrix multiplication in the context of quantum communication complexity. Bshouty [Nader H. Bshouty, 2013] investigates the multilinear complexity of d-fold multiplication in commutative algebras to understand the size of so-called testers. The study of bilinear complexity is a classical topic in algebraic complexity theory, starting with the work by Strassen. However, there has been no systematic study of the multilinear complexity of multilinear maps. In the present work, we systematically investigate the multilinear complexity of d-fold multiplication in arbitrary associative algebras. We prove a multilinear generalization of the famous Alder-Strassen theorem, which is a lower bound for the bilinear complexity of the (2-fold) multiplication in an associative algebra. We show that the multilinear complexity of the d-fold multiplication has a lower bound of d ⋅ dim A - (d-1)t, where t is the number of maximal twosided ideals in A. This is optimal in the sense that there are algebras for which this lower bound is tight. Furthermore, we prove the following dichotomy that the quotient algebra A/rad A determines the complexity of the d-fold multiplication in A: When the semisimple algebra A/rad A is commutative, then the multilinear complexity of the d-fold multiplication in A is polynomial in d. On the other hand, when A/rad A is noncommutative, then the multilinear complexity of the d-fold multiplication in A is exponential in d.
Cite as

Markus Bläser, Hendrik Mayer, and Devansh Shringi. On the Multilinear Complexity of Associative Algebras. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blaser_et_al:LIPIcs.STACS.2023.12,
  author =	{Bl\"{a}ser, Markus and Mayer, Hendrik and Shringi, Devansh},
  title =	{{On the Multilinear Complexity of Associative Algebras}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.12},
  URN =		{urn:nbn:de:0030-drops-176645},
  doi =		{10.4230/LIPIcs.STACS.2023.12},
  annote =	{Keywords: Multilinear computations, associative algebras, matrix multiplication, Alder-Strassen theorem}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.48
Larger Corner-Free Sets from Combinatorial Degenerations

Authors: Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021). We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in F₂ⁿ × F₂ⁿ of size Ω(3.39ⁿ/poly(n)), which improves the previous lower bound Ω(2.82ⁿ) of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound 4^{n - o(n)}. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group G, three players need to determine whether their inputs x₁, x₂, x₃ ∈ G sum to zero. We find that the NOF communication complexity of the Eval problem over F₂ⁿ is at most 0.24n + 𝒪(log n), which improves the previous upper bound 0.5n + 𝒪(log n).
Cite as

Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam. Larger Corner-Free Sets from Combinatorial Degenerations. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{christandl_et_al:LIPIcs.ITCS.2022.48,
  author =	{Christandl, Matthias and Fawzi, Omar and Ta, Hoang and Zuiddam, Jeroen},
  title =	{{Larger Corner-Free Sets from Combinatorial Degenerations}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{48:1--48:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.48},
  URN =		{urn:nbn:de:0030-drops-156441},
  doi =		{10.4230/LIPIcs.ITCS.2022.48},
  annote =	{Keywords: Corner-free sets, communication complexity, number on the forehead, combinatorial degeneration, hypergraphs, Shannon capacity, eval problem}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.12
Limits on the Universal Method for Matrix Multiplication

Authors: Josh Alman

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
In this work, we prove limitations on the known methods for designing matrix multiplication algorithms. Alman and Vassilevska Williams [Alman and Williams, 2018] recently defined the Universal Method, which substantially generalizes all the known approaches including Strassen’s Laser Method [V. Strassen, 1987] and Cohn and Umans' Group Theoretic Method [Cohn and Umans, 2003]. We prove concrete lower bounds on the algorithms one can design by applying the Universal Method to many different tensors. Our proofs use new tools for upper bounding the asymptotic slice rank of a wide range of tensors. Our main result is that the Universal method applied to any Coppersmith-Winograd tensor CW_q cannot yield a bound on omega, the exponent of matrix multiplication, better than 2.16805. By comparison, it was previously only known that the weaker "Galactic Method" applied to CW_q could not achieve an exponent of 2. We also study the Laser Method (which is, in principle, a highly special case of the Universal Method) and prove that it is "complete" for matrix multiplication algorithms: when it applies to a tensor T, it achieves omega = 2 if and only if it is possible for the Universal method applied to T to achieve omega = 2. Hence, the Laser Method, which was originally used as an algorithmic tool, can also be seen as a lower bounding tool. For example, in their landmark paper, Coppersmith and Winograd [Coppersmith and Winograd, 1990] achieved a bound of omega <= 2.376, by applying the Laser Method to CW_q. By our result, the fact that they did not achieve omega=2 implies a lower bound on the Universal Method applied to CW_q. Indeed, if it were possible for the Universal Method applied to CW_q to achieve omega=2, then Coppersmith and Winograd’s application of the Laser Method would have achieved omega=2.
Cite as

Josh Alman. Limits on the Universal Method for Matrix Multiplication. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 12:1-12:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{alman:LIPIcs.CCC.2019.12,
  author =	{Alman, Josh},
  title =	{{Limits on the Universal Method for Matrix Multiplication}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{12:1--12:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.12},
  URN =		{urn:nbn:de:0030-drops-108347},
  doi =		{10.4230/LIPIcs.CCC.2019.12},
  annote =	{Keywords: Matrix Multiplication, Laser Method, Slice Rank}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.26
Barriers for Fast Matrix Multiplication from Irreversibility

Authors: Matthias Christandl, Péter Vrana, and Jeroen Zuiddam

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent omega, is a central problem in algebraic complexity theory. The best upper bounds on omega, leading to the state-of-the-art omega <= 2.37.., have been obtained via the laser method of Strassen and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on omega. We introduce a new and more general barrier, providing stronger limitations than in previous work. Concretely, we introduce the notion of "irreversibility" of a tensor and we prove (in some precise sense) that any approach that uses an irreversible tensor in an intermediate step (e.g., as a starting tensor in the laser method) cannot give omega = 2. In quantitative terms, we prove that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor. The quantum functionals and Strassen support functionals give (so far, the best) lower bounds on irreversibility. We provide lower bounds on the irreversibility of key intermediate tensors, including the small and big Coppersmith - Winograd tensors, that improve limitations shown in previous work. Finally, we discuss barriers on the group-theoretic approach in terms of "monomial" irreversibility.
Cite as

Matthias Christandl, Péter Vrana, and Jeroen Zuiddam. Barriers for Fast Matrix Multiplication from Irreversibility. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{christandl_et_al:LIPIcs.CCC.2019.26,
  author =	{Christandl, Matthias and Vrana, P\'{e}ter and Zuiddam, Jeroen},
  title =	{{Barriers for Fast Matrix Multiplication from Irreversibility}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.26},
  URN =		{urn:nbn:de:0030-drops-108487},
  doi =		{10.4230/LIPIcs.CCC.2019.26},
  annote =	{Keywords: Matrix multiplication exponent, barriers, laser method}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.24
Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication

Authors: Harry Buhrman, Matthias Christandl, and Jeroen Zuiddam

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the support rank of the iterated matrix multiplication tensor. We employ Strassen’s laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds.
Cite as

Harry Buhrman, Matthias Christandl, and Jeroen Zuiddam. Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buhrman_et_al:LIPIcs.ITCS.2017.24,
  author =	{Buhrman, Harry and Christandl, Matthias and Zuiddam, Jeroen},
  title =	{{Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{24:1--24:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.24},
  URN =		{urn:nbn:de:0030-drops-81812},
  doi =		{10.4230/LIPIcs.ITCS.2017.24},
  annote =	{Keywords: quantum communication complexity, broadcast channel, number-in-hand, matrix multiplication, support rank}
}
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