2 Search Results for "Freksen, Casper Benjamin"


Document
Track A: Algorithms, Complexity and Games
Lower Bounds for Multiplication via Network Coding

Authors: Peyman Afshani, Casper Benjamin Freksen, Lior Kamma, and Kasper Green Larsen

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)


Abstract
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, very recently proved by Harvey and van der Hoeven (2019), shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n). In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size Omega(n lg n), thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant’s conjectures.

Cite as

Peyman Afshani, Casper Benjamin Freksen, Lior Kamma, and Kasper Green Larsen. Lower Bounds for Multiplication via Network Coding. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 10:1-10:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{afshani_et_al:LIPIcs.ICALP.2019.10,
  author =	{Afshani, Peyman and Freksen, Casper Benjamin and Kamma, Lior and Larsen, Kasper Green},
  title =	{{Lower Bounds for Multiplication via Network Coding}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{10:1--10:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.10},
  URN =		{urn:nbn:de:0030-drops-105861},
  doi =		{10.4230/LIPIcs.ICALP.2019.10},
  annote =	{Keywords: Circuit Complexity, Circuit Lower Bounds, Multiplication, Network Coding, Fine-Grained Complexity}
}
Document
On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss Transforms

Authors: Casper Benjamin Freksen and Kasper Green Larsen

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)


Abstract
The Johnson-Lindenstrauss lemma is one of the corner stone results in dimensionality reduction. It says that given N, for any set of N, vectors X \subset R^n, there exists a mapping f : X --> R^m such that f(X) preserves all pairwise distances between vectors in X to within(1 ± \eps) if m = O(\eps^{-2} lg N). Much effort has gone into developing fast embedding algorithms, with the Fast Johnson-Lindenstrauss transform of Ailon and Chazelle being one of the most well-known techniques. The current fastest algorithm that yields the optimal m = O(\eps{-2}lg N) dimensions has an embedding time of O(n lg n + \eps^{-2} lg^3 N). An exciting approach towards improving this, due to Hinrichs and Vybíral, is to use a random m times n Toeplitz matrix for the embedding. Using Fast Fourier Transform, the embedding of a vector can then be computed in O(n lg m) time. The big question is of course whether m = O(\eps^{-2} lg N) dimensions suffice for this technique. If so, this would end a decades long quest to obtain faster and faster Johnson-Lindenstrauss transforms. The current best analysis of the embedding of Hinrichs and Vybíral shows that m = O(\eps^{-2} lg^2 N) dimensions suffice. The main result of this paper, is a proof that this analysis unfortunately cannot be tightened any further, i.e., there exists a set of N vectors requiring m = \Omega(\eps^{-2} lg^2 N) for the Toeplitz approach to work.

Cite as

Casper Benjamin Freksen and Kasper Green Larsen. On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss Transforms. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 32:1-32:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{freksen_et_al:LIPIcs.ISAAC.2017.32,
  author =	{Freksen, Casper Benjamin and Larsen, Kasper Green},
  title =	{{On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss Transforms}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{32:1--32:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.32},
  URN =		{urn:nbn:de:0030-drops-82540},
  doi =		{10.4230/LIPIcs.ISAAC.2017.32},
  annote =	{Keywords: dimensionality reduction, Johnson-Lindenstrauss, Toeplitz matrices}
}
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