3 Search Results for "Panova, Greta"

Document
DOI: 10.4230/LIPIcs.CCC.2024.35
Finer-Grained Hardness of Kernel Density Estimation

Authors: Josh Alman and Yunfeng Guan

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
In batch Kernel Density Estimation (KDE) for a kernel function f : ℝ^m × ℝ^m → ℝ, we are given as input 2n points x^{(1)}, …, x^{(n)}, y^{(1)}, …, y^{(n)} ∈ ℝ^m in dimension m, as well as a vector v ∈ ℝⁿ. These inputs implicitly define the n × n kernel matrix K given by K[i,j] = f(x^{(i)}, y^{(j)}). The goal is to compute a vector v ∈ ℝⁿ which approximates K w, i.e., with || Kw - v||_∞ < ε ||w||₁. For illustrative purposes, consider the Gaussian kernel f(x,y) : = e^{-||x-y||₂²}. The classic approach to this problem is the famous Fast Multipole Method (FMM), which runs in time n ⋅ O(log^m(ε^{-1})) and is particularly effective in low dimensions because of its exponential dependence on m. Recently, as the higher-dimensional case m ≥ Ω(log n) has seen more applications in machine learning and statistics, new algorithms have focused on this setting: an algorithm using discrepancy theory, which runs in time O(n / ε), and an algorithm based on the polynomial method, which achieves inverse polynomial accuracy in almost linear time when the input points have bounded square diameter B < o(log n). A recent line of work has proved fine-grained lower bounds, with the goal of showing that the "curse of dimensionality" arising in FMM is necessary assuming the Strong Exponential Time Hypothesis (SETH). Backurs et al. [NeurIPS 2017] first showed the hardness of a variety of Empirical Risk Minimization problems including KDE for Gaussian-like kernels in the case with high dimension m = Ω(log n) and large scale B = Ω(log n). Alman et al. [FOCS 2020] later developed new reductions in roughly this same parameter regime, leading to lower bounds for more general kernels, but only for very small error ε < 2^{- log^{Ω(1)} (n)}. In this paper, we refine the approach of Alman et al. to show new lower bounds in all parameter regimes, closing gaps between the known algorithms and lower bounds. For example: - In the setting where m = Clog n and B = o(log n), we prove Gaussian KDE requires n^{2-o(1)} time to achieve additive error ε < Ω(m/B)^{-m}, matching the performance of the polynomial method up to low-order terms. - In the low dimensional setting m = o(log n), we show that Gaussian KDE requires n^{2-o(1)} time to achieve ε such that log log (ε^{-1}) > ̃ Ω ((log n)/m), matching the error bound achievable by FMM up to low-order terms. To our knowledge, no nontrivial lower bound was previously known in this regime. Our approach also generalizes to any parameter regime and any kernel. For example, we achieve similar fine-grained hardness results for any kernel with slowly-decaying Taylor coefficients such as the Cauchy kernel. Our new lower bounds make use of an intricate analysis of the "counting matrix", a special case of the kernel matrix focused on carefully-chosen evaluation points. As a key technical lemma, we give a novel approach to bounding the entries of its inverse by using Schur polynomials from algebraic combinatorics.
Cite as

Josh Alman and Yunfeng Guan. Finer-Grained Hardness of Kernel Density Estimation. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 35:1-35:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alman_et_al:LIPIcs.CCC.2024.35,
  author =	{Alman, Josh and Guan, Yunfeng},
  title =	{{Finer-Grained Hardness of Kernel Density Estimation}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{35:1--35:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.35},
  URN =		{urn:nbn:de:0030-drops-204311},
  doi =		{10.4230/LIPIcs.CCC.2024.35},
  annote =	{Keywords: Kernel Density Estimation, Fine-Grained Complexity, Schur Polynomials}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.134
Functional Closure Properties of Finite ℕ-Weighted Automata

Authors: Julian Dörfler and Christian Ikenmeyer

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We determine all functional closure properties of finite ℕ-weighted automata, even all multivariate ones, and in particular all multivariate polynomials. We also determine all univariate closure properties in the promise setting, and all multivariate closure properties under certain assumptions on the promise, in particular we determine all multivariate closure properties where the output vector lies on a monotone algebraic graph variety.
Cite as

Julian Dörfler and Christian Ikenmeyer. Functional Closure Properties of Finite ℕ-Weighted Automata. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 134:1-134:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dorfler_et_al:LIPIcs.ICALP.2024.134,
  author =	{D\"{o}rfler, Julian and Ikenmeyer, Christian},
  title =	{{Functional Closure Properties of Finite \mathbb{N}-Weighted Automata}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{134:1--134:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.134},
  URN =		{urn:nbn:de:0030-drops-202777},
  doi =		{10.4230/LIPIcs.ICALP.2024.134},
  annote =	{Keywords: Finite automata, weighted automata, counting, closure properties, algebraic varieties}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.51
On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions

Authors: Julian Dörfler, Christian Ikenmeyer, and Greta Panova

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

Abstract
Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. We provide the first toy setting in which a separation can be achieved for a family of polynomials via these multiplicities. Mulmuley and Sohoni’s papers also conjecture that the vanishing behavior of multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. We provide first finite settings where a separation via multiplicities can be achieved, while the separation via occurrences is provably impossible. These settings are surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite’s reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases.
Cite as

Julian Dörfler, Christian Ikenmeyer, and Greta Panova. On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dorfler_et_al:LIPIcs.ICALP.2019.51,
  author =	{D\"{o}rfler, Julian and Ikenmeyer, Christian and Panova, Greta},
  title =	{{On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{51:1--51:14},
  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.2019.51},
  URN =		{urn:nbn:de:0030-drops-106276},
  doi =		{10.4230/LIPIcs.ICALP.2019.51},
  annote =	{Keywords: Algebraic complexity theory, geometric complexity theory, Waring rank, plethysm coefficients, occurrence obstructions, multiplicity obstructions}
}
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